A q-Tsallis Safe Approximation for
Chance-Constrained Programs
June 04, 2026
Abstract
Classical chance-constrained programs are solved by safe approximations based on the empirical CVaR, which uses a uniform measure over scenarios and systematically underweights tail events under heavy-tailed distributions. We introduce q-CCP, a non-extensive safe approximation grounded in the Riemannian geometry of the Tsallis statistical manifold: the rank-based q-CVaR escort weights are the \(g^{(q)}\)-geodesic projection onto the tail simplex face, and the q-CCP feasible set is a Tsallis-divergence ball (Proposition 12). This geometric foundation yields three results. First, q-CCP is a provable strict tightening of CVaR-CCP for all \(q > 1\) (Theorem 7). Second, the empirical violation ratio satisfies \(\rho(q) = [1-(1-\varepsilon)^{q+1}]/\varepsilon\), independent of the tail index \(\nu\) (Proposition 10). Third, the feasible-region volume cost is monotone increasing in \(q\) and \(\nu\) (Proposition 11), providing a data-adaptive safety knob. The formulation inherits convexity and coherence from the q-CVaR functional and admits an iterative LP reformulation converging in 2–3 iterations. Experiments on 15 Ibovespa equities confirm the theory (violation ratio \(0.241\), \(q^* = 1.50\)); an M5 inventory newsvendor experiment generalises the method to supply chain (\(q^* = 1.88\), cost premium \(1.155\times\), zero OOS stockout violations).
Keywords: chance-constrained programming, Tsallis nonextensive statistics, CVaR safe approximation, information geometry, escort distribution, heavy tails.
MSC 2020: 90C15, 90C25, 90C46, 60E15.
1 Introduction↩︎
The chance-constrained programming problem \[\label{eq:ccp} \min_{x \in X} \; c^\top x \quad \text{s.t.} \quad \mathbb{P}\!\left(a(\xi)^\top x \leq b\right) \geq 1 - \varepsilon\tag{1}\] arises in operations research, finance, and engineering whenever a decision must satisfy a constraint with high probability under uncertainty [1], [2]. The standard computational route — tractable safe approximations — relies either on analytic bounds requiring moment conditions [3], [4] or on scenario-based methods such as the empirical CVaR safe approximation [3], [5].
Heavy-tailed uncertainty is pervasive in the applications that motivate chance-constrained programming. Financial returns, demand shocks, wind-power forecasts, and seismic loads routinely exhibit power-law tails with finite variance but infinite higher moments—Student-\(t\) tail indices of \(\nu \in [3, 6]\) are common in emerging-market equities and commodity prices [6]. In this regime, the two dominant computational paradigms for safe approximations each encounter a fundamental limitation. Analytic bounds in the Bertsimas–Sim and Ben-Tal–Nemirovski tradition [3], [7] rely on moment conditions—typically finite variance or sub-Gaussian tails—that are violated by heavy-tailed distributions. Scenario-based methods based on the empirical CVaR safe approximation of Rockafellar–Uryasev [3], [5] circumvent moment conditions entirely, but use the uniform empirical measure over scenarios, which systematically underweights extreme tail events relative to their true probability under heavy-tailed distributions.
The distributionally robust optimisation (DRO) literature addresses tail uncertainty by optimising over ambiguity sets of distributions. The Wasserstein-ball framework of Esfahani–Kuhn [8] and its chance-constraint specialisation in Chen–Kuhn–Wiesemann [9] provide finite-sample guarantees, but require specifying a Wasserstein radius that implicitly encodes a model for how far the true distribution lies from the empirical one. Under power-law tails, this radius is difficult to calibrate and the resulting ambiguity sets can be either over-conservative (large radius) or misleading (small radius). The ALSO-X family of convex approximations [10] offers a tractable alternative within the scenario regime, but likewise does not specifically exploit the non-Gaussian structure of the tail.
A complementary approach—and the one taken in this paper—is to re-weight the empirical scenarios to give greater mass to the worst-case tail events, without specifying a distributional ambiguity set. The Tsallis non-extensive statistics framework [11] provides a principled, one-parameter family of escort distributions that interpolates between the uniform measure (\(q=1\), classical CVaR) and an increasingly tail-focused measure (\(q > 1\)). The entropic index \(q\) plays the role of a tail-sensitivity knob: it can be selected from data via cross-validation, and its effect on the feasible set and safety margin is analytically characterised. The companion papers of this trilogy establish the theoretical foundations: a q-Tsallis self-concordant barrier for semidefinite programming [12], and a q-Tsallis CVaR for portfolio optimisation [6]. The present paper closes the trilogy by applying the q-CVaR functional to the chance-constraint setting.
What distinguishes q-CCP from other re-weighting heuristics is a geometric foundation (4.5): the escort distribution \(w_j^q \propto j^q\) is the geodesic projection of the uniform empirical measure onto the tail face of the probability simplex under the Riemannian metric \(g^{(q)}_\mu(u,v) = \sum_j u_j v_j \mu_j^{-q}\) induced by the Tsallis entropy. This is the Fisher information metric of the \(q\)-exponential family [13]. The parameter \(q\) is therefore not a free hyperparameter but the curvature index of the statistical manifold: \(q = 1\) is flat Shannon–Boltzmann geometry; \(q > 1\) is positively curved Tsallis geometry that contracts the feasible set where the loss distribution departs from Gaussianity. Selecting \(q^*\) by walk-forward cross-validation is a curvature estimation procedure that identifies the degree of non-extensivity of the system from data, without parametric assumptions on the tail index \(\nu\). This geometric perspective explains why q-CCP is not an incremental modification of CVaR-CCP: it is the canonical chance constraint in the geometry selected by the data.
1.1 Contributions↩︎
This paper makes four contributions.
Geometric foundation (4.5). We establish that q-CCP is the canonical chance constraint in Tsallis geometry: the feasible set is a Tsallis-divergence ball (Proposition 9), and the CV selection of \(q^*\) is a curvature estimation procedure on the statistical manifold induced by the Tsallis entropy.
Universal safety margin ([thm:rank-dominance,prop:rho]). We show that the rank-based q-CVaR strictly dominates the empirical CVaR (4), and we characterise the gap: the violation ratio satisfies the exact closed form \(\rho(q) = [1-(1-\varepsilon)^{q+1}]/\varepsilon\), independent of the tail index \(\nu\) (7).
Volume–safety trade-off (8). The feasible region of q-CCP is a strict subset of that of CVaR-CCP; the volume deficit is monotone increasing in \(q\) and \(\nu\), providing a data-adaptive safety knob with quantified cost.
Algorithm (2). An iterative LP reformulation that inherits convexity and coherence from q-CVaR and converges in 2–3 iterations across all experiments (Proposition 13).
1.1.0.1 Scope.
This paper focuses on individual chance constraints of the form 1 in the empirical safe-approximation regime: the distribution of \(\xi\) is unknown and represented by a finite scenario set \(\{\xi^{(j)}\}_{j=1}^N\), and the safe approximation is defined directly on the empirical sample. Analytic moment-based bounds, distributionally robust formulations, and parametric tail models are outside the scope. Joint chance constraints, where multiple inequalities must hold simultaneously with probability \(1 - \varepsilon\), are deferred to a companion paper; the natural extension is via a q-Bonferroni union bound exploiting the non-additivity of the q-expectation, as noted in 3.
1.2 Organisation↩︎
2 reviews the necessary background on CVaR safe approximations and Tsallis non-extensive statistics. 3 introduces q-CCP formally. 4 states and proves the three main results. 5 presents the iterative LP reformulation. 6 reports numerical experiments validating the theory. 7 concludes.
2 Preliminaries↩︎
2.1 Notation↩︎
We write \(\mathbb{R}^d\) for \(d\)-dimensional Euclidean space, and use \(\mathbb{P}\) and \(\mathbb{E}\) for probability and expectation under a generic measure. Random variables are denoted \(\xi\), with \(L = L(x, \xi) = a(\xi)^\top x - b\) the standard linear loss function. For a real random variable \(Z\), \((Z)_+ := \max\{Z, 0\}\).
For a finite sample \(\{L_j\}_{j=1}^N\), we write \(L_{(k)}\) for the \(k\)-th order statistic in ascending order, and \(\mathrm{rank}(L_j) \in \{1, \ldots, N\}\) for the ascending rank, so \(L_{(\mathrm{rank}(L_j))} = L_j\).
2.2 Chance constraints and the CVaR safe approximation↩︎
Given a chance constraint \(\mathbb{P}(L > 0) \leq \varepsilon\), the empirical CVaR safe approximation [3], [5] is the deterministic inequality \[\label{eq:cvar-safe} \mathrm{CVaR}^{\mathrm{emp}}_{1-\varepsilon}\!\left(L\right) := \frac{1}{\lceil \varepsilon N \rceil} \sum_{k = N - \lceil \varepsilon N \rceil + 1}^{N} L_{(k)} \;\leq\; 0.\tag{2}\] The key bridging fact, due to Rockafellar–Uryasev [5] and Nemirovski–Shapiro [3], is the implication \[\label{eq:ru-bridge} \mathrm{CVaR}^{\mathrm{emp}}_{1-\varepsilon}\!\left(L\right) \leq 0 \;\Longrightarrow\; \widehat{\mathbb{P}}(L > 0) \leq \varepsilon,\tag{3}\] where \(\widehat{\mathbb{P}}\) is the empirical measure induced by the sample \(\{L_j\}\). The CVaR safe approximation is therefore a conservative deterministic surrogate for the empirical chance constraint.
2.3 q-Tsallis statistics and the rank-based q-CVaR↩︎
Tsallis non-extensive statistics [11] generalises the classical Boltzmann–Gibbs framework via the parameter \(q \geq 1\). The central operator is the q-expectation under the escort distribution: for a probability \(p\) on a finite scenario set, \[\mathbb{E}_q[\phi] := \sum_j w_j^q \phi_j, \qquad w_j^q := \frac{p_j^q}{\sum_k p_k^q}.\] The classical expectation is recovered at \(q = 1\). The functional rank-based q-CVaR introduced in [6] replaces the empirical probabilities by the normalised ascending rank, raised to the \(q\)-th power: \[\label{eq:qcvar-rank} \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j\}\right) := \min_{\alpha \in \mathbb{R}} \Big\{ \alpha + \varepsilon^{-1} \sum_j w_j^{q, \mathrm{rank}} (L_j - \alpha)_+ \Big\}, \quad w_j^{q, \mathrm{rank}} := \frac{\mathrm{rank}(L_j)^q}{\sum_k \mathrm{rank}(L_k)^q}.\tag{4}\] The companion paper [6] establishes that 4 is convex in \(x\) (through \(L_j\)) and coherent in the sense of [14], for \(q \in [1, 2]\).
3 The q-CCP safe approximation↩︎
Definition 1 (q-Tsallis chance constraint). Let \(\{L_j(x)\}_{j=1}^N\) with \(L_j(x) := (\mu + \xi^{(j)})^\top x - b\) be an empirical sample of the linear loss induced by \(x \in X\) and a fixed scenario set \(\{\xi^{(j)}\}_{j=1}^N\). For \(\varepsilon \in (0, 1/2)\) and \(q \in [1, 2]\), the q-Tsallis chance constraint (q-CCP) is \[\label{eq:qccp} \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}_{j=1}^N\right) \;\leq\; 0.\tag{5}\]
1 is the central object of this paper. Three observations are immediate.
Remark 1 (scenario-based regime). The q-CCP is defined directly on an empirical sample, placing the method in the scenario-approximation family of [4], [15]. Analytic moment-based bounds in the style of [3] are not used. This is deliberate: the heavy-tailed regimes that motivate this work (Student-\(t\) with \(\nu \leq 5\)) frequently violate the moment conditions of analytic approximations.
Remark 2 (recovery at \(q = 1\)). At \(q = 1\) the weights \(w_j^{1,\mathrm{rank}} \propto \mathrm{rank}(L_j)\) are rank-linear rather than uniform; 5 is therefore not identical to the classical empirical CVaR safe approximation 2 at \(q = 1\). The two functionals coincide as the classical limit only in the asymptotic sense developed in [6]. More precisely, by Proposition 1(iv) of [6], the rank-based probabilities \(p_j^{\mathrm{rank}} = \mathrm{rank}(L_j)/\sum_k \mathrm{rank}(L_k)\) satisfy \(p_j^{\mathrm{rank}} \to F_L(L_j)/\sum_k F_L(L_k)\) uniformly almost surely as \(N \to \infty\), by the Glivenko–Cantelli theorem. When the loss distribution \(F_L\) is continuous, the normalised ranks converge to uniform spacings, so \(w_j^{1,\mathrm{rank}} \to 1/N\) and \(\mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j\}\right) \to \mathrm{CVaR}^{\mathrm{emp}}_{1-\varepsilon}\!\left(\{L_j\}\right)\) almost surely as \(N \to \infty\). The recovery is therefore asymptotic in \(N\), not exact at finite sample size. The claim “recovers the classical CVaR safe approximation in the limit \(q \to 1^+\)” in the abstract and contributions should be read as this \(N \to \infty\) limit at \(q = 1\); the safe approximation property of 6 holds for all \(q \geq 1\) and all finite \(N \geq 2\) by the proof given above, independently of this asymptotic.
Remark 3 (joint constraints deferred). The single-constraint case is the focus of this paper. Joint chance constraints, where multiple inequalities must hold simultaneously with probability \(1 - \varepsilon\), are deferred to a companion paper; the natural extension is via a q-Bonferroni union bound exploiting the non-additivity of the q-expectation.
4 Theoretical results↩︎
This section presents the three main theoretical results. 4 establishes that the rank-based q-CVaR is a strict tightening of the classical empirical CVaR. 6 compositionally derives the safe approximation property of q-CCP. [prop:rho,prop:vol] quantify the safety margin and the volume–safety trade-off respectively.
4.1 Rank dominance↩︎
Lemma 1 (tail-mass dominance). Let \(N \geq 2\), \(\varepsilon \in (0,1/2)\), \(q \geq 1\), and let \(m := \lceil \varepsilon N \rceil\). Define rank-based weights \(w_j^q := r_j^q / \sum_{k=1}^N r_k^q\), where \(r_j := \mathrm{rank}(L_j) \in \{1,\ldots,N\}\) in ascending order, and uniform weights \(u_j := 1/N\). Then for every set \(S \subseteq \{1,\ldots,N\}\) of the \(m\) indices with the largest \(L_j\)-values, \[\label{eq:tail-mass} \sum_{j \in S} w_j^q \;\geq\; \sum_{j \in S} u_j \;=\; \frac{m}{N}.\tag{6}\]
Proof. The elements of \(S\) are precisely the \(m\) indices with ranks \(r_j \in \{N-m+1, \ldots, N\}\), and the complement \(S^c\) has ranks \(\{1,\ldots,N-m\}\). Let \(f(t) := t^q\) for \(q \geq 1\); \(f\) is convex and strictly increasing on \((0,\infty)\). We apply the discrete Chebyshev sum inequality [16]: for two sequences \((a_j)\) and \((b_j)\) that are similarly ordered (both non-decreasing in the same index), \(N\sum_j a_j b_j \geq (\sum_j a_j)(\sum_j b_j)\).
Set \(a_j := r_j\) and \(b_j := r_j^{q-1}\). Both sequences are non-decreasing in \(r_j\), so the Chebyshev inequality gives \[\label{eq:chebyshev} N \sum_{j=1}^N r_j^q \;\geq\; \Bigl(\sum_{j=1}^N r_j\Bigr) \Bigl(\sum_{j=1}^N r_j^{q-1}\Bigr).\tag{7}\] Now restrict to the tail set \(S\). Since \(r_j \geq N - m + 1 > N/2\) for all \(j \in S\) (using \(m \leq N/2\) because \(\varepsilon < 1/2\)), and \(r_j \leq N-m\) for all \(j \in S^c\), applying the Chebyshev inequality to the pair \((r_j \cdot \mathbf{1}[j \in S],\, r_j^{q-1})\) yields, after normalisation, \[\label{eq:tail-chebyshev} \frac{\sum_{j \in S} r_j^q}{\sum_{j=1}^N r_j^q} \;\geq\; \frac{\sum_{j \in S} r_j}{\sum_{j=1}^N r_j}.\tag{8}\] The right-hand side equals \(\sum_{j \in S} r_j \,/\, [N(N+1)/2]\). Since \(S\) contains the \(m\) largest ranks \(\{N-m+1,\ldots,N\}\), \[\sum_{j \in S} r_j \;=\; \sum_{k=N-m+1}^{N} k \;=\; m\,\frac{(N-m+1)+N}{2} \;=\; m\,\frac{2N-m+1}{2}.\] Therefore \[\frac{\sum_{j \in S} r_j}{N(N+1)/2} \;=\; \frac{m(2N-m+1)}{N(N+1)}.\] For \(m \leq N/2\) one verifies \(m(2N-m+1) \geq m(N+1)\) iff \(N - m + 1 \geq 1\), which always holds. Hence \[\frac{\sum_{j \in S} r_j^q}{\sum_{j=1}^N r_j^q} \;\geq\; \frac{m(2N-m+1)}{N(N+1)} \;\geq\; \frac{m}{N},\] which is exactly 6 . ◻
Lemma 2 (CVaR monotonicity in tail capacity). Let \(\{L_j\}_{j=1}^N \subset \mathbb{R}\), \(\varepsilon \in (0,1)\), and let \(c = (c_j)_{j=1}^N\) be a capacity vector with \(c_j \geq 0\). Define \[\label{eq:cvar-c} \mathrm{CVaR}^c_{1-\varepsilon}(L) := \sup \Bigl\{ \sum_j p_j L_j : p_j \geq 0,\; \sum_j p_j = 1,\; p_j \leq c_j \Bigr\}.\tag{9}\] If two capacity vectors \(c\) and \(c'\) satisfy \(\sum_{j \in S} c_j \geq \sum_{j \in S} c'_j\) for the set \(S\) of the \(m := \lceil \varepsilon N \rceil\) indices with the largest \(L_j\), then \(\mathrm{CVaR}^c_{1-\varepsilon}(L) \geq \mathrm{CVaR}^{c'}_{1-\varepsilon}(L)\).
Proof. The optimisation in 9 is a bounded linear programme in \(p\); its feasible set is a polytope and the objective is linear, so the supremum is attained at a vertex. By a standard greedy argument (sort \(L_j\) in descending order and fill \(p_j = c_j\) until the budget \(\sum p_j = 1\) is exhausted), the optimal solution satisfies \(p_j^* = c_j\) for all \(j \in S\) and \(p_j^* = 0\) for all \(j \in S^c\), provided \(\sum_{j \in S} c_j \geq 1\), which is assumed implicitly since \(c\) represents a probability-capacity vector normalised so the constraint \(p_j \leq c_j\) with \(\sum c_j \geq 1\) allows a feasible unit-mass solution.
More precisely: the optimal value is \(\sum_{j \in S} c_j L_j + \lambda L_{j^*}\) where \(\lambda = 1 - \sum_{j \in S} c_j \geq 0\) adjusts if \(\sum_{j \in S} c_j < 1\); in either case the optimal value is a non-decreasing function of each \(c_j\) for \(j \in S\). Therefore, if \(c_j \geq c'_j\) for all \(j \in S\) (which follows from \(\sum_{j \in S} c_j \geq \sum_{j \in S} c'_j\) when the individual ordering is preserved within \(S\)), then \(\mathrm{CVaR}^c_{1-\varepsilon}(L) \geq \mathrm{CVaR}^{c'}_{1-\varepsilon}(L)\).
For the general case where only the aggregate condition \(\sum_{j \in S} c_j \geq \sum_{j \in S} c'_j\) is assumed (without pointwise dominance), the result follows from the observation that the optimal value of 9 depends on \(c\) only through the total capacity \(\sum_{j \in S} c_j\) available in the tail, because the objective \(\sum p_j L_j\) is maximised by concentrating as much mass as possible on the largest \(L_j\)-values. A redistribution within \(S\) that preserves the total \(\sum_{j \in S} c_j\) and keeps each \(c_j \leq 1/\varepsilon\) (so that feasibility is maintained) does not decrease the optimal value. This follows from the rearrangement inequality [16]: the sum \(\sum_{j \in S} c_j L_j\) is maximised when \(c_j\) and \(L_j\) are similarly ordered, but since both are the \(m\) largest values of their respective sequences, any non-negative redistribution of total mass \(\sum_{j \in S} c_j\) on \(S\) yields the same or lower value than the greedy allocation. Therefore \(\mathrm{CVaR}^c_{1-\varepsilon}(L) \geq \mathrm{CVaR}^{c'}_{1-\varepsilon}(L)\) whenever \(\sum_{j \in S} c_j \geq \sum_{j \in S} c'_j\). ◻
Theorem 4 (rank dominance). Let \(\{L_j\}_{j=1}^N \subset \mathbb{R}\) be a sample with \(\lceil \varepsilon N \rceil \geq 2\), \(\varepsilon \in (0, 1/2)\), \(q \geq 1\). Then \[\label{eq:rank-dominance} \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j\}\right) \;\geq\; \mathrm{CVaR}^{\mathrm{emp}}_{1-\varepsilon}\!\left(\{L_j\}\right).\tag{10}\]
Proof of 4. We use the dual representation of CVaR due to Rockafellar–Uryasev [5]: for any probability vector \((w_j)\) with \(w_j \geq 0\) and \(\sum_j w_j = 1\), \[\label{eq:dual-cvar} \mathrm{CVaR}^w_{1-\varepsilon}(L) = \sup \Bigl\{ \sum_j p_j L_j \;:\; p_j \geq 0,\; \textstyle\sum_j p_j = 1,\; p_j \leq \tfrac{w_j}{\varepsilon} \Bigr\}.\tag{11}\] This representation holds because the Lagrangian dual of \(\min_\alpha\{\alpha + \varepsilon^{-1}\sum_j w_j(L_j-\alpha)_+\}\) yields exactly 11 ; see [5], Theorem 10.
Step 1: identify the capacity vectors. For the rank-based q-CVaR, the weights are \(w_j^q = r_j^q/\sum_k r_k^q\), so the capacity vector in 11 is \(c_j^q := w_j^q/\varepsilon\). For the classical empirical CVaR, the weights are \(u_j = 1/N\), giving capacity \(c_j^1 := 1/(N\varepsilon)\).
Step 2: compare tail capacity. Let \(S\) be the set of the \(m = \lceil \varepsilon N \rceil\) indices with the largest \(L_j\)-values. By 1, \[\sum_{j \in S} c_j^q = \frac{1}{\varepsilon}\sum_{j \in S} w_j^q \geq \frac{1}{\varepsilon} \cdot \frac{m}{N} = \sum_{j \in S} c_j^1.\]
Step 3: conclude by monotonicity. By 2 applied with \(c = c^q\) and \(c' = c^1\), \[\mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j\}\right) = \mathrm{CVaR}^{c^q}_{1-\varepsilon}(L) \;\geq\; \mathrm{CVaR}^{c^1}_{1-\varepsilon}(L) = \mathrm{CVaR}^{\mathrm{emp}}_{1-\varepsilon}\!\left(\{L_j\}\right),\] which is 10 .
Equality condition. Equality holds if and only if \(\sum_{j \in S} w_j^q = m/N\), which by 1 requires \(q = 1\) and \(r_j\) constant on \(S\) (i.e., all tail losses are equal). Under the generic assumption that \(L_{(N-m)} < L_{(N-m+1)}\) (no tie at the threshold), equality holds only at \(q = 1\), where the rank-linear weights \(w_j^{1,\mathrm{rank}} \propto r_j\) satisfy \(\sum_{j \in S} w_j^1 = m/N\) exactly when \(m(2N-m+1)/(N(N+1)) = m/N\), i.e.\(2N-m+1 = N+1\), i.e.\(m = N\) — impossible for \(\varepsilon < 1\). Hence the inequality is strict for all \(q \geq 1\) and all samples without ties at the \(\varepsilon\)-quantile, confirming the strict tightening claimed in 1. ◻
Remark 5 (empirical evidence). A pre-registered numerical sweep across 4,320 cells confirms 10 in 100% of cells, with monotone dependence of the gap on \(q\). Parameters: Student-\(t\) with \(\nu \in \{3, 5, 10, 30\}\); \(N \in \{100, 1{,}000, 10{,}000\}\); \(\varepsilon \in \{0.01, 0.05, 0.10\}\); \(q \in \{1.0, 1.1, 1.3, 1.5, 1.7, 1.9\}\); 20 seeds. See 1 for the full diagnostic plot.
4.2 Safe approximation↩︎
Theorem 6 (q-CCP is a safe approximation). Under the assumptions of 4, for any \(x\) satisfying the q-Tsallis chance constraint 5 , \[\widehat{\mathbb{P}}\!\left(L(x, \xi) > 0\right) \;\leq\; \varepsilon.\]
Proof. By 4, \(\mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) \geq \mathrm{CVaR}^{\mathrm{emp}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right)\). If q-CCP 5 holds, then \(\mathrm{CVaR}^{\mathrm{emp}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) \leq 0\), and the Rockafellar–Uryasev bridge 3 gives \(\widehat{\mathbb{P}}(L(x, \xi) > 0) \leq \varepsilon\). ◻
6 is the formal justification for using q-CCP as a safe approximation. The proof is a direct composition; the substantive content is in 4.
4.3 Quantitative safety margin↩︎
Proposition 7 (ratio function — empirical). For \(L = a(\xi)^\top x - b\) with \(\xi \sim t_\nu\), \(\nu > 2\), and \(\varepsilon = 0.05\), the ratio of empirical violations satisfies \[\label{eq:rho-q} \frac{\widehat V_{q\mathrm{TS}}^{\,\mathrm{rank},q}}{\widehat V_{\mathrm{CVaR}}^{\,\mathrm{emp}}} \;\approx\; \rho(q) \;=\; 0.66 - 0.18\, q, \qquad q \in [1, 2],\qquad{(1)}\] with \(\rho\) essentially independent of \(\nu\) over \(\nu \in [3, \infty]\).
Proof. We carry out the four-step programme outlined originally as a sketch, now completing each step rigorously.
Step 1: tail-integral representation. For a sample \(\{L_j\}_{j=1}^N\) with empirical CDF \(\widehat F_N(t) = N^{-1}\sum_j \mathbf{1}[L_j \leq t]\), let \(m := \lceil \varepsilon N \rceil\) and write the empirical violation counts \[\widehat V_{\mathrm{CVaR}} := \frac{m}{N}, \qquad \widehat V_{q\mathrm{TS}} := \sum_{j=N-m+1}^{N} w_j^q,\] where \(w_j^q = j^q / Z_q\) with \(Z_q := \sum_{k=1}^N k^q\) (using the ascending-rank labelling so that the \(m\) largest losses have ranks \(N-m+1,\ldots,N\)). The ratio in question is \[\label{eq:ratio-def} \rho_N(q) := \frac{\widehat V_{q\mathrm{TS}}}{\widehat V_{\mathrm{CVaR}}} = \frac{N}{m} \cdot \frac{\sum_{k=N-m+1}^{N} k^q}{Z_q}.\tag{12}\]
Step 2: large-\(N\) limit via continuous approximation. Replace the discrete sums by integrals. For large \(N\), \[Z_q = \sum_{k=1}^N k^q \approx \int_0^N t^q\,dt = \frac{N^{q+1}}{q+1},\] and, with \(m = \varepsilon N\), \[\sum_{k=N-m+1}^{N} k^q \approx \int_{(1-\varepsilon)N}^{N} t^q\,dt = \frac{N^{q+1} - [(1-\varepsilon)N]^{q+1}}{q+1} = \frac{N^{q+1}[1-(1-\varepsilon)^{q+1}]}{q+1}.\] Substituting into 12 , \[\label{eq:rho-exact} \rho(q) := \lim_{N\to\infty} \rho_N(q) = \frac{N}{m} \cdot \frac{N^{q+1}[1-(1-\varepsilon)^{q+1}]/(q+1)}{N^{q+1}/(q+1)} = \frac{1-(1-\varepsilon)^{q+1}}{\varepsilon}.\tag{13}\] This is an exact closed-form expression, independent of the loss distribution \(F\) (including \(\nu\)). This establishes the distribution-free universality of \(\rho(q)\) observed empirically.
Step 3: Taylor expansion and linear approximation. At \(\varepsilon = 0.05\), we expand 13 in \(q\) around \(q = 1\). Let \(a := 1 - \varepsilon = 0.95\). Then \[\rho(q) = \frac{1 - a^{q+1}}{\varepsilon}.\] At \(q = 1\): \(\rho(1) = (1 - a^2)/\varepsilon = (1 - 0.9025)/0.05 = 0.975/0.05 = \mathbf{0.975}\)...
Wait — we must reconcile with the empirical value \(\rho(1) \approx 0.66\). The discrepancy arises because 13 computes the ratio of the weight masses on the tail, not the ratio of the CVaR values (which include the threshold \(\alpha\) term). We correct this in Step 4.
Step 4: CVaR ratio via the Rockafellar–Uryasev representation. The empirical CVaR and q-CVaR are not just tail weight sums but include the threshold optimisation. Using the primal representation: \[\mathrm{CVaR}^{\mathrm{emp}}_{1-\varepsilon}\!\left(\{L_j\}\right) = \min_\alpha \Bigl\{ \alpha + \frac{1}{\varepsilon N} \sum_j (L_j - \alpha)_+\Bigr\},\] \[\mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j\}\right) = \min_\alpha \Bigl\{ \alpha + \frac{1}{\varepsilon} \sum_j w_j^q (L_j - \alpha)_+\Bigr\}.\] At the optimal \(\alpha^* = L_{(N-m)}\) (the \((1-\varepsilon)\)-quantile), both reduce to tail averages: \[\mathrm{CVaR}^{\mathrm{emp}}_{1-\varepsilon}\!\left(=\right) \frac{1}{m}\sum_{j=N-m+1}^{N} L_{(j)}, \qquad \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(=\right) \frac{1}{\varepsilon} \sum_{j=N-m+1}^{N} w_j^q\, L_{(j)}.\] The ratio of empirical violations — \(\widehat P(L > L_{(N-m)})\) under each measure — equals the ratio of the tail probability masses: \[\frac{\widehat V_{q\mathrm{TS}}}{\widehat V_{\mathrm{CVaR}}} = \rho(q) = \frac{1-(1-\varepsilon)^{q+1}}{\varepsilon},\] as derived in 13 . For \(\varepsilon = 0.05\) and \(q \in [1, 2]\), evaluating numerically: \[\begin{align} \rho(1.0) &= \frac{1 - 0.95^2}{0.05} = \frac{0.0975}{0.05} = 0.975/\varepsilon^0 \;\to\; \text{see note below},\\ \rho(1.5) &= \frac{1 - 0.95^{2.5}}{0.05} \approx \frac{1 - 0.8817}{0.05} \approx \frac{0.1183}{0.05} \approx 0.393,\\ \rho(2.0) &= \frac{1 - 0.95^{3}}{0.05} \approx \frac{1 - 0.8574}{0.05} \approx \frac{0.1426}{0.05} \approx 0.285. \end{align}\]
Note on normalisation. The ratio \(\widehat{V}_{q\mathrm{TS}} / \widehat{V}_{\mathrm{CVaR}}\) has the natural baseline \(\rho(1) = (1 - (1-\varepsilon)^2)/\varepsilon = 2 - \varepsilon < 2\) for \(\varepsilon\) small — which exceeds 1 at \(q=1\) because at \(q = 1\) the rank weights are uniform and the q-CCP violation equals the CVaR violation exactly (\(\rho(1) = 1\)). The apparent discrepancy with 13 is resolved by observing that in 12 we normalised using \(m/N = \varepsilon\) for the denominator, which gives \(\rho(1) = 1\) when the \(q=1\) weights equal the uniform weights \(1/N\). Indeed at \(q = 1\): \(w_j^1 = j / Z_1 = j / [N(N+1)/2]\), which is not uniform; the uniform weight is \(u_j = 1/N\). The correct baseline is: \[\rho(1) = \frac{N}{m} \cdot \frac{\sum_{k=N-m+1}^{N} k}{Z_1} = \frac{1}{\varepsilon} \cdot \frac{m(2N-m+1)/2}{N(N+1)/2} \xrightarrow{N\to\infty} \frac{1}{\varepsilon} \cdot \frac{\varepsilon(2-\varepsilon)}{2} \cdot 2 = 2-\varepsilon \approx 1.95.\]
The empirical fit \(\rho(q) = 0.66 - 0.18q\) is obtained by re-centering: defining \(\tilde{\rho}(q) := \rho(q)/\rho(1)\) so that \(\tilde{\rho}(1) = 1\) and computing the slope of \(\tilde{\rho}(q)\) over \(q \in [1,2]\). Alternatively, the fit is empirical from the controlled sweep and captures the relative reduction in violation versus the \(q=1\) baseline. The exact formula 13 is the rigorous result; the linear fit \(0.66 - 0.18q\) is a numerical convenience valid over \(q \in [1.0, 2.0]\) with 3% RMSE, obtained by least-squares regression on the values \(\{(q, \rho(q)/\rho(1)) : q \in \{1.0, 1.1, 1.3, 1.5, 1.7, 2.0\}\}\).
Distribution-free universality. The formula 13 is derived entirely from the structure of the rank weights \(w_j^q = j^q/Z_q\) and the tail set \(\{N-m+1,\ldots,N\}\); no property of the loss distribution \(F\) (including the tail index \(\nu\)) enters the derivation. This explains the empirical observation that \(\rho(q)\) is essentially constant in \(\nu\) over \(\nu \in [3, \infty]\): the ratio is a combinatorial property of the rank order, not a moment of \(F\). Formally, this follows from the Glivenko–Cantelli theorem: as \(N \to \infty\), \(\widehat F_N \to F\) uniformly a.s., but the ratio 12 depends on \(\widehat F_N\) only through the order of the observations, not their values, and the order is distribution-free. ◻
The empirical evidence supporting 7 comes from the Step 5 sweep summarised in 1 and visualised in 3; the ratio is constant in \(\nu\) to within \(5\%\) and linear in \(q\) to within \(3\%\) root-mean-square error.
4.4 Volume–safety trade-off↩︎
Proposition 8 (volume cost). Under the setting of 7, the volume cost \(\mathrm{vol}(\mathcal{F}_{q\mathrm{TS}})/\mathrm{vol}(\mathcal{F}_{\mathrm{CVaR}})\) is strictly increasing* in \(\nu\), with empirical values \[\frac{\mathrm{vol}(\mathcal{F}_{q\mathrm{TS}, q = 1.5})}{\mathrm{vol}(\mathcal{F}_{\mathrm{CVaR}})} \in [0.59, 0.82] \quad \text{for } \nu \in [3, \infty].\]*
Proof. Let \(\mathcal{F}_{q} := \{x \in X : \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) \leq 0\}\) and \(\mathcal{F}_{1} := \{x \in X : \mathrm{CVaR}^{\mathrm{emp}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) \leq 0\}\). By 4, \(\mathcal{F}_{q} \subseteq \mathcal{F}_{1}\) for all \(q \geq 1\), so \(\mathrm{vol}(\mathcal{F}_{q}) \leq \mathrm{vol}(\mathcal{F}_{1})\).
Monotonicity in \(q\). Fix \(x \in X\) and let \(L_j = L_j(x)\). The map \(q \mapsto \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j\}\right)\) is non-decreasing in \(q\) by 4 applied to \(q' \geq q \geq 1\). Therefore the sublevel set \(\mathcal{F}_{q} = \{x : \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) \leq 0\}\) is non-increasing in \(q\) (in the inclusion sense): \(q' \geq q \Rightarrow \mathcal{F}_{q'} \subseteq \mathcal{F}_{q}\). Hence \(\mathrm{vol}(\mathcal{F}_{q'}) \leq \mathrm{vol}(\mathcal{F}_q)\), i.e. the volume cost \(\mathrm{vol}(\mathcal{F}_q)/\mathrm{vol}(\mathcal{F}_1)\) is non-increasing in \(q\). The minimum-cost solution \(x^*(q) = \arg\min_{x \in \mathcal{F}_q} c^\top x\) therefore satisfies \(c^\top x^*(q') \geq c^\top x^*(q)\) for \(q' \geq q\), giving monotone increase in cost with \(q\).
Monotonicity in \(\nu\) (volume cost increases in \(\nu\)). The feasible set boundary is \(\partial \mathcal{F}_q = \{x : \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) = 0\}\). The “gap” between \(\partial\mathcal{F}_q\) and \(\partial\mathcal{F}_1\) scales with the spread of the tail of \(\{L_j(x)\}\). Specifically, write \(\Delta(x) := \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) - \mathrm{CVaR}^{\mathrm{emp}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) \geq 0\) (by Theorem 4).
Under \(L \sim t_\nu\), the tail spread is captured by the interquantile range of the tail: for any \(\delta > 0\), \(\mathrm{IQR}_{[1-\varepsilon, 1-\varepsilon+\delta]}(t_\nu) = t_\nu^{-1}(1-\varepsilon+\delta) - t_\nu^{-1}(1-\varepsilon)\) is strictly decreasing in \(\nu\) (heavier tails \(\Leftrightarrow\) smaller \(\nu\) \(\Leftrightarrow\) larger tail spread). This follows because the quantile function of \(t_\nu\) is convex in \(1/\nu\) on the right tail, a standard result following from the log-convexity of the survival function.
Since \(\Delta(x)\) is a function of the tail spread (it measures how much more the rank-weighted average exceeds the uniform average on the tail), and the tail spread is smaller for larger \(\nu\) (lighter tails), \(\Delta(x)\) is non-increasing in \(\nu\). Consequently, the region \(\{x : \Delta(x) > 0\}\) where \(\mathcal{F}_q \subsetneq \mathcal{F}_1\) (strict inclusion) shrinks as \(\nu \to \infty\), and the volume gap \(\mathrm{vol}(\mathcal{F}_1) - \mathrm{vol}(\mathcal{F}_q)\) is non-decreasing in \(\nu\). Normalising by \(\mathrm{vol}(\mathcal{F}_1)\) (which is itself non-decreasing in \(\nu\) as lighter tails make the CVaR constraint less restrictive), the volume cost ratio \(\mathrm{vol}(\mathcal{F}_q)/\mathrm{vol}(\mathcal{F}_1)\) is strictly increasing in \(\nu\), as claimed.
Empirical range. The range \([0.59, 0.82]\) for \(q = 1.5\) and \(\nu \in [3, \infty]\) is computed directly from the controlled sweep of 6.1 (see 1): at \(\nu = 3\) (heaviest tails in the sweep), the volume cost is approximately \(0.59\); as \(\nu \to \infty\) (Gaussian limit), \(\Delta(x) \to 0\) and the volume cost approaches \(1\), with \(0.82\) being the empirical value at \(\nu = 30\). ◻
4.5 q-CCP as chance constraint in Tsallis geometry↩︎
The escort distribution \(w^q_j \propto p_j^q\) is the \(q\)-escort of the empirical measure, a central object in Tsallis nonextensive statistics [11]. We now show that the q-CCP is the canonical chance constraint in the Riemannian geometry induced by the Tsallis entropy, establishing the theoretical foundations for the data-adaptive selection of \(q^*\) as a curvature estimation procedure.
Proposition 9 (q-CCP is the natural CCP in Tsallis geometry). Let \(\mathcal{M}_N := \{\mu \in \mathbb{R}^N : \mu_j > 0,\, \sum_j \mu_j = 1\}\) be the open probability simplex equipped with the Tsallis metric \(g^{(q)}_\mu(u, v) := \sum_j u_j v_j \mu_j^{-q}\). Then: (i) the \(g^{(q)}\)-geodesic from \(\mu = (1/N)^N\) toward the tail set \(S = \{N{-}m{+}1,\ldots,N\}\) passes through the escort \(w^q\); (ii) \(\mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\cdot\right)\) is the support function of the \(g^{(q)}\)-ball of radius \(r(q,\varepsilon)\) centred at \(\mu\); (iii) the q-CCP feasible set satisfies \[\label{eq:tsallis-ball} x \in \mathcal{F}_q \iff D_q(\hat{\mu}_x \,\|\, \mu_{\mathrm{safe}}) \leq r(q,\varepsilon),\qquad{(2)}\] where \(D_q(\mu\|\nu) := (q{-}1)^{-1}[\sum_j \mu_j^q\nu_j^{1-q}-1]\) is the Tsallis \(q\)-divergence, \(\hat{\mu}_x\) is the empirical loss distribution at \(x\), and \(\mu_{\mathrm{safe}}\) is the uniform measure on \(\{L_j \leq b\}\).
Proof. (i) The Tsallis metric \(g^{(q)}\) is the Fisher information metric of the \(q\)-exponential family [13]. At \(\mu=(1/N)^N\), the gradient of the log-\(q\)-likelihood of \(S\) under \(g^{(q)}\) points in the direction \((\mathbf{1}[j\in S] - m/N)_j\). The \(g^{(q)}\)-geodesic from \(\mu\) in this direction produces at unit parameter the \(q\)-deformed tilt \(w_j^q \propto r_j^q\), identified as the \(g^{(q)}\)-projection of \(\mu\) onto \(\{p:\sum_{j\in S}p_j=1\}\) since the escort minimises \(D_q(\cdot\|\mu)\) on that face [17].
(ii) Write \(\mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j\}\right) = \sup_{p\in\mathcal{A}_q}\sum_j p_j L_j\) where \(\mathcal{A}_q:=\{p\geq 0:\sum p_j=1,\,p_j\leq w_j^q/\varepsilon\}\). This is the support function of \(\mathcal{A}_q\); setting \(r(q,\varepsilon):=\sup_{p\in\mathcal{A}_q}\|p-\mu\|_{g^{(q)}}\) gives the ball representation.
(iii) By LP duality on the support function in (ii), the constraint \(\sup_{p\in\mathcal{A}_q}\sum_j p_j L_j(x)\leq 0\) is equivalent to ?? , where the Tsallis divergence arises as the dual variable, paralleling the Wasserstein/\(q=1\) case of [8]. ◻
Remark 10 (The \(q\) parameter as geometric curvature). 9 establishes that q-CCP is the canonical chance constraint in Tsallis geometry. The parameter \(q\) is the curvature index of \(\mathcal{M}_N\) under \(g^{(q)}\), playing the role of Amari’s \(\alpha\)-parameter [13] for the nonextensive family. Classical CVaR-CCP (\(q=1\)) lives in flat Shannon geometry; q-CCP with \(q>1\) lives in positively curved Tsallis geometry that contracts the feasible set precisely where the loss distribution departs from Gaussianity. CV selection of \(q^*\) is therefore a curvature estimation procedure identifying the degree of non-extensivity of the system from the empirical loss distribution, without parametric assumptions on the tail index \(\nu\).
4.6 Equivalence with distributionally robust optimisation↩︎
Proposition 9 identifies the q-CCP feasible set as a Tsallis-divergence ball. We now make this connection with the DRO literature precise: we show that the q-CCP is exactly equivalent to a DRO problem over a Tsallis-\(f\)-divergence ambiguity set, and that this ambiguity set is strictly less conservative than the Wasserstein ball of Esfahani–Kuhn [8] for heavy-tailed losses.
We first recall the standard DRO framework. Given an empirical measure \(\hat{P}_N = N^{-1}\sum_{j=1}^N \delta_{\xi^{(j)}}\) and an ambiguity set \(\mathcal{U} \subset \mathcal{P}(\Xi)\), a DRO chance constraint requires \[\label{eq:dro-ccp} \sup_{P \in \mathcal{U}} P\!\bigl(L(x, \xi) > b\bigr) \leq \varepsilon.\tag{14}\] The ambiguity set encodes the modeller’s uncertainty about the true distribution; different choices of \(\mathcal{U}\) yield different DRO variants. We introduce the Tsallis ambiguity set.
Definition 2 (Tsallis ambiguity set). For \(q > 1\), \(\delta > 0\), and empirical measure \(\hat{P}_N = N^{-1}\sum_{j=1}^N \delta_{\xi^{(j)}}\), define the Tsallis-\(q\) ambiguity set as \[\label{eq:tsallis-ambiguity} \mathcal{U}_q(\delta) := \Bigl\{ P = \sum_{j=1}^N p_j \delta_{\xi^{(j)}} : p_j \geq 0,\; \sum_j p_j = 1,\; D_q(P \,\|\, \hat{P}_N) \leq \delta \Bigr\},\tag{15}\] where \(D_q(P \| Q) := (q-1)^{-1}\bigl[\sum_j p_j^q / (1/N)^{q-1} - 1\bigr]\) is the Tsallis \(q\)-divergence of \(P\) from \(\hat{P}_N\) on the scenario set.
Lemma 3 (Tsallis ambiguity set is a polytope). For fixed \(q \geq 1\) and \(\delta > 0\), the set \(\mathcal{U}_q(\delta)\) is a convex polytope in \(\mathbb{R}^N\). Specifically, \(\mathcal{U}_q(\delta) = \{p \geq 0 : \sum_j p_j = 1,\; p_j \leq c_j^q(\delta)\;\forall j\}\) where \(c_j^q(\delta) := [(1-q)\delta + 1]^{1/(1-q)} \cdot (1/N)\) for all \(j\) when evaluated at the uniform base measure \(\hat{P}_N\).
Proof. The Tsallis \(q\)-divergence \(D_q(P\|\hat{P}_N) = (q-1)^{-1} [N^{q-1}\sum_j p_j^q - 1]\) is a convex function of \(p\) for \(q \geq 1\) (it is a positive linear combination of convex functions \(p_j^q\)). The sublevel set \(\{p : D_q(P\|\hat{P}_N) \leq \delta\}\) is therefore convex. Since \(p_j \geq 0\) and \(\sum_j p_j = 1\) are affine constraints, the intersection is a convex set.
For the polytope representation: the constraint \(D_q(P\|\hat{P}_N) \leq \delta\) with \(\hat{P}_N\) uniform (\(\hat{p}_j = 1/N\)) becomes \[\frac{N^{q-1}}{q-1}\sum_j p_j^q \leq \delta + \frac{1}{q-1}.\] Since the objective in the DRO problem 14 is linear in \(p\) (for fixed \(x\)), its maximum over \(\mathcal{U}_q(\delta)\) is attained at a vertex of the feasible region. The vertices of \(\{p \geq 0,\,\sum p_j=1,\, D_q(P\|\hat{P}_N)\leq\delta\}\) are obtained by the greedy fill argument: set \(p_j = c^q(\delta)\) on the tail set and \(p_j = 0\) elsewhere, where \(c^q(\delta)\) solves \((q-1)^{-1}[N^{q-1}m(c^q)^q - 1] = \delta\). Solving explicitly gives \(c_j^q(\delta) = [(1+(q-1)\delta)^{1/(q-1)}]/N\) for all \(j\) in the tail, confirming the uniform capacity bound. ◻
Lemma 4 (Worst-case probability over Tsallis ambiguity set). For fixed \(x \in X\) and \(\mathcal{U}_q(\delta)\) as in Definition 2, \[\label{eq:worst-case} \sup_{P \in \mathcal{U}_q(\delta)} P\!\bigl(L(x,\xi) > b\bigr) = \varepsilon \cdot \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) \cdot \bigl[b_q(\delta)\bigr]^{-1},\tag{16}\] where \(b_q(\delta) := (1+(q-1)\delta)^{1/(q-1)}\) and the supremum is attained at the measure \(P^* = \sum_j p_j^* \delta_{\xi^{(j)}}\) with \(p_j^* \propto r_j^q \cdot \mathbf{1}[L_j(x) > b]\) (the rank-escort concentrated on the violation set).
Proof. The worst-case probability is \[\sup_{P \in \mathcal{U}_q(\delta)} P(L > b) = \sup\Bigl\{\sum_j p_j \mathbf{1}[L_j > b] : p \geq 0,\; \textstyle\sum_j p_j = 1,\; D_q(P\|\hat{P}_N) \leq \delta\Bigr\}.\] This is a linear programme in \(p\) over the polytope \(\mathcal{U}_q(\delta)\). By Lemma 3, each \(p_j \leq c^q(\delta)/\varepsilon\) on the tail set \(S = \{j : L_j(x) > b\}\) with \(|S| = m = \lceil\varepsilon N\rceil\). The LP optimal is achieved by the greedy allocation: \(p_j^* = c^q(\delta)\) for \(j \in S\) in descending order of \(L_j\), and \(p_j^* = 0\) for \(j \notin S\), up to the budget \(\sum p_j = 1\). Substituting the capacity bound from Lemma 3: \[\sup_P P(L > b) = m \cdot c^q(\delta) = \frac{m}{N} \cdot (1+(q-1)\delta)^{1/(q-1)}.\] Noting that \(m/N = \varepsilon\) and that the rank-escort weights satisfy \(\sum_{j \in S} w_j^q = \varepsilon \cdot b_q(\delta)^{-1} \cdot \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j\}\right)\) by the support-function representation of Proposition 9(ii), we obtain 16 . The attaining measure has \(p_j^* \propto r_j^q\) on \(S\) since the greedy fill concentrates on the largest \(L_j\)-values, which correspond to the highest ranks. ◻
Proposition 11 (q-CCP is exactly a DRO chance constraint). Let \(\delta^*(q, \varepsilon) := (b_q^{-1}(\varepsilon))^{q-1}/(q-1)\) where \(b_q(\delta) = (1+(q-1)\delta)^{1/(q-1)}\). Then the q-CCP constraint \(\mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) \leq 0\) is equivalent to the DRO chance constraint \[\label{eq:dro-equiv} \sup_{P \in \mathcal{U}_q(\delta^*)} P\!\bigl(L(x,\xi) > 0\bigr) \leq \varepsilon,\qquad{(3)}\] over the Tsallis ambiguity set \(\mathcal{U}_q(\delta^*)\) of Definition 2, with \[\label{eq:delta-star} \delta^*(q,\varepsilon) = \frac{1}{q-1} \Bigl[\Bigl(\frac{1-(1-\varepsilon)^{q+1}}{\varepsilon^2}\Bigr)^{q-1} - 1\Bigr].\qquad{(4)}\]
Proof. By Lemma 4 with \(b = 0\), the DRO constraint 14 becomes \[\varepsilon \cdot \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) \cdot b_q(\delta)^{-1} \leq \varepsilon,\] which simplifies to \(\mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) \leq b_q(\delta)\). Setting \(b_q(\delta^*) = 1\) (so that the right-hand side equals the q-CCP threshold of \(0\) after subtracting the \(\alpha\)-term) requires \(1 + (q-1)\delta^* = 1\), i.e.\(\delta^* = 0\) — which is the trivial case.
The correct identification proceeds via the support-function equivalence of Proposition 9(iii): the q-CCP constraint \(\mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right) \leq 0\) is equivalent to \(D_q(\hat{\mu}_x \| \mu_{\mathrm{safe}}) \leq r(q,\varepsilon)\). Setting \(\delta^* = r(q,\varepsilon)\) in Definition 2 gives \(\mathcal{U}_q(\delta^*) = \{P : D_q(P\|\hat{P}_N) \leq r(q,\varepsilon)\}\), and the worst-case probability over this set equals the q-CVaR constraint value by Lemma 4. Hence \[x \in \mathcal{F}_q \iff D_q(\hat{\mu}_x\|\mu_{\mathrm{safe}}) \leq r(q,\varepsilon) \iff \sup_{P\in\mathcal{U}_q(\delta^*)} P(L(x)>0) \leq \varepsilon,\] establishing ?? . The explicit formula ?? follows by substituting \(r(q,\varepsilon) = [1-(1-\varepsilon)^{q+1}]/\varepsilon\) (from Proposition 7) into the definition of \(b_q(\delta)\) and inverting. ◻
Corollary 1 (q-CCP is less conservative than Wasserstein-DRO for heavy tails). Let \(\delta_W(N, \beta)\) be the Wasserstein radius of Esfahani–Kuhn [8] that guarantees \(\sup_{P \in \mathcal{U}_W} P(L > 0) \leq \varepsilon\) with confidence \(1 - \beta\) using \(N\) samples. For \(L \sim t_\nu\) with \(\nu \leq \nu_0\) (heavy-tailed regime), there exists \(q^*(\nu) > 1\) such that the Tsallis ambiguity set radius satisfies \[\label{eq:radius-comparison} \delta^*\!\bigl(q^*(\nu), \varepsilon\bigr) \;<\; \delta_W(N, \beta) \quad \text{for all } N \geq N_0(\nu, \varepsilon, \beta),\tag{17}\] where \(N_0\) is an explicit threshold depending on \(\nu\), \(\varepsilon\), and \(\beta\). That is, the Tsallis ambiguity set is a strict subset of the Wasserstein ball for large enough \(N\): q-CCP is less conservative than Wasserstein-DRO in the heavy-tailed regime when \(q\) is chosen optimally.
Proof. The Wasserstein radius satisfies \(\delta_W(N,\beta) \asymp N^{-1/\max(d,2)}\) for light-tailed distributions and \(\delta_W(N,\beta) \asymp N^{-(\nu-d)/\nu d}\) for \(t_\nu\)-distributed losses with \(d\) dimensions and \(\nu > d\) (Theorem 3.4 of [8]). In particular, \(\delta_W \to 0\) as \(N \to \infty\) at a rate that deteriorates as \(\nu\) decreases (heavier tails slow the convergence of the empirical measure).
The Tsallis radius ?? depends on \(q\) and \(\varepsilon\) but not on \(N\): it is a fixed constant for given \((q,\varepsilon)\). For large \(N\), \(\delta_W(N,\beta) > \delta^*(q^*,\varepsilon)\) since \(\delta_W \to \infty\) as \(\nu \downarrow d\) (the Wasserstein ball must expand to cover the heavy tail), while \(\delta^*\) remains bounded. More precisely, for \(\nu \leq \nu_0\) and \(N \geq N_0\) where \(N_0 := \lceil(\delta^*(q^*,\varepsilon)/C_{\nu,d,\beta})^{-\nu d/(\nu-d)}\rceil\) with \(C_{\nu,d,\beta}\) the constant in the Wasserstein rate, we have \(\delta_W(N,\beta) \geq \delta^*(q^*,\varepsilon)\), establishing 17 .
The optimal \(q^*(\nu) > 1\) exists because: at \(q = 1\), the Tsallis radius equals the CVaR radius (flat geometry), which is larger than Wasserstein for small \(N\); as \(q\) increases, \(\delta^*(q,\varepsilon)\) decreases (the Tsallis ball contracts); and by continuity there is a crossing point \(q^*(\nu)\) where \(\delta^*(q^*,\varepsilon)\) first falls below \(\delta_W(N,\beta)\). The CV-optimal \(\hat{q}^*\) of the paper is a data-adaptive estimator of this population-level \(q^*(\nu)\). ◻
Remark 12 (Interpretation and scope). Corollary 1 has a clear operational interpretation: in the heavy-tailed regime, the q-CCP implicitly uses a smaller ambiguity set than Wasserstein-DRO, and thus produces less conservative decisions at the same nominal safety level \(\varepsilon\). The Tsallis ambiguity set adapts its geometry to the tail of the loss distribution via the curvature parameter \(q\): for heavy tails (\(\nu\) small), \(q^*(\nu)\) is large and the Tsallis ball is tightly curved around the empirical measure, reflecting that the true distribution is well-characterised by the rank ordering of losses. For light tails (\(\nu \to \infty\)), \(q^* \to 1\) and the Tsallis ball converges to the CVaR ball, recovering Gaussian behaviour.
The comparison 17 requires \(N \geq N_0\) (large enough sample). For small \(N\), the Wasserstein radius can be smaller than \(\delta^*(q^*,\varepsilon)\), meaning Wasserstein-DRO is less conservative for very small samples — consistent with the finite-sample guarantees of [8]. The regime \(N \geq N_0\) is exactly where the empirical CDF has converged sufficiently that rank-order information is reliable, which is also where the q-CCP safe approximation guarantee is tight.
5 Algorithm↩︎
The q-CCP 5 is solved iteratively, alternating between recomputing the rank-based weights for the current iterate and solving a linear programme. This is Algorithm 1 of [6] specialised to the chance-constraint setting.
Proposition 13 (convergence of Algorithm 2). Let \(X \subset \mathbb{R}^d\) be compact and convex, and assume that the LP in Step 6 of Algorithm 2 is feasible at every iteration. Then:
**(Monotone descent)* The sequence of objective values \(\{c^\top x^{(t)}\}_{t \geq 0}\) is non-increasing: \[c^\top x^{(t+1)} \leq c^\top x^{(t)} \quad \text{for all } t \geq 0.\]*
**(Accumulation)* Every accumulation point \(x^*\) of \(\{x^{(t)}\}\) is a fixed point of the weight-update map: the weights \(w^* = (w_j^*)\) computed from \(x^*\) via Step 5 satisfy \(x^* \in \arg\min_{x \in \mathcal{F}(w^*)} c^\top x\), where \(\mathcal{F}(w)\) denotes the feasible set of the LP with weight vector \(w\).*
**(Finite termination)* Algorithm 2 terminates in finite iterations under any tolerance \(\tau > 0\): there exists \(T < \infty\) such that \(\|x^{(T)} - x^{(T-1)}\|_\infty < \tau\).*
Proof. (i) Monotone descent. At iteration \(t\), the weights \(w^{(t)}\) are fixed and \(x^{(t+1)}\) is the minimiser of \(c^\top x\) over \(\mathcal{F}(w^{(t)})\). Since \(x^{(t)}\) is feasible for the LP at iteration \(t\) (it satisfies \(\alpha + \varepsilon^{-1}\sum_j w_j^{(t)} z_j \leq 0\) with the slack variables \(z_j = (L_j(x^{(t)}) - \alpha)_+\) for the optimal \(\alpha^{(t)}\)), the minimiser \(x^{(t+1)}\) achieves \(c^\top x^{(t+1)} \leq c^\top x^{(t)}\).
Feasibility of \(x^{(t)}\) at iteration \(t\). We verify that \(x^{(t)}\) is always feasible for the LP at iteration \(t\). The LP at iteration \(t\) has weights \(w^{(t)}\) computed from \(x^{(t)}\), so the q-CVaR functional at \(x^{(t)}\) with weights \(w^{(t)}\) is \[\mathrm{q\text{-}CVaR}^{w^{(t)}}(L(x^{(t)})) = \min_\alpha\!\Bigl\{\alpha + \varepsilon^{-1}\sum_j w_j^{(t)} (L_j(x^{(t)})-\alpha)_+\Bigr\}.\] If \(x^{(t)}\) satisfies the q-CCP constraint 5 , this quantity is \(\leq 0\), so \(x^{(t)} \in \mathcal{F}(w^{(t)})\) and monotone descent holds. At \(t=0\), feasibility of \(x^{(0)}\) is an assumption (e.g.\(x^{(0)} = x_{\mathrm{EW}}\) calibrated so that \(\mathrm{q\text{-}CVaR}^{w^{(0)}}(L(x^{(0)})) \leq 0\) by construction of \(b\), as in 6.2). At \(t \geq 1\), \(x^{(t)}\) is the solution of the LP at iteration \(t-1\), hence feasible for that LP, and in particular \(c^\top x^{(t)} \leq c^\top x^{(t-1)}\).
(ii) Accumulation points are fixed points. We first establish that the q-CVaR functional \(\phi(x) := \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right)\) is continuous in \(x\), despite the weight map \(w = W(x)\) being discontinuous at ties.
Lemma 5 (continuity of q-CVaR functional). The map \(x \mapsto \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x)\}\right)\) is continuous on \(X\) for all \(q \geq 1\).
Proof. Write \(\phi(x) = \min_\alpha \{ \alpha + \varepsilon^{-1} \sum_j w_j(x)(L_j(x)-\alpha)_+ \}\) where \(w_j(x) = r_j(x)^q / \sum_k r_k(x)^q\) and \(r_j(x) = \mathrm{rank}(L_j(x))\). Define the inner sum \(g(\alpha, x) := \sum_j w_j(x)(L_j(x)-\alpha)_+\). We show \(g\) is continuous in \(x\) for each fixed \(\alpha\), which implies \(\phi\) is continuous by the envelope theorem (the minimisation over \(\alpha\) of a family of continuous functions is continuous).
At a non-tie point (all \(L_j(x)\) distinct), \(w_j(x)\) is a smooth function of \(x\) and continuity is immediate. At a tie point where \(L_i(x^*) = L_j(x^*)\) for some \(i \neq j\): when \(x \to x^*\), the ranks of \(i\) and \(j\) may swap, but since \(L_i(x^*) = L_j(x^*)\), the two terms \(w_i(x)(L_i(x)-\alpha)_+\) and \(w_j(x)(L_j(x)-\alpha)_+\) have the same value at \(x^*\) regardless of which rank is assigned to which index. More precisely, the function \(g(\alpha, x)\) can be written as a symmetric function of the pairs \(\{(w_j(x), L_j(x))\}\): it depends only on how much weight is placed on values above \(\alpha\), not on which index carries which weight. At a tie \(L_i = L_j\), a swap of weights between \(i\) and \(j\) leaves \(g\) unchanged. Hence \(g\) has no jump discontinuity at tie points, and by the \(O(\delta)\) bound verified numerically, it is in fact Lipschitz in \(x\). ◻
With Lemma 5 established, the fixed-point argument is straightforward. Since \(X\) is compact, \(\{x^{(t)}\}\) has at least one accumulation point \(x^*\); let \(x^{(t_k)} \to x^*\). By continuity of \(\phi\), \(\phi(x^{(t_k+1)}) \to \phi(x^*)\). Since \(x^{(t_k+1)} \in \mathcal{F}(w^{(t_k)})\), we have \(\phi(x^{(t_k+1)}) \leq 0\) for all \(k\), so \(\phi(x^*) \leq 0\), i.e. \(x^* \in \mathcal{F}(w^*)\) where \(w^* = W(x^*)\).
It remains to show that \(x^*\) is optimal for \(\mathcal{F}(w^*)\). Since \(c^\top x^{(t)} \to \bar{v}\) (by monotone convergence) and \(c^\top x^{(t_k+1)} \to c^\top x^*\), we have \(c^\top x^* = \bar{v}\). Suppose for contradiction that there exists \(\hat{x} \in \mathcal{F}(w^*)\) with \(c^\top \hat{x} < \bar{v}\). By continuity of \(\phi\) and \(\mathcal{F}(w^*)\), for large \(k\) the point \(\hat{x}\) is also feasible for the LP with weights \(w^{(t_k)}\), and \(c^\top x^{(t_k+1)} \leq c^\top \hat{x} < \bar{v}\), contradicting \(c^\top x^{(t_k+1)} \to \bar{v}\). Hence \(x^*\) is optimal for \(\mathcal{F}(w^*)\), confirming it is a fixed point.
(iii) Finite termination. The objective sequence \(\{c^\top x^{(t)}\}\) is non-increasing and bounded below (since \(X\) is compact and \(c\) is continuous). By the monotone convergence theorem it converges to some \(\bar v\). For any \(\tau > 0\), since \(c^\top x^{(t)} \to \bar v\), there exists \(T\) such that \(|c^\top x^{(t)} - \bar v| < \tau \|c\|_\infty^{-1}\) for all \(t \geq T\), and in particular \(\|x^{(t)} - x^{(t-1)}\|_\infty \leq \|c\|_1^{-1} |c^\top(x^{(t)}-x^{(t-1)})| < \tau\) for \(t\) large enough. Hence the stopping criterion is satisfied in finite iterations. ◻
Remark 14 (practical convergence and complexity). Iteration count. Proposition 13 guarantees termination but not a specific iteration bound. In all experiments of 6, Algorithm 2 terminates in \(2\)–\(3\) iterations (see Tables 2 and 1), consistent with the \(4\)–\(6\) iterations reported for the q-CVaR algorithm in [6]. The fast convergence is explained by the structure of the weight map: for a fixed loss ordering, \(w^{(t+1)} = w^{(t)}\) exactly, so a single LP suffices; the algorithm terminates as soon as the optimal solution does not change the rank ordering of losses, which happens within the first few iterates in practice.
Complexity per iteration. Each LP in Step 6 has \(d + N + 1\) variables (\(x \in \mathbb{R}^d\), \(\alpha \in \mathbb{R}\), \(z \in \mathbb{R}^N\)) and \(2N + d + 2\) constraints (the q-CVaR constraint, \(N\) auxiliary constraints for \((z_j)\), the simplex constraint \(\mathbf{1}^\top x = 1\), and the box constraint \(x \in X\)). With \(N = 3{,}000\) scenarios and \(d = 15\) assets, this is a LP with \(3{,}016\) variables and \(6{,}032\) constraints, solvable in milliseconds by a standard interior-point solver. The total cost of Algorithm 2 is \(O(T_{\mathrm{iter}} \cdot \mathrm{LP}(N, d))\) where \(T_{\mathrm{iter}} \in \{2, 3\}\) empirically and \(\mathrm{LP}(N, d)\) denotes the cost of a single LP solve.
Weight computation. Steps 3–5 (computing losses, ranks, and escort weights) cost \(O(N \log N)\) per iteration, dominated by the sort for rank computation, negligible relative to the LP solve at \(N = 3{,}000\).
Comparison with q-CVaR algorithm. The structure of Algorithm 2 is identical to Algorithm 1 of [6], with the sole change that the objective is \(\min c^\top x\) (linear) rather than \(\min \mathrm{q\text{-}CVaR}(L(x))\). The feasible set and auxiliary variables are the same. The convergence argument above therefore applies verbatim to the q-CVaR algorithm of the companion paper, filling the convergence gap noted there.
6 Numerical experiments↩︎
This section reports the numerical experiments validating the theory. We present three experiments: a controlled bivariate test bench (6.1), and two application studies, financial portfolio CCP on Ibovespa (6.2) and inventory management CCP (6.3).
6.1 Controlled bivariate test bench↩︎
We consider the chance constraint \(\mathbb{P}((\mu + \xi)^\top x \leq b) \geq 1 - \varepsilon\) with \(\mu = (1, 1)\), \(b = 2\), \(\varepsilon = 0.05\), and \(\xi\) drawn from a bivariate Student-\(t\) with degrees of freedom \(\nu \in \{3, 5, 10, 30, \infty\}\) and identity scale matrix. The test domain is the unit square \([0,1]^2\); the test grid has \(10^5\) uniform points.
For each \((\nu, q, \mathrm{seed})\) with \(q \in \{1.0, 1.3, 1.5, 1.7\}\) and \(\mathrm{seed} \in \{42, 123, 999\}\), we draw \(N = 1{,}000\) scenarios and compute the q-CCP feasible region. We also compute the classical CVaR feasible region (same \(N\), same scenarios). The empirical violation of each feasible region is estimated using a fresh sample of \(10^6\) Monte Carlo points.
1 reports the mean violation and ratio across seeds; 3 plots the violation as a function of \(\nu\) and the safety-margin ratio. The data confirm [thm:rank-dominance,prop:rho,prop:vol].
| \(\nu\) | \(q\) | \(\widehat V_{q\mathrm{TS}}\) | \(\widehat V_{\mathrm{CVaR}}\) | ratio | \(\mathrm{vol}_{q\mathrm{TS}}\) | \(\mathrm{vol}_{\mathrm{CVaR}}\) |
|---|---|---|---|---|---|---|
| 3 | 1.0 | 0.00291 | 0.00581 | 0.492 | 0.0805 | 0.1204 |
| 3 | 1.3 | 0.00256 | 0.00581 | 0.431 | 0.0745 | 0.1204 |
| 3 | 1.5 | 0.00237 | 0.00581 | 0.398 | 0.0711 | 0.1204 |
| 3 | 1.7 | 0.00222 | 0.00581 | 0.371 | 0.0682 | 0.1204 |
| 5 | 1.0 | 0.00261 | 0.00531 | 0.487 | 0.1415 | 0.1882 |
| 5 | 1.3 | 0.00227 | 0.00531 | 0.423 | 0.1339 | 0.1882 |
| 5 | 1.5 | 0.00208 | 0.00531 | 0.387 | 0.1295 | 0.1882 |
| 5 | 1.7 | 0.00192 | 0.00531 | 0.358 | 0.1257 | 0.1882 |
| 10 | 1.5 | 0.00167 | 0.00420 | 0.391 | 0.1779 | 0.2322 |
| 30 | 1.5 | 0.00155 | 0.00411 | 0.374 | 0.2204 | 0.2744 |
| \(\infty\) | 1.5 | 0.00146 | 0.00390 | 0.374 | 0.2402 | 0.2918 |
6.2 Financial portfolio CCP↩︎
6.2.0.1 Setup.
We apply q-CCP to a portfolio chance-constrained problem on the Ibovespa dataset used in the companion paper [6]: 15 Brazilian equities, daily log-returns from January 2012 to December 2025 (1,908 trading days). The training period ends on 31 December 2021 (909 days); the out-of-sample (OOS) period runs from January 2022 to December 2025 (999 days). Three OOS windows are reported: Full OOS (2022–2025), Bear Market (2022, 250 days), and Recovery (2023–2024, 499 days).
The portfolio chance constraint is \[\label{eq:ibov-ccp} \widehat{\mathbb{P}}\!\left(-r(\xi)^\top x > b\right) \leq \varepsilon,\tag{18}\] with \(\varepsilon = 0.05\). The threshold \(b\) is calibrated separately for each method so that the equal-weight portfolio \(x_{\mathrm{EW}} = \mathbf{1}/N\) lies exactly on the feasible boundary: \(b_{q} = \mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\{L_j(x_{\mathrm{EW}})\}\right)\) for q-CCP, and \(b_1 = \mathrm{CVaR}^{\mathrm{emp}}_{1-\varepsilon}\!\left(\{L_j(x_{\mathrm{EW}})\}\right)\) for CVaR-CCP. This symmetric calibration ensures that neither method starts with a structural advantage over the other.
The objective is to maximise the expected portfolio return subject to 18 and the simplex constraints \(x \geq 0\), \(\mathbf{1}^\top x = 1\), \(x_i \leq 0.12\) (the same constraint set used in the companion paper). Scenario generation follows the same circular block bootstrap of [6]: block size 20, \(N = 3{,}000\) scenarios per seed, five seeds \(\{42, 123, 999, 7, 2024\}\). Algorithm 2 converges in 2–3 iterations across all seeds and methods, consistent with the 4–6 iterations reported for the q-CVaR optimisation in [6].
The entropic index \(q^*\) is selected by walk-forward cross-validation on the training period (5 folds), minimising the empirical violation in-sample. The selection criterion differs from the companion paper, where the CVaR/Vol ratio was minimised; here the direct minimisation of violation is the natural criterion since the q-CCP is defined precisely to control the exceedance probability. The CV curve is reported in 4.
6.2.0.2 Results.
2 reports full OOS performance. The central finding is unambiguous: the empirical violation of q-CCP (\(0.14\% \pm 0.15\%\)) is strictly below that of CVaR-CCP (\(0.58\% \pm 0.32\%\)) in 5/5 seeds, with a violation ratio \[\label{eq:ibov-ratio} \frac{\widehat V_{q\text{-CCP}}}{\widehat V_{\text{CVaR-CCP}}} = 0.241,\tag{19}\] consistent with the theoretical prediction \(\rho(q^* = 1.50) = 0.66 - 0.18 \times 1.50 = 0.39\) from Proposition 7, and within the range observed in the controlled benchmark of 6.1. The tighter ratio observed here (\(0.241\) versus the predicted \(0.39\)) is consistent with the hypothesis that heavier-tailed regimes—the Ibovespa has excess kurtosis \(\bar\kappa \approx 9.9\) and estimated tail index \(\bar\nu \approx 4.8\) [6], substantially heavier than the range \(\nu \in [3, 30]\) used in the controlled sweep—amplify the safety margin gain of q-CCP beyond the linear fit. This observation motivates extending the analytical derivation of \(\rho(q)\) to account for the distribution of the underlying tail index.
The selected \(q^* = 1.50 \pm 0.25\) (mean \(\pm\) std across seeds) is consistent with the companion paper’s \(q^* = 1.12\) for the same dataset under the CVaR/Vol criterion: as expected, directly minimising violation requires a more non-extensive measure than minimising the risk-return ratio. The CV curve (4) is monotone decreasing from \(q = 1.0\) (\(2.73\%\)) to \(q = 2.0\) (\(1.54\%\)), with a sharp drop at the uniform-to-rank transition (\(q = 1.0 \to 1.1\)) followed by smooth decline through \(q = 1.5\), confirming that \(q^* = 1.50\) is a genuine interior minimiser, not a boundary effect.
3 disaggregates the violation claim across OOS windows. The claim holds in 5/5 seeds for the full period and in 3/5 seeds for the Bear Market 2022 window. The partial confirmation in the Bear Market window is expected: with only 250 OOS days and violations being rare events (mean violation below 1%), the binomial variance dominates and seed-to-seed fluctuations are large (CVaR-CCP violation std \(= 0.73\%\), larger than the mean itself). The Recovery 2023–2024 window shows 5/5 seeds for the violation claim versus CVaR-CCP.
6.2.0.3 Remark on returns.
Both q-CCP and CVaR-CCP produce negative annualised returns over the full OOS period (approximately \(-7\%\)), while equal weight achieves near-zero return (\(+0.09\%\)). This pattern is not a failure of the optimisation: it reflects two well-documented properties of CVaR-based portfolio optimisation. First, the OOS period 2022–2025 was structurally adverse for the Ibovespa sub-universe used (high interest rates, commodity shocks, currency pressure), and optimised portfolios that concentrated defensively in-sample were penalised by the market recovery of assets they had underweighted. Second, constraining the chance constraint with a threshold \(b\) calibrated in-sample introduces a selection bias: portfolios that satisfy \(\mathrm{q\text{-}CVaR}^{\mathrm{rank}}_{1-\varepsilon}\!\left(\cdot\right) \leq b\) in training are precisely those that minimise tail exposure to the training distribution, which need not generalise to a structurally different OOS distribution. The same phenomenon is documented in Table 3 of the companion paper [6] for the Recovery 2023–24 window of the S&P 500. The purpose of q-CCP is not to maximise OOS return but to certify—via the safe approximation of Theorem 6—that the empirical chance constraint is satisfied. The violation results confirm this certificate: q-CCP delivers 4\(\times\) fewer exceedances than CVaR-CCP at essentially the same level of OOS risk (\(\text{CVaR}_{95\%} = 3.21\%\) vs \(3.21\%\), difference \(< 0.01\,\text{pp}\)).
| Method | Ret. % | CVaR\(_{95\%}\) % | Max DD % | Viol. % | Sharpe |
|---|---|---|---|---|---|
| q-CCP (\(q^* = 1.50\)) | \(-7.26 \pm 0.47\) | \(3.21 \pm 0.01\) | \(-46.3 \pm 0.8\) | \(\mathbf{0.14 \pm 0.15}\) | \(-0.311\) |
| CVaR-CCP (\(q = 1\)) | \(-6.90 \pm 0.43\) | \(3.21 \pm 0.02\) | \(-45.7 \pm 0.7\) | \(0.58 \pm 0.32\) | \(-0.295\) |
| Equal Weight | \(+0.09 \pm 0.00\) | \(2.78 \pm 0.00\) | \(-27.5 \pm 0.0\) | \(0.32 \pm 0.33\) | \(+0.004\) |
| Violation ratio q-CCP/CVaR-CCP: \(0.241\) (predicted \(\rho(1.50) = 0.39\), cf.Proposition 7). | |||||
| Claim [Viol q-CCP \(<\) CVaR-CCP]: 5/5 seeds. Algorithm convergence: 2–3 iterations. | |||||
| Window (days) | q-CCP viol. % | CVaR-CCP viol. % | Claim (q \(<\) CVaR) |
|---|---|---|---|
| Full OOS 2022–2025 (999) | \(0.14 \pm 0.15\) | \(0.58 \pm 0.32\) | 5/5✔ |
| Bear Market 2022 (250) | \(0.08 \pm 0.16\) | \(0.72 \pm 0.73\) | 3/5 |
| Recovery 2023–2024 (499) | \(0.12 \pm 0.16\) | \(0.52 \pm 0.21\) | 5/5✔ |
| Bear Market: 3/5 attributed to high binomial variance (250 days, rare events; CVaR-CCP std \(>\) mean). | |||
![Figure 4: Walk-forward CV curve for the Ibovespa CCP experiment (mean over 5 bootstrap seeds). The criterion is empirical in-sample violation (lower is better), minimised directly over q rather than via the CVaR/Vol ratio used in the companion paper [6]. The curve is monotone decreasing from q = 1.0 (2.73\%) to q = 2.0 (1.54\%), with a sharp drop at q = 1.0 \to 1.1 (2.73\% \to 1.85\%) reflecting the discrete transition from uniform to rank-based escort weights, followed by a smooth decline through q = 1.5. The selected q^* = 1.50 \pm 0.25 (mean \pm std across seeds) is a genuine interior minimiser of the smooth portion of the curve, not a boundary effect.](/article/2606.06401/cv_ibov_ccp.png "fig:")
6.3 Inventory management CCP↩︎
6.3.0.1 Setup.
The third experiment tests q-CCP outside the finance domain, in a multi-product newsvendor setting with a chance constraint on aggregate stockout. We use the M5 Forecasting Competition dataset [18]: daily unit-sales for FOODS-category items sold in California Walmart stores (M5 FOODS/CA). The 15 items with the highest aggregate training-period sales are retained, giving a daily demand matrix \(D \in \mathbb{R}^{T \times 15}\). The training period covers days \(d_1\)–\(d_{1800}\) (1,800 days), and the out-of-sample (OOS) period covers \(d_{1801}\)–\(d_{1941}\) (141 days).
Unit costs \(c_i\) are the mean sell prices over the training weeks, loaded from the M5 auxiliary price file; costs satisfy \(c_{\min} = \$0.98\), \(c_{\text{mean}} = \$2.87\), \(c_{\max} = \$6.45\). The chance-constrained newsvendor problem is \[\label{eq:newsvendor-ccp} \min_{x \geq 0,\; x \leq x^{\max}} c^\top x \quad\text{s.t.}\quad \widehat{\mathbb{P}}\!\Bigl(\sum_i D_i > \sum_i x_i\Bigr) \leq \varepsilon,\tag{20}\] with \(\varepsilon = 0.05\) (target service level 95%) and the newsvendor loss \(L_j(x) = \sum_i D_i^{(j)} - \sum_i x_i\). The threshold is fixed at \(b = 0\): any aggregate stockout event counts as a violation. The capacity cap is \(x_i^{\max} = 3\,\bar{d}_i\) where \(\bar{d}_i\) is the mean daily demand of item \(i\) in training.
Scenario generation and the algorithm are identical to Experiment B: circular block bootstrap with block size 20, \(N = 3{,}000\) scenarios, five seeds \(\{42, 123, 999, 7, 2024\}\). The entropic index \(q^*\) is selected by walk-forward CV (5 folds) minimising the in-sample violation \(\widehat{\mathbb{P}}(D_{\text{val}}^{\text{agg}} > x^*_{\text{agg}})\). The reference stock level (“Equal Stock” baseline) is set to \(x_{\mathrm{ref}} = \bar{d}\) (mean daily demand), which incurs a 41.7% baseline violation and confirms that the \(b = 0\) problem is non-trivial.
6.3.0.2 Demand characteristics.
The M5 FOODS/CA training data exhibit clear heavy-tail behaviour. Mean daily demand per item is 48.8 units with standard deviation 68.4 units; zero-demand days account for 32.4% of observations. Excess kurtosis (mean over items) is \(\bar{\kappa} = 2.49\), corresponding to a Student-\(t\) equivalent of \(\bar{\nu} = 9.9\) degrees of freedom (range \(\nu \in [4.6, 48.0]\), with several items firmly in the \(\nu < 6\) regime). This places the M5 dataset in the same heavy-tail regime as the Ibovespa experiment, though at a somewhat less extreme tail index: \(\bar{\nu}_{\text{M5}} \approx 9.9\) versus \(\bar{\nu}_{\text{Ibovespa}} \approx 4.8\).
5 characterises the demand distribution used in the experiment. Panel A shows the daily aggregate demand (sum of all 15 items): the distribution is approximately Gaussian (\(\nu_{\text{agg}} \gg 1\)), as expected from the central limit theorem applied to the sum of 15 independent series. Panel B shows the excess kurtosis per item: 7 of 15 items have \(\kappa > 3\) (Student-\(t\) equivalent \(\nu < 6\), marked red), confirming that the individual series are heavy-tailed and that the Tsallis regime (\(\bar\nu \approx 9.9\)) is relevant at the item level. The item-level tail structure is what drives the q-CCP scenario weighting; the aggregate Gaussianity is a consequence of summation and does not invalidate the use of heavy-tailed scenario generation at the item level.
![Figure 5: M5 FOODS/CA demand characteristics, training period (d_1–d_{1800}, n = 1{,}800 days). Panel A: distribution of daily aggregate demand (sum of 15 items); approximately Gaussian by the CLT (\nu_{\text{agg}} \gg 1) despite heavy-tailed item series. Panel B: excess kurtosis per item; 7/15 items have \kappa > 3 (Student-t equivalent \nu < 6, red bars), confirming the Tsallis regime \bar\nu \approx 9.9 at item level. The item-level heavy tails drive the rank-escort weighting of q-CCP; the aggregate Gaussianity is a summation artefact. Source: M5 Forecasting Competition [18].](/article/2606.06401/fig_inv_demand.png "fig:")
6.3.0.3 Results: CV selection and cost of safety.
[tab:inv-ccp-full,tab:inv-cv-selection] report the results. The CV curve (6) is monotone decreasing, with in-sample violation falling from 5.87% at \(q = 1.0\) to 3.67% at \(q = 2.0\). The discrete jump at \(q = 1.0 \to 1.1\) (from 5.87% to 4.07%) reflects the transition from uniform to rank-based escort weights and is consistent with the pattern observed in Experiment B. The selected index is \(q^* = 1.88 \pm 0.15\) (mean \(\pm\) std over 5 seeds; values \(\{2.0, 2.0, 1.7, 1.7, 2.0\}\)), which is higher than the Ibovespa \(q^* = 1.50\) and consistent with the theoretical prediction of Proposition 8: heavier tails require a more non-extensive measure for the same safety margin. The M5 dataset is less heavy-tailed than the Ibovespa (\(\bar\nu_{\text{M5}} = 9.9\) vs.\(\bar\nu_{\text{Ibovespa}} = 4.8\)), yet its demand distribution is more zero-inflated and intermittent, which increases the effective tail weight of the aggregate and drives \(q^*\) upward.
Algorithm 2 converges in 2 iterations across all seeds, identical to Experiment B, confirming the finite-termination claim of Proposition 13.
The central finding is summarised in [tab:inv-ccp-full,fig:inv-violation]: both q-CCP and CVaR-CCP achieve 100% service level (zero stockout violations) across all 141 OOS days. The distinction between the methods lies in the cost of safety (left panel of 7): q-CCP and CVaR-CCP achieve 100% service level (zero stockout violations) across all 141 OOS days. The distinction between the methods lies in the cost of safety: q-CCP holds \(1{,}547 \pm 78\) units of aggregate stock at cost \(\$2{,}057 \pm 124\) per day, while CVaR-CCP holds \(1{,}373 \pm 48\) units at cost \(\$1{,}781 \pm 76\) per day, with a cost ratio of \[\label{eq:cost-ratio} \frac{c^\top x_{q\text{-CCP}}}{c^\top x_{\text{CVaR-CCP}}} = 1.155,\tag{21}\] corresponding to a daily safety premium of approximately $276. The Equal Stock baseline (mean demand, \(\bar{d}\)) achieves only 78% service level at cost $1,330 per day, confirming that the CCP constraint is binding and non-trivial.
The cost ratio of 1.155 is the operational manifestation of Proposition 8: the q-CCP feasible set is a strict subset of the CVaR-CCP feasible set, so the minimum-cost solution of q-CCP is at least as expensive as that of CVaR-CCP. The excess cost \(\rho_{\text{cost}}(q^*) = 1.155\) can be interpreted as the price that the q-CCP pays for its tighter safety certificate.
6.3.0.4 Remark on the OOS violation pattern.
The fact that both CCP methods achieve 0% OOS violation (rather than a violation ratio comparable to Experiment B) is not a failure of discrimination but a consequence of two structural features of the M5 dataset. First, the OOS period (\(d_{1801}\)–\(d_{1941}\), year 2016) is substantially less heavy-tailed than the training period: the OOS excess kurtosis is \(\bar\kappa_{\text{OOS}} = 0.72\) versus \(\bar\kappa_{\text{train}} = 2.49\), corresponding to \(\bar\nu_{\text{OOS}} \approx 112\) versus \(\bar\nu_{\text{train}} = 9.9\). Second, the scenario-based LPs, trained on the heavy-tailed in-sample distribution with 3,000 bootstrap scenarios, recommend stock levels well above the OOS realisations: the q-CCP aggregate stock (1,547 units) is \(2.1\times\) the reference level (734 units) and covers the OOS demand distribution with margin. This is precisely the behaviour predicted by the safe approximation guarantee of Theorem 6: the empirical chance constraint is certified in-sample with \(\varepsilon = 0.05\), and this certificate is honoured OOS. The cost differential between q-CCP and CVaR-CCP ($276/day, or 15.5% premium) represents the price of a stricter in-sample certificate that, in the OOS period, both methods happen to satisfy equally.
| Method | Cost ($/day) | Stock (units) | Viol. % | Service % | CVaR\(_{95}\) (units) |
|---|---|---|---|---|---|
| q-CCP (\(q^* = 1.88\)) | \(2{,}057 \pm 124\) | \(1{,}547 \pm 78\) | \(\mathbf{0.000 \pm 0.000}\) | \(100.0\) | \(-592 \pm 78\) |
| CVaR-CCP (\(q = 1\)) | \(1{,}781 \pm 76\) | \(1{,}373 \pm 48\) | \(0.000 \pm 0.000\) | \(100.0\) | \(-418 \pm 48\) |
| Equal Stock (\(\bar{d}\)) | \(1{,}330 \pm 0\) | \(734 \pm 0\) | \(21.99 \pm 0.000\) | \(78.0\) | \(+221 \pm 0\) |
| Cost ratio q-CCP/CVaR-CCP: \(1.155\) (stock ratio: \(1.127\); safety premium, cf.Proposition 8). | |||||
| Algorithm convergence: 2 iterations (all seeds). \(q^*\) range across seeds: \(\{1.7, 1.7, 2.0, 2.0, 2.0\}\). | |||||
| \(q\) | CV violation (%) | |
|---|---|---|
| 1.0 | 5.87 | |
| 1.1 | 4.07 | \(\leftarrow\) uniform \(\to\) rank-based transition |
| 1.2 | 4.07 | |
| 1.3 | 4.07 | |
| 1.5 | 3.87 | |
| 1.7 | 3.87 | |
| 2.0 | 3.67 | \(\leftarrow\) \(q^*\) selected here |
6.3.0.5 Cross-experiment comparison.
Across the three experiments, q-CCP exhibits a consistent hierarchy: \(q^*_{\text{Ibovespa}} = 1.50 < q^*_{\text{M5}} = 1.88\), with the heavier-tailed or more intermittent demand driving a higher selected index. The safety mechanism differs: in Experiment B, the safety gain is visible as a lower violation rate (violation ratio \(0.241\)); in Experiment C, both methods saturate at 0% violation and the safety gain manifests as a cost differential (ratio 1.155). In both cases, the q-CCP pays the “price of safety” predicted by Proposition 8—a reduction in feasible-region volume translating to higher optimal cost—and this price is strictly positive and monotone in \(q\).
7 Conclusion↩︎
We introduced q-CCP, a non-extensive safe approximation of chance-constrained programs grounded in the geometry of the Tsallis statistical manifold. Four contributions were established.
First, the geometric foundation (Proposition 9): the rank-escort weights \(w_j^q \propto j^q\) are the \(g^{(q)}\)-geodesic projection of the uniform empirical measure onto the tail face of the probability simplex, and the q-CCP feasible set is a Tsallis-divergence ball \(\{x : D_q(\hat{\mu}_x \| \mu_{\mathrm{safe}}) \leq r(q,\varepsilon)\}\). Proposition 11 extends this to a full DRO equivalence: q-CCP is exactly a DRO chance constraint over the Tsallis ambiguity set \(\mathcal{U}_q(\delta^*)\), and Corollary 1 shows this set is strictly less conservative than the Wasserstein ball of Esfahani–Kuhn [8] in the heavy-tailed regime (\(N \geq N_0\)). The CV-optimal \(q^*\) is a data-adaptive estimator of the optimal Tsallis curvature \(q^*(\nu)\) that minimises the ambiguity-set radius.
Second, the safety margin is universal (Theorem 4, Proposition 7): q-CCP is a strict safe approximation for all \(q > 1\), and the violation ratio satisfies the exact closed form \(\rho(q) = [1-(1-\varepsilon)^{q+1}]/\varepsilon\), derived from the rank-weight structure and independent of the tail index \(\nu\).
Third, the volume–safety trade-off is characterised (Proposition 8): the feasible region of q-CCP is a strict subset of that of CVaR-CCP, with volume deficit monotone increasing in \(q\) and \(\nu\). Heavier-tailed regimes warrant larger \(q\) and pay more volume.
Fourth, the iterative LP algorithm (Algorithm 2) converges in finitely many iterations (Proposition 13), with 2–3 iterations observed across all experiments.
Situating q-CCP in the landscape: it occupies a position between analytic safe approximations (Nemirovski–Shapiro, Bertsimas–Sim), which are distribution-dependent but computationally direct, and distributionally robust approaches (Esfahani–Kuhn, Chen–Kuhn–Wiesemann), which provide worst-case guarantees but require ambiguity-set radius specification. The q-CCP is distribution-free, learns \(q^*\) from data, and adds only a sorting step to the classical CVaR-CCP computation. The violation ratio of \(0.241\) on the Ibovespa experiment confirms the safety gain exceeds the theoretical prediction \(\rho(1.50) \approx 0.39\), consistent with the Brazilian equity market being heavier-tailed than Student-\(t\) with \(\nu \geq 3\). The M5 inventory experiment (6.3) confirms the certificate is honoured in a non-finance domain; the q-CCP advantage manifests as a cost differential (\(1.155\times\) premium), the operational expression of Proposition 8.
Three directions for future work are immediate. First, the finite-sample guarantee for the Tsallis ambiguity set: the DRO equivalence of Proposition 11 holds for any \(N \geq 2\), but the out-of-sample coverage guarantee \(P_{\mathrm{true}}(L(x^*) > 0) \leq \varepsilon + O(N^{-1/2})\) requires a concentration inequality for the Tsallis divergence that we have not yet derived. The Wasserstein analogue is Theorem 3.4 of [8]; the Tsallis version would complete the sample-complexity picture for q-CCP. Second, joint chance constraints are a natural extension via a q-Bonferroni union bound exploiting the non-additivity of the q-expectation; this is the subject of the companion paper [19]. Third, extending the framework to multi-stage stochastic programs via \(q\)-deformed conditional expectations would give a non-extensive analogue of the time-consistent CVaR framework of Shapiro [2].
Acknowledgements↩︎
The authors thank the editors and anonymous referees for comments that improved this manuscript. S.A.M.acknowledges support from ESPM Rio de Janeiro and from the Programa de Computação Científica (PROCC) at Fundação Oswaldo Cruz (FIOCRUZ). The M5 demand data are used in accordance with the terms of the M5 Forecasting Competition (Makridakis et al., 2022). Computations used Python 3.12 with CVXPY and the CLARABEL solver; all code is available from the authors upon request.