cvar-based constraint reformulation

This paper addresses the optimal asset allocation problem for (re)insurers subject to regulatory Conditional Value-at-Risk (CVaR) constraints. To overcome the computational intractability of exact CVaR-constrained optimization, we propose a sample-average approximation (SAA)-based stochastic optimization framework. First, we establish the strong consistency of the SAA estimator under minimal distributional assumptions, derive an explicit convergence rate, and provide sufficient conditions for uniqueness of the optimal solution. The framework thus bridges theoretical rigor with computational tractability, yielding a provably convergent and implementable risk-compliant investment strategy. It enhances capital efficiency and portfolio robustness while ensuring regulatory compliance and prudent risk management.

In few-shot settings, tail risk measures—such as Conditional Value-at-Risk (CVaR)—are severely biased due to data sparsity in distributional tails. To address this, this paper introduces the first systematic integration of Extreme Value Theory (EVT) into Distributionally Robust Optimization (DRO), yielding a theoretically grounded Tail-DRO framework. Our method parametrically models the tail via the Generalized Pareto Distribution (GPD), constructing an uncertainty set that balances representativeness and conservatism; it requires only scalar parameter calibration and naturally extends to multivariate settings and diverse tail-risk metrics. We establish theoretical guarantees on feasibility under limited samples. Empirical evaluation on synthetic and real-world financial/insurance datasets demonstrates that our approach yields significantly more robust and unbiased worst-case CVaR estimates compared to conventional DRO baselines, achieving consistent performance gains across all benchmarks.

This study addresses the Pandora’s box problem and prophet inequalities under the Conditional Value-at-Risk (CVaR) risk measure. By employing a variational reduction, the risk-aware Pandora’s box problem is transformed into a one-dimensional index policy, for which an exact Weitzman-type solution is established. Regarding prophet inequalities, the work demonstrates that no constant-factor approximation can be guaranteed without distributional assumptions; however, under continuous distributions satisfying the increasing failure rate average (IFRA) condition, a threshold policy yields an explicit constant-factor approximation guarantee. This paper provides the first systematic characterization of theoretical limits for these two sequential decision problems within the CVaR framework, revealing that risk-sensitive objectives can undermine classical approximation guarantees and identifying new feasibility conditions that depend critically on distributional structure.

This paper addresses chance-constrained optimization (CCO) problems with stochastic constraints by proposing a novel framework termed Conformal Prediction Programming (CPP). CPP is the first to integrate conformal prediction into CCO, leveraging the quantile lemma to reformulate chance constraints as deterministic counterparts, thereby providing both prior and posterior guarantees on constraint satisfaction. The framework accommodates distributional shifts, class-conditional constraints, and joint chance constraints, and yields variants including Robust CPP and Mondrian CPP. Algorithmically, it encompasses three implementations: CPP-MIP (mixed-integer programming), CPP-Bilevel (bilevel optimization), and CPP-Discarding (sample-discard-based optimization). Extensive experiments across multiple case studies demonstrate that CPP significantly outperforms conventional scenario-based methods—achieving superior constraint satisfaction rates while maintaining high solution quality—without compromising theoretical rigor or computational scalability.

Existing methods struggle to accurately capture the asymmetric and heavy-tailed characteristics of renewable energy forecast errors, often leading to inefficient reserve allocation or underestimated risk. This work proposes a source-agnostic framework that avoids prespecifying error distribution forms: leveraging historical forecast errors or probabilistic forecasts, it employs nonparametric density estimation to construct conditional error distributions for load, wind, and photovoltaic generation, which are then aggregated into a net load error distribution. The expected uncovered imbalance is quantified via Conditional Value-at-Risk (CVaR), and upward and downward reserve capacities are determined according to coverage and risk criteria. These reserves are incorporated as deterministic constraints into the production cost model, circumventing the need for scenario generation and stochastic optimization. Experiments on a synthetic NYISO dataset demonstrate that the proposed method significantly reduces reserve requirements while meeting target coverage levels, effectively mitigating the tail-risk distortion inherent in Gaussian assumptions.

Controlling conditional value-at-risk (CVaR)—a nonlinear tail risk measure—remains challenging in non-stationary and even adversarial environments. This work proposes the first distribution-free online framework that extends conformal inference to adversarial settings, achieving asymptotically exact CVaR control under arbitrary data-generating processes by integrating the variational representation of CVaR with online learning. Theoretical analysis establishes finite-sample conservativeness and asymptotic tightness guarantees. Empirical evaluations on portfolio optimization and toxicity mitigation in large language models demonstrate the method’s effectiveness, consistently attaining target CVaR levels with robustness across diverse scenarios.

This work addresses the challenge that when the loss function depends on decision variables, the regularity properties—such as continuity and differentiability—of Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are generally not guaranteed, thereby hindering theoretical and algorithmic advances in risk-aware optimization. Focusing on such decision-dependent losses, the paper establishes simple yet rigorous sufficient conditions under which CVaR is proven for the first time to be continuously differentiable, and provides an explicit expression for its gradient. By integrating tools from perturbation analysis of probability measures, path-differentiability, and real analysis, the study develops a unified theoretical framework that ensures the continuity of VaR and the continuous differentiability of CVaR, thereby furnishing reliable gradient information and convergence guarantees for optimization problems involving tail risk.

This study addresses the problem of robust information structure design in settings where agents’ utility coefficients are unknown. It introduces Calibrated Bayesian Correlated Equilibrium (Cal-BCE) as a solution concept and establishes a revelation principle and a joint decentralization theorem tailored to environments with utility uncertainty, elucidating how distinct risk criteria constrain cross-agent action covariances. Within a linear-quadratic-Gaussian framework, the authors leverage Hadamard invertibility conditions and employ second-order cone and semidefinite programming techniques to reformulate the original nonconvex problem into a convex optimization form incorporating probabilistic and Conditional Value-at-Risk (CVaR) constraints. Empirical experiments using 15 industry ETFs demonstrate that probability-based optimization enhances average welfare, while CVaR-based optimization strengthens tail-risk protection, revealing a clear trade-off between the two objectives.