quantile-based reformulation

This work addresses constraint satisfaction problems involving random variables, aiming to find deterministic parameters that maximize the probability of satisfying all constraints under uncertainty. To this end, the authors propose a novel approach that integrates oracle-based stochastic gradient descent with interval arithmetic: the former efficiently explores high-quality parameter configurations, while the latter provides rigorous and successively tighter lower bounds on the satisfaction probability. This is the first method to synergistically combine these two techniques for solving stochastic constraints, offering both high-probability convergence guarantees and computational reliability. Experimental results on stochastic satisfiability modulo theories (SSMT) and stochastic trajectory planning tasks demonstrate that the proposed method efficiently generates sequences of reliable lower bounds that closely approximate the true optimal value.

This work addresses resource allocation under high-reliability requirements (e.g., power distribution, emergency response, cloud resource scheduling), where conventional distributionally robust optimization (DRO) methods suffer exponential distortion in the asymptotic scaling laws of optimal cost and decisions under joint chance constraints. Method: We propose a novel DRO paradigm that preserves asymptotic scaling consistency. Our approach employs an *f*-divergence ambiguity set coupled with marginal distribution modeling to accurately characterize scaling behavior as the constraint satisfaction probability approaches one. Contribution/Results: For the first time, we achieve exact asymptotic scaling of optimal decisions and cost under near-certainty constraints, bounding decision conservatism within constant or logarithmic factors. Moreover, we break the classical 1/*N* violation-probability lower bound—achieving sub-Pareto-optimal estimation. Theoretically and empirically, our method attains violation probabilities significantly below Ω(1/*N*) with *N* samples, markedly enhancing both decision accuracy and economic efficiency in high-reliability settings.

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.

To address scenario-based decision-making under uncertainty, this paper proposes a risk-controllable decision-making method based on scenario compression. To overcome the looseness and strong distributional assumptions inherent in existing risk upper bounds, we derive the first compression-size-dependent risk bound that requires no additional assumptions—integrating stochastic geometric analysis, compression set theory, and refined probabilistic inequalities with optimization. This bound significantly improves tightness: for identical numbers of scenarios and prescribed risk tolerance levels, the upper bound on decision failure probability is reduced by 20–40% on average. The method ensures theoretical rigor while maintaining broad applicability across diverse data-driven robust decision-making settings, thereby providing a more reliable risk-quantification framework for robust optimization under uncertainty.

This study addresses the feasibility determination problem under subjective probability constraints within a finite set of alternative systems. The authors propose a statistical inference method that operates directly on Bernoulli simulation outputs, uniquely integrating multi-threshold subjective constraints with Bernoulli observations without relying on normal approximations. To handle extreme scenarios—such as when all systems are feasible or none are—the method incorporates two heuristic strategies that dynamically adjust thresholds during execution. The resulting batch-mean-independent testing algorithm maintains rigorous statistical validity while significantly outperforming existing approaches designed for normally distributed data. Empirical experiments demonstrate the method’s computational efficiency and robust adaptability across diverse problem settings.

This work addresses the challenges of model inaccuracy and ambiguous state distributions in nonlinear systems by proposing a distributionally robust optimization-based chance-constrained control framework. The approach constructs an ambiguity set using relative entropy constraints and derives an upper bound on risk expectations via the variational representation of the exponential integral. It further integrates nonlinear covariance propagation with adaptive determination of the ambiguity set radius based on second-order dynamic truncation error. Notably, the method recovers nominal risk in the zero-divergence limit, thereby overcoming the restrictive assumptions of Gaussianity prevalent in conventional approaches. Validation on spacecraft stochastic guidance tasks demonstrates that the proposed framework effectively enforces probabilistic safety constraints under distributional uncertainty.

This work addresses the feedback loop between decisions and data generation in decision-dependent chance-constrained optimization by proposing the first model-free scenario optimization framework that incorporates a performative feedback mechanism. The framework alternates between data generation and optimization to seek a self-consistent equilibrium, and establishes the existence of performative solutions via Kakutani’s fixed-point theorem. An algorithm combining stochastic fixed-point iterations, logarithmically increasing sample scheduling, and scenario approximation is further developed, guaranteeing almost sure convergence. Empirical validation on a large language model jailbreaking defense task demonstrates the method’s effectiveness and practicality, successfully achieving co-evolution and stable convergence between the defensive classifier and the distribution of adversarial prompts.

This work addresses the limitations of existing uncertainty quantification methods in regression tasks, which predominantly rely on pointwise predictive risk and struggle to characterize uncertainty under non-conditional-expectation objectives. The authors propose QUEST, a novel framework that introduces highest density regions (HDRs) into uncertainty quantification, measuring uncertainty via the Lebesgue measure of the HDR’s support set and incorporating a robustness parameter α to capture the concentration of probability mass around distributional modes. This metric satisfies key axioms such as monotonicity and translation invariance, overcoming fundamental shortcomings of conventional scoring rules. Empirical evaluations on selective prediction benchmarks demonstrate that QUEST significantly outperforms standard uncertainty measures—such as predictive variance and differential entropy—in capturing both aleatoric and epistemic uncertainty.

This work addresses the reliability risks in Gaussian mixture models arising from structural misspecification by introducing a Wasserstein-2-based continuous ambiguity set that overcomes the conventional limitation of finite support. This framework enables endogenous optimization of the number of mixture components, means, and covariances under the worst-case distribution. By integrating Bures–Wasserstein geometry, strong duality theory, and semi-infinite programming, the authors reformulate chance-constrained problems and develop an efficient algorithm combining adaptive cutting planes with block-alternating local search, which attains any prescribed accuracy within finitely many steps. Applied to energy allocation in electric vehicle charging stations, the method not only guarantees target reliability levels but also yields structurally distinct resource allocation strategies compared to nominal solutions.

This study addresses the challenges of non-convex objective optimization and the high adaptation costs associated with multiple objectives in chance-constrained programming by proposing the D3Opt framework. This method decouples constraints from objectives through a risk-conditioned diffusion model combined with a particle Feynman–Kac correction, while freezing the diffusion prior to enable derivative-free optimization. The core contribution lies in achieving, for the first time, a “learn feasibility once, reuse across multiple objectives” paradigm that eliminates the need for retraining when encountering new objectives. Experimental results demonstrate that D3Opt efficiently handles both smooth and non-smooth objectives, exhibiting superior objective generalization capability and optimization efficiency under fixed constraints.