Stochastic Programming

First-order Methods for Unconstrained Vector Optimization Problems: A Unified Majorization-Minimization Perspective

  • Jian Chen
    • Categories Multi-Criteria Optimization, Nonlinear Optimization, Stochastic Programming Tags , , , ,

    In this paper, we develop a unified majorization-minimization scheme and convergence analysis with first-order surrogate functions for unconstrained vector optimization problems (VOPs). By selecting different surrogate functions, the unified method can be reduced to various existing first-order methods. The unified convergence analysis reveals that the slow convergence of the steepest descent method is primarily attributed … Read more

    A Gauge Set Framework for Flexible Robustness Design

  • Ningji Wei
  • Xian Yu
  • Peter Yun Zhang
    • Categories Robust Optimization, Stochastic Programming

    This paper proposes a unified framework for designing robustness in optimization under uncertainty using gauge sets, convex sets that generalize distance and capture how distributions may deviate from a nominal reference. Representing robustness through a gauge set reweighting formulation brings many classical robustness paradigms under a single convex-analytic perspective. The corresponding dual problem, the upper … Read more

    Primal-dual resampling for solution validation in convex stochastic programming

  • Yi Chu
  • Susan R. Hunter
  • Raghu Pasupathy
    • Categories Convex Optimization, Optimization in Data Science, Stochastic Programming Tags , ,

    Suppose we wish to determine the quality of a candidate solution to a convex stochastic program in which the objective function is a statistical functional parameterized by the decision variable and known deterministic constraints may be present. Inspired by stopping criteria in primal-dual and interior-point methods, we develop cancellation theorems that characterize the convergence of … Read more

    A Theoretical Framework for Auxiliary-Loss-Free Load Balancing of Sparse Mixture-of-Experts in Large-Scale AI Models

  • X.Y. Han
  • Yuan Zhong
    • Categories Convex and Nonsmooth Optimization, Optimization in Data Science, Stochastic Programming Tags , , , , , , ,

    In large-scale AI training, Sparse Mixture-of-Experts (s-MoE) layers enable scaling by activating only a small subset of experts per token. An operational challenge in this design is load balancing: routing tokens to minimize the number of idle experts, which is important for the efficient utilization of costly GPUs and for the thorough training of architecture … Read more

    On Solving Chance-Constrained Models with Gaussian Mixture Distribution

  • Shibshankar Dey
  • Sanjay Mehrotra
  • Anirudh Subramanyam
    • Categories Nonlinear Optimization, Stochastic Programming, Yield Management Tags , , , , , ,

    We study linear chance-constrained problems where the coefficients follow a Gaussian mixture distribution. We provide mixed-binary quadratic programs that give inner and outer approximations of the chance constraint based on piecewise linear approximations of the standard normal cumulative density function. We show that $O\left(\sqrt{\ln(1/\tau)/\tau} \right)$ pieces are sufficient to attain $\tau$-accuracy in the chance constraint. … Read more

    Stochastic Mixed-Integer Programming: A Survey

  • Ward Romeijnders
  • Yihang Zhang
  • Suvrajeet Sen
    • Categories Stochastic Programming Tags ,

    The goal of this survey is to provide a road-map for exploring the growing area of stochastic mixed-integer programming (SMIP) models and algorithms. We provide a comprehensive overview of existing decomposition algorithms for two-stage SMIPs, including Dantzig-Wolfe decomposition, dual decomposition, Lagrangian cuts, and decomposition approaches using parametric cutting planes and scaled cuts. Moreover, we explicitly … Read more

    Stronger cuts for Benders’ decomposition for stochastic Unit Commitment Problems based on interval variables

  • Azema
  • Vincent Leclère
  • Wim van Ackooij
    • Categories (Mixed) Integer Linear Programming, Energy, Stochastic Programming Tags , ,

    The Stochastic Unit Commitment (SUC) problem models the scheduling of power generation units under uncertainty, typically using a two-stage stochastic program with integer first-stage and continuous second-stage variables. We propose a new Benders decomposition approach that leverages an extended formulation based on interval variables, enabling decomposition by both unit and time interval under mild technical … Read more

    Distributionally Robust Optimization with Integer Recourse: Convex Reformulations and Critical Recourse Decisions

  • Yiling Zhang
    • Categories Integer Programming, Robust Optimization, Stochastic Programming Tags , , , ,

    The paper studies distributionally robust optimization models with integer recourse. We develop a unified framework that provides finite tight convex relaxations under conic moment-based ambiguity sets and Wasserstein ambiguity sets.  They provide tractable primal representations without relying on sampling or semi-infinite optimization. Beyond tractability, the relaxations offer interpretability that captures the criticality of recourse decisions. … Read more

    Two-Stage Data-Driven Contextual Robust Optimization: An End-to-End Learning Approach for Online Energy Applications

  • Carlos Gamboa
  • Alexandre Street
  • Davi Valladão
  • Bernardo K. Pagnoncelli
    • Categories Energy, Robust Optimization, Stochastic Programming Tags , , ,

    Traditional end-to-end contextual robust optimization models are trained for specific contextual data, requiring complete retraining whenever new contextual information arrives. This limitation hampers their use in online decision-making problems such as energy scheduling, where multiperiod optimization must be solved every few minutes. In this paper, we propose a novel Data-Driven Contextual Uncertainty Set, which gives … Read more