Stochastic Programming

Convergence analysis of sampling-based decomposition methods for risk-averse multistage stochastic convex programs

  • Vincent Guigues
    • Categories Convex Optimization, Stochastic Programming

    We consider a class of sampling-based decomposition methods to solve risk-averse multistage stochastic convex programs. We prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse functions. This formula can be used to obtain an efficient implementation of Stochastic Dual Dynamic Programming applied to convex nonlinear problems. … Read more

    A Generalization of Benders’ Algorithm for Two-Stage Stochastic Optimization Problems With Mixed Integer Recourse

  • Anahita Hassanzadeh
  • Ted K. Ralphs
    • Categories (Mixed) Integer Linear Programming, Stochastic Programming Tags ,

    We describe a generalization of Benders’ method for solving two-stage stochastic linear optimization problems in which there are both continuous and integer variables in the first and second stages. Benders’ method relies on finding effective lower approximations for the value function of the second-stage problem. In this setting, the value function is a discontinuous, non-convex, … Read more

    On the Value Function of a Mixed Integer Linear Optimization Problem and an Algorithm for its Construction

  • Anahita Hassanzadeh
  • Ted K. Ralphs
    • Categories (Mixed) Integer Linear Programming, Stochastic Programming Tags , , ,

    This paper addresses the value function of a general mixed integer linear optimization problem (MILP). The value function describes the change in optimal objective value as the right-hand side is varied and understanding its structure is central to solving a variety of important classes of optimization problems. We propose a discrete representation of the MILP … Read more

    Cut Generation for Optimization Problems with Multivariate Risk Constraints

  • Simge Küçükyavuz
  • Nilay Noyan
    • Categories Integer Programming, Multi-Criteria Optimization, Stochastic Programming

    We consider a class of multicriteria stochastic optimization problems that features benchmarking constraints based on conditional value-at-risk and second-order stochastic dominance. We develop alternative mixed-integer programming formulations and solution methods for cut generation problems arising in optimization under such multivariate risk constraints. We give the complete linear description of two non-convex substructures appearing in these … Read more

    Hypotheses testing on the optimal values of several risk-neutral or risk-averse convex stochastic programs and application to hypotheses testing on several risk measure values

  • Vincent Guigues
    • Categories Convex Optimization, Stochastic Programming

    Given an arbitrary number of risk-averse or risk-neutral convex stochastic programs, we study hypotheses testing problems aiming at comparing the optimal values of these stochastic programs on the basis of samples of the underlying random vectors. We propose non-asymptotic tests based on confidence intervals on the optimal values of the stochastic programs obtained using the … Read more

    Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs

  • Shabbir Ahmed
  • James R. Luedtke
  • Yongjia Song
  • Weijun Xie
    • Categories Stochastic Programming Tags , ,

    We propose two new Lagrangian dual problems for chance-constrained stochastic programs based on relaxing nonanticipativity constraints. We compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed-integer programming (MIP) formulation. For a given dual solution, the associated Lagrangian relaxation … Read more

    A scalable bounding method for multi-stage stochastic integer programs

  • Osman Y. Ozaltin
  • Burhaneddin Sandikçi
    • Categories Integer Programming, Parallel Algorithms, Stochastic Programming Tags , , , ,

    Many dynamic decision problems involving uncertainty can be appropriately modeled as multi-stage stochastic programs. However, most practical instances are so large and/or complex that it is impossible to solve them on a single computer, especially due to memory limitations. Extending the work of Sandikci et al. (2013) on two-stage stochastic mixed-integer-programs (SMIPs), this paper develops … Read more

    Stochastic Topology Design Optimization for Continuous Elastic Materials

  • Miguel Carrasco
  • Benjamin Ivorra
  • Ángel M. Ramos
    • Categories Mechanical Engineering, Multidisciplinary Design Optimization, Stochastic Programming

    In this paper, we develop a stochastic model for topology optimization. We find robust structures that minimize the compliance for a given main load having a stochastic behavior. We propose a model that takes into account the expected value of the compliance and its variance. We show that, similarly to the case of truss structures, … Read more

    Power-Capacity and Ramp-Capability Reserves for Wind Integration in Power-Based UC

  • Ross Baldick
  • Germán Morales-España
  • Javier García-González
  • Andres Ramos
    • Categories Applications - OR and Management Sciences, Robust Optimization, Stochastic Programming Tags , , , ,

    This paper proposes a power-based network-constrained unit commitment (UC) model as an alternative to the traditional deterministic UCs to deal with wind generation uncertainty. The formulation draws a clear distinction between power-capacity and ramp-capability reserves to deal with wind production uncertainty. These power and ramp requirements can be obtained from wind forecast information. The model … Read more

    An Accelerated Proximal Coordinate Gradient Method and its Application to Regularized Empirical Risk Minimization

  • Qihang Lin
  • Zhaosong Lu
  • Lin Xiao
    • Categories Applications - Science and Engineering, Convex and Nonsmooth Optimization, Stochastic Programming Tags , ,

    We consider the problem of minimizing the sum of two convex functions: one is smooth and given by a gradient oracle, and the other is separable over blocks of coordinates and has a simple known structure over each block. We develop an accelerated randomized proximal coordinate gradient (APCG) method for minimizing such convex composite functions. … Read more