Stochastic Programming

A preconditioning technique for Schur complement systems arising in stochastic optimization

  • Mihai Anitescu
  • Cosmin G. Petra
    • Categories Parallel Algorithms, Stochastic Programming

    Deterministic sample average approximations of stochastic programming problems with recourse are suitable for a scenario-based, treelike parallelization with interior-point methods and a Schur complement mechanism. However, the direct linear solves involving the Schur complement matrix are expensive, and adversely a ect the scalability of this approach. In this paper we propose a stochastic preconditioner to address … Read more

    Risk Adjusted Budget Allocation Models with Application in Homeland Security

  • Tito Homem-de-Mello
  • Jian Hu
  • Sanjay Mehrotra
    • Categories Multi-Criteria Optimization, Robust Optimization, Stochastic Programming Tags , , , , ,

    This paper presents and studies several models for multi-criterion budget allocation problems under uncertainty. We start by introducing a robust weighted objective model, which is developed further using the concept of stochastic dominance to incorporate risk averseness of the decision maker. A budget minimization variant of this model is also presented. We use a Sample … Read more

    Mathematical Programming Approaches for Generating p-Efficient Points

  • Miguel A. Lejeune
  • Nilay Noyan
    • Categories Stochastic Programming

    Probabilistically constrained problems, in which the random variables are finitely distributed, are non-convex in general and hard to solve. The p-efficiency concept has been widely used to develop efficient methods to solve such problems. Those methods require the generation of p-efficient points (pLEPs) and use an enumeration scheme to identify pLEPs. In this paper, we … Read more

    Convex approximations in stochastic programming by semidefinite programming

  • István Deák
  • Imre Pólik
  • Tamás Terlaky
  • András Prékopa
    • Categories Semi-definite Programming, Stochastic Programming Tags , , ,

    The following question arises in stochastic programming: how can one approximate a noisy convex function with a convex quadratic function that is optimal in some sense. Using several approaches for constructing convex approximations we present some optimization models yielding convex quadratic regressions that are optimal approximations in $L_1$, $L_\infty$ and $L_2$ norm. Extensive numerical experiments … Read more

    Dual Averaging Methods for Regularized Stochastic Learning and Online Optimization

  • Lin Xiao
    • Categories Convex and Nonsmooth Optimization, Stochastic Programming Tags , , , ,

    We consider regularized stochastic learning and online optimization problems, where the objective function is the sum of two convex terms: one is the loss function of the learning task, and the other is a simple regularization term such as $\ell_1$-norm for promoting sparsity. We develop extensions of Nesterov’s dual averaging method, that can exploit the … Read more

    A decomposition-based warm-start method for stochastic programming

  • Marco Colombo
  • Andreas Grothey
    • Categories Stochastic Programming Tags , ,

    In this paper we propose a warm-start technique for interior point methods applicable to multi-stage stochastic programming problems. The main idea is to generate an initial point for the interior point solver by decomposing the barrier problem associated with the deterministic equivalent at the sec- ond stage and using a concatenation of the solutions of … Read more

    Kusuoka Representation of Higher Order Dual Risk Measures

  • Darinka Dentcheva
  • Andrzej Ruszczyński
  • Spiridon Penev
    • Categories Stochastic Programming Tags , , , , , ,

    We derive representations of higher order dual measures of risk in $\mathcal{L}^p$ spaces as suprema of integrals of Average Values at Risk with respect to probability measures on $(0,1]$ (Kusuoka representations). The suprema are taken over convex sets of probability measures. The sets are described by constraints on the dual norms of certain transformations of … Read more

    Stability Analysis of Two Stage Stochastic Mathematical Programs with Complementarity Constraints via NLP-Regularization

  • Gui-Hua Lin
  • Yongchao Liu
  • Huifu Xu
    • Categories Complementarity and Variational Inequalities, Stochastic Programming Tags , , , ,

    This paper presents numerical approximation schemes for a two stage stochastic programming problem where the second stage problem has a general nonlinear complementarity constraint: first, the complementarity constraint is approximated by a parameterized system of inequalities with a well-known regularization approach (SIOPT, Vol.11, 918-936) in deterministic mathematical programs with equilibrium constraints; the distribution of the … Read more

    Two-Stage Stochastic Programming Involving CVaR with an Application to Disaster Management

  • Nilay Noyan
    • Categories Applications - Science and Engineering, Stochastic Programming

    Traditional two-stage stochastic programming is risk-neutral; that is, it considers the expectation as the preference criterion while comparing the random variables (e.g., total cost) to identify the best decisions. However, in the presence of variability risk measures should be incorporated into decision making problems in order to model its effects. In this study, we consider … Read more

    Economic Impacts of Advanced Weather Forecasting on Energy System Operations

  • Mihai Anitescu
  • Emil M. Constantinescu
  • Victor M. Zavala
    • Categories Basic Sciences Applications, Statistics, Stochastic Programming

    We analyze the impacts of adopting advanced weather forecasting systems at different levels of the decision-making hierarchy of the power grid. Using case studies, we show that state-of-the-art numerical weather prediction (NWP) models can provide high-precision forecasts and uncertainty information that can significantly enhance the performance of planning, scheduling, energy management, and feedback control systems. … Read more