First and second order optimality conditions for optimal control problems of state constrained integral equations

  • Joseph Frédéric Bonnans
  • Xavier Dupuis
  • Constanza De La Vega
    • Categories Infinite Dimensional Optimization

    This paper deals with optimal control problems of integral equations, with initial-final and running state constraints. The order of a running state constraint is defined in the setting of integral dynamics, and we work here with constraints of arbitrary high orders. First and second-order necessary conditions of optimality are obtained, as well as second-order sufficient … Read more

    On the bilinearity rank of a proper cone and Lyapunov-like transformations

  • M. Seetharama Gowda
  • Jiyuan Tao
    • Categories Complementarity and Variational Inequalities, Linear, Cone and Semidefinite Programming Tags , , ,

    A real square matrix Q is a bilinear complementarity relation on a proper cone K in R^n if x in K, s in K^* with x perpendicular to s implies x^{T}Qs=0, where K^* is the dual of K. The bilinearity rank of K is the dimension of the space of all bilinear complementarity relations on … Read more

    A Probabilistic Model for Minmax Regret in Combinatorial Optimization

  • Karthik Natarajan
  • Kim-Chuan Toh
  • Dongjian Shi
    • Categories Combinatorial Optimization

    In this paper, we propose a probabilistic model for minimizing the anticipated regret in combinatorial optimization problems with distributional uncertainty in the objective coefficients. The interval uncertainty representation of data is supplemented with information on the marginal distributions. As a decision criterion, we minimize the worst-case conditional value-at-risk of regret. The proposed model includes the … Read more

    Sparse Approximation via Penalty Decomposition Methods

  • Zhaosong Lu
  • Yong Zhang
    • Categories (Mixed) Integer Nonlinear Programming, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    In this paper we consider sparse approximation problems, that is, general $l_0$ minimization problems with the $l_0$-“norm” of a vector being a part of constraints or objective function. In particular, we first study the first-order optimality conditions for these problems. We then propose penalty decomposition (PD) methods for solving them in which a sequence of … Read more

    Multi-Variate McCormick Relaxations

  • Alexander Mitsos
  • Angelos Tsoukalas
    • Categories Global Optimization Theory Tags , , , , , ,

    G. P. McCormick [Math Prog 1976] provides the framework for convex/concave relaxations of factorable functions, via rules for the product of functions and compositions of the form F(f(z)), where F is a univariate function. Herein, the composition theorem is generalized to allow multivariate outer functions F, and theory for the propagation of subgradients is presented. … Read more

    On the non-homogeneity of completely positive cones

  • M. Seetharama Gowda
  • Roman Sznajder
    • Categories Linear, Cone and Semidefinite Programming Tags , , ,

    Given a closed cone C in the Euclidean n-space, the completely positive cone of C is the convex cone K generated by matrices of the form uu^T as u varies over C. Examples of completely positive cones include the positive semidefinite cone (when C is the entire Euclidean n-space) and the cone of completely positive … Read more

    Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope

  • Marianna Eisenberg-Nagy
  • Monique Laurent
  • Antonios Varvitsiotis
    • Categories Graphs and Matroids, Semi-definite Programming Tags , , , , , ,

    We study a new geometric graph parameter $\egd(G)$, defined as the smallest integer $r\ge 1$ for which any partial symmetric matrix which is completable to a correlation matrix and whose entries are specified at the positions of the edges of $G$, can be completed to a matrix in the convex hull of correlation matrices of … Read more

    Tilt stability, uniform quadratic growth, and strong metric regularity of the subdifferential.

  • Dmitriy Drusvyatskiy
  • Adrian Lewis
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , , , ,

    We prove that uniform second order growth, tilt stability, and strong metric regularity of the subdifferential — three notions that have appeared in entirely different settings — are all essentially equivalent for any lower-semicontinuous, extended-real-valued function. CitationCornell University, School of Operations Research and Information Engineering, 206 Rhodes Hall Cornell University Ithaca, NY 14853. May 2012.ArticleDownload … Read more