Nonlinear Optimization

The complexity of simple models – a study of worst and typical hard cases for the Standard Quadratic Optimization Problem

  • Immanuel M. Bomze
  • Werner Schachinger
  • Reinhard Ullrich
    • Categories Game Theory, Global Optimization Theory, Quadratic Programming

    In a Standard Quadratic Optimization Problem (StQP), a possibly indefinite quadratic form (the simplest nonlinear function) is extremized over the standard simplex, the simplest polytope. Despite this simplicity, the nonconvex instances of this problem class allow for remarkably rich patterns of coexisting local solutions, which are closely related to practical difficulties in solving StQPs globally. … Read more

    Virtuous smoothing for global optimization

  • Jon Lee
  • Daphne Skipper
    • Categories Convex and Nonsmooth Optimization, Global Optimization, Nonlinear Optimization Tags , , , ,

    In the context of global optimization and mixed-integer non-linear programming, generalizing a technique of D’Ambrosio, Fampa, Lee and Vigerske for handling the square-root function, we develop a virtuous smoothing method, using cubics, aimed at functions having some limited non-smoothness. Our results pertain to root functions ($w^p$ with $0

    Global optimal control with the direct multiple shooting method

  • Holger Diedam
  • Sebastian Sager
    • Categories Global Optimization Applications, Global Optimization Theory, Systems governed by Differential Equations Optimization Tags , ,

    We propose to solve global optimal control problems with a new algorithm that is based on Bock’s direct multiple shooting method. We provide conditions and numerical evidence for a significant overall runtime reduction compared to the standard single shooting approach. CitationOptimal Control Applications and Methods, Vol. 39 (2), pp. 449–470, 2017 DOI 10.1002/oca.2324 Article online … Read more

    A numerical evaluation of the bounded degree sum-of-squares hierarchy of Lasserre, Toh, and Yang on the pooling problem

  • Joachim Dahl
  • Etienne de Klerk
  • Ahmadreza Marandi
    • Categories Nonlinear Optimization, Semi-definite Programming Tags , , ,

    The bounded degree sum-of-squares (BSOS) hierarchy of Lasserre, Toh, and Yang [EURO J. Comput. Optim., 2015] constructs lower bounds for a general polynomial optimization problem with compact feasible set, by solving a sequence of semi-definite programming (SDP) problems. Lasserre, Toh, and Yang prove that these lower bounds converge to the optimal value of the original … Read more

    Structured Nonconvex and Nonsmooth Optimization: Algorithms and Iteration Complexity Analysis

  • Bo Jiang
  • Tianyi Lin
  • Shiqian Ma
  • Shuzhong Zhang
    • Categories Nonlinear Optimization Tags , , , , ,

    Nonconvex optimization problems are frequently encountered in much of statistics, business, science and engineering, but they are not yet widely recognized as a technology. A reason for this relatively low degree of popularity is the lack of a well developed system of theory and algorithms to support the applications, as is the case for its … Read more

    Mathematical Programs with Equilibrium Constraints: A sequential optimality condition, new constraint qualifications and algorithmic consequences.

  • Alberto Ramos
    • Categories Constrained Nonlinear Optimization

    Mathematical programs with equilibrium (or complementarity) constraints, MPECs for short, are a difficult class of constrained optimization problems. The feasible set has a very special structure and violates most of the standard constraint qualifications (CQs). Thus, the Karush-Kuhn-Tucker (KKT) conditions are not necessarily satisfied by minimizers and the convergence assumptions of many methods for solving … Read more

    Algorithms for stochastic optimization with expectation constraints

  • Guanghui Lan
  • Zhiqiang Zhou
    • Categories Bound-constrained Optimization, Convex and Nonsmooth Optimization, Stochastic Programming Tags , , , ,

    This paper considers the problem of minimizing an expectation function over a closed convex set, coupled with an expectation constraint on either decision variables or problem parameters. We first present a new stochastic approximation (SA) type algorithm, namely the cooperative SA (CSA), to handle problems with the expectation constraint on devision variables. We show that … Read more

    Approximation Properties and Tight Bounds for Constrained Mixed-Integer Optimal Control

  • Christian Kirches
  • Felix Lenders
  • Paul Manns
    • Categories Approximation Algorithms, Complementarity and Variational Inequalities, Systems governed by Differential Equations Optimization

    We extend recent work on mixed-integer nonlinear optimal control prob- lems (MIOCPs) to the case of integer control functions subject to constraints. Promi- nent examples of such systems include problems with restrictions on the number of switches permitted, or problems that minimize switch cost. We extend a theorem due to [Sager et al., Math. Prog. … Read more

    A predictor-corrector path-following algorithm for dual-degenerate parametric optimization problems

  • Johannes Jäschke
  • Vyacheslav Kungurtsev
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

    Most path-following algorithms for tracing a solution path of a parametric nonlinear optimization problem are only certifiably convergent under strong regularity assumptions about the problem functions, in particular, the linear independence of the constraint gradients at the solutions, which implies a unique multiplier solution for every nonlinear program. In this paper we propose and prove … Read more

    A new algebraic analysis to linear mixed models

  • Yongge Tian
    • Categories Constrained Nonlinear Optimization, Statistics Tags , , , , , ,

    This article presents a new investigation to the linear mixed model $\by = \bX \bbe + \bZ\bga + \bve$ with fixed effect $\bX\bbe$ and random effect $\bZ\bga$ under a general assumption via some novel algebraic tools in matrix theory, and reveals a variety of deep and profound properties hidden behind the linear mixed model. We … Read more