Nonlinear Optimization

Intrinsic Representation of Tangent Vectors and Vector transport on Matrix Manifolds

  • Pierre-Antoine Absil
  • Kyle A. Gallivan
  • Wen Huang
    • Categories Applications - Science and Engineering, Linear, Cone and Semidefinite Programming, Nonlinear Optimization

    In Riemannian optimization problems, commonly encountered manifolds are $d$-dimensional matrix manifolds whose tangent spaces can be represented by $d$-dimensional linear subspaces of a $w$-dimensional Euclidean space, where $w > d$. Therefore, representing tangent vectors by $w$-dimensional vectors has been commonly used in practice. However, using $w$-dimensional vectors may be the most natural but may not … 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

    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

    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

    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