Nonlinear Optimization

New sequential optimality conditions for mathematical problems with complementarity constraints and algorithmic consequences

  • Roberto Andreani
  • Gabriel Haeser
  • Paulo J. S. Silva
  • Leonardo D. Secchin
    • Categories Nonlinear Optimization

    In recent years, the theoretical convergence of iterative methods for solving nonlinear constrained optimization problems has been addressed using sequential optimality conditions, which are satisfied by minimizers independently of constraint qualifications (CQs). Even though there is a considerable literature devoted to sequential conditions for standard nonlinear optimization, the same is not true for Mathematical Problems … Read more

    Quasi-Newton approaches to Interior Point Methods for quadratic problems

  • Jacek Gondzio
  • Francisco N. C. Sobral
    • Categories Nonlinear Systems and Least-Squares, Quadratic Programming Tags , , , ,

    Interior Point Methods (IPM) rely on the Newton method for solving systems of nonlinear equations. Solving the linear systems which arise from this approach is the most computationally expensive task of an interior point iteration. If, due to problem’s inner structure, there are special techniques for efficiently solving linear systems, IPMs enjoy fast convergence and … Read more

    On the Relation between MPECs and Optimization Problems in Abs-Normal Form

  • Lisa Hegerhorst-Schultchen
  • Christian Kirches
  • Marc C. Steinbach
    • Categories Constrained Nonlinear Optimization Tags , , , ,

    We show that the problem of unconstrained minimization of a function in abs-normal form is equivalent to identifying a certain stationary point of a counterpart Mathematical Program with Equilibrium Constraints (MPEC). Hence, concepts introduced for the abs-normal forms turn out to be closely related to established concepts in the theory of MPECs. We give a … Read more

    Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms

  • John W. Pearson
  • Margherita Porcelli
  • Martin Stoll
    • Categories Control Applications, Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , , , ,

    PDE-constrained optimization problems with control or state constraints are challenging from an analytical as well as numerical perspective. The combination of these constraints with a sparsity-promoting L1 term within the objective function requires sophisticated optimization methods. We propose the use of an Interior Point scheme applied to a smoothed reformulation of the discretized problem, and … Read more

    On the Complexity of Detecting Convexity over a Box

  • Amir Ali Ahmadi
  • Georgina Hall
    • Categories Convex Optimization, Global Optimization, Nonlinear Optimization Tags , , ,

    It has recently been shown that the problem of testing global convexity of polynomials of degree four is {strongly} NP-hard, answering an open question of N.Z. Shor. This result is minimal in the degree of the polynomial when global convexity is of concern. In a number of applications however, one is interested in testing convexity … Read more

    Selection of variables in parallel space decomposition for the mesh adaptive direct search algorithm

  • Stéphane Alarie
  • Nadir Amaioua
  • Charles Audet
  • Sébastien Le Digabel
  • Louis-Alexandre Leclaire
    • Categories Nonlinear Optimization, Parallel Algorithms

    The parallel space decomposition of the Mesh Adaptive Direct Search algorithm (PSDMADS proposed in 2008) is an asynchronous parallel method for constrained derivative-free optimization with large number of variables. It uses a simple generic strategy to decompose a problem into smaller dimension subproblems. The present work explores new strategies for selecting subset of variables defining … Read more

    Efficient Optimization Algorithms for Robust Principal Component Analysis and Its Variants

  • Necdet Serhat Aybat
  • Shiqian Ma
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , , ,

    Robust PCA has drawn significant attention in the last decade due to its success in numerous application domains, ranging from bio-informatics, statistics, and machine learning to image and video processing in computer vision. Robust PCA and its variants such as sparse PCA and stable PCA can be formulated as optimization problems with exploitable special structures. … Read more

    Sensitivity Analysis for Nonlinear Programming in CasADi

  • J A E Andersson
  • J B Rawlings
    • Categories Modeling Languages and Systems, Quadratic Programming Tags , , , ,

    We present an extension of the CasADi numerical optimization framework that allows arbitrary order NLP sensitivities to be calculated automatically and efficiently. The approach, which can be used together with any NLP solver available in CasADi, is based on a sparse QR factorization and an implementation of a primal-dual active set method. The whole toolchain … Read more

    A limited-memory optimization method using the infinitely many times repeated BNS update and conjugate directions

  • Ladislav Lukšan
  • Jan Vlček
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , , , , ,

    To improve the performance of the limited-memory variable metric L-BFGS method for large scale unconstrained optimization, repeating of some BFGS updates was proposed in [1, 2]. But the suitable extra updates need to be selected carefully, since the repeating process can be time consuming. We show that for the limited-memory variable metric BNS method, matrix … Read more

    A family of spectral gradient methods for optimization

  • Yu-Hong Dai
  • Yakui Huang
  • Xin-Wei Liu
    • Categories Convex Optimization, Unconstrained Optimization

    We propose a family of spectral gradient methods, whose stepsize is determined by a convex combination of the short Barzilai-Borwein (BB) stepsize and the long BB stepsize. It is shown that each member of the family shares certain quasi-Newton property in the sense of least squares. The family also includes some other gradient methods as … Read more