Constrained Nonlinear Optimization

A second-order optimality condition with first- and second-order complementarity associated with global convergence of algorithms

  • Gabriel Haeser
    • Categories Constrained Nonlinear Optimization Tags , , ,

    We develop a new notion of second-order complementarity with respect to the tangent subspace related to second-order necessary optimality conditions by the introduction of so-called tangent multipliers. We prove that around a local minimizer, a second-order stationarity residual can be driven to zero while controlling the growth of Lagrange multipliers and tangent multipliers, which gives … Read more

    Can linear superiorization be useful for linear optimization problems?

  • Yair Censor
    • Categories Constrained Nonlinear Optimization, Linear Programming, Problem Solving Environments Tags , , , , , , , , ,

    Linear superiorization considers linear programming problems but instead of attempting to solve them with linear optimization methods it employs perturbation resilient feasibility-seeking algorithms and steers them toward reduced (not necessarily minimal) target function values. The two questions that we set out to explore experimentally are (i) Does linear superiorization provide a feasible point whose linear … Read more

    Linear superiorization for infeasible linear programming

  • Yair Censor
  • Yehuda Zur
    • Categories Constrained Nonlinear Optimization, Linear Programming

    Linear superiorization (abbreviated: LinSup) considers linear programming (LP) problems wherein the constraints as well as the objective function are linear. It allows to steer the iterates of a feasibility-seeking iterative process toward feasible points that have lower (not necessarily minimal) values of the objective function than points that would have been reached by the same … Read more

    A New First-order Algorithmic Framework for Optimization Problems with Orthogonality Constraints

  • Xiaojun Chen
  • Bin Gao
  • Xin Liu
  • Ya-xiang Yuan
    • Categories Basic Sciences Applications, Constrained Nonlinear Optimization Tags , , , , ,

    In this paper, we consider a class of optimization problems with orthogonality constraints, the feasible region of which is called the Stiefel manifold. Our new framework combines a function value reduction step with a correction step. Different from the existing approaches, the function value reduction step of our algorithmic framework searches along the standard Euclidean … Read more

    Low-complexity method for hybrid MPC with local guarantees

  • Alexander Domahidi
  • Angelos Georghiou
  • Damian Frick
  • Juan L. Jerez
  • Manfred Morari
    • Categories Constrained Nonlinear Optimization, Control Applications Tags , , ,

    Model predictive control problems for constrained hybrid systems are usually cast as mixed-integer optimization problems (MIP). However, commercial MIP solvers are designed to run on desktop computing platforms and are not suited for embedded applications which are typically restricted by limited computational power and memory. To alleviate these restrictions, we develop a novel low-complexity, iterative … Read more

    A Study of the Difference-of-Convex Approach for Solving Linear Programs with Complementarity Constraints

  • Francisco Jara-Moroni
  • Jong-Shi Pang
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , ,

    This paper studies the difference-of-convex (DC) penalty formulations and the associated difference-of-convex algorithm (DCA) for computing stationary solutions of linear programs with complementarity constraints (LPCCs). We focus on three such formulations and establish connections between their stationary solutions and those of the LPCC. Improvements of the DCA are proposed to remedy some drawbacks in a … Read more

    A Sequential Algorithm for Solving Nonlinear Optimization Problems with Chance Constraints

  • Frank E. Curtis
  • Victor M. Zavala
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , , ,

    An algorithm is presented for solving nonlinear optimization problems with chance constraints, i.e., those in which a constraint involving an uncertain parameter must be satisfied with at least a minimum probability. In particular, the algorithm is designed to solve cardinality-constrained nonlinear optimization problems that arise in sample average approximations of chance-constrained problems, as well as … Read more

    Second-order optimality and beyond: characterization and evaluation complexity in convexly-constrained nonlinear optimization

  • Coralia Cartis
  • Nicholas I. M. Gould
  • Philippe L. Toint
    • Categories Bound-constrained Optimization, Constrained Nonlinear Optimization, Unconstrained Optimization Tags , , ,

    High-order optimality conditions for convexly-constrained nonlinear optimization problems are analyzed. A corresponding (expensive) measure of criticality for arbitrary order is proposed and extended to define high-order $\epsilon$-approximate critical points. This new measure is then used within a conceptual trust-region algorithm to show that, if derivatives of the objective function up to order $q \geq 1$ … Read more

    Online First-Order Framework for Robust Convex Optimization

  • Nam Ho-Nguyen
  • Fatma Kılınç-Karzan
    • Categories Constrained Nonlinear Optimization, Robust Optimization Tags , , ,

    Robust optimization (RO) has emerged as one of the leading paradigms to efficiently model parameter uncertainty. The recent connections between RO and problems in statistics and machine learning domains demand for solving RO problems in ever more larger scale. However, the traditional approaches for solving RO formulations based on building and solving robust counterparts or … Read more

    A derivative-free trust-region augmented Lagrangian algorithm

  • Charles Audet
  • Sébastien Le Digabel
  • Mathilde Peyrega
    • Categories Constrained Nonlinear Optimization

    We present a new derivative-free trust-region (DFTR) algorithm to solve general nonlinear constrained problems with the use of an augmented Lagrangian method. No derivatives are used, neither for the objective function nor for the constraints. An augmented Lagrangian method, known as an effective tool to solve equality and inequality constrained optimization problems with derivatives, is … Read more