Nonlinear Optimization

New optimality conditions for the semivectorial bilevel optimization problem

  • Stephan Dempe
  • Alain Zemkoho
  • Nazih Gadhi
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    The paper is concerned with the optimistic formulation of a bilevel optimization problem with multiobjective lower-level problem. Considering the scalarization approach for the multiobjective program, we transform our problem into a scalar-objective optimization problem with inequality constraints by means of the well-known optimal value reformulation. Completely detailed rst-order necessary optimality conditions are then derived in … Read more

    Sample Size Selection in Optimization Methods for Machine Learning

  • Richard H. Byrd
  • Gillian M Chin
  • Jorge Nocedal
  • Yuchen Wu
    • Categories Nonlinear Optimization, Stochastic Programming Tags , ,

    This paper presents a methodology for using varying sample sizes in batch-type optimization methods for large scale machine learning problems. The first part of the paper deals with the delicate issue of dynamic sample selection in the evaluation of the function and gradient. We propose a criterion for increasing the sample size based on variance … Read more

    Optimization problems with value function objectives

  • Alain Zemkoho
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    The family of optimization problems with value function objectives includes the minmax programming problem and the bilevel optimization problem. In this paper, we derive necessary optimality conditions for this class of problems. The main focus is on the case where the functions involved are nonsmooth and the constraints are the very general operator constraints. Citation … Read more

    Regularized Sequential Quadratic Programming

  • Philip E. Gill
  • Daniel P. Robinson
    • Categories Constrained Nonlinear Optimization Tags , , , , ,

    We present the formulation and analysis of a new sequential quadratic programming (\SQP) method for general nonlinearly constrained optimization. The method pairs a primal-dual generalized augmented Lagrangian merit function with a \emph{flexible} line search to obtain a sequence of improving estimates of the solution. This function is a primal-dual variant of the augmented Lagrangian proposed … Read more

    A SIMPLE APPROACH TO OPTIMALITY CONDITIONS IN MINMAX PROGRAMMING

  • Alain Zemkoho
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization

    Considering the minmax programming problem, lower and upper subdi erential optimality conditions, in the sense of Mordukhovich, are derived. The approach here, mainly based on the nonsmooth dual objects of Mordukhovich, is completely di erent from that of most of the previous works where generalizations of the alternative theorem of Farkas have been applied. The results obtained … Read more

    Higher-Order Confidence Intervals for Stochastic Programming using Bootstrapping

  • Mihai Anitescu
  • Cosmin G. Petra
    • Categories Constrained Nonlinear Optimization, Stochastic Programming Tags , ,

    We study the problem of constructing confidence intervals for the optimal value of a stochastic programming problem by using bootstrapping. Bootstrapping is a resampling method used in the statistical inference of unknown parameters for which only a small number of samples can be obtained. One such parameter is the optimal value of a stochastic optimization … Read more

    Differentiable exact penalty functions for nonlinear second-order cone programs

  • Ellen H. Fukuda
  • Paulo J. S. Silva
  • Masao Fukushima
    • Categories Nonlinear Optimization, Second-Order Cone Programming Tags , , ,

    We propose a method to solve nonlinear second-order cone programs (SOCPs), based on a continuously differentiable exact penalty function. The construction of the penalty function is given by incorporating a multipliers estimate in the augmented Lagrangian for SOCPs. Under the nondegeneracy assumption and the strong second-order sufficient condition, we show that a generalized Newton method … Read more

    A robust Kantorovich’s theorem on inexact Newton method with relative residual error tolerance

  • Orizon Pereira Ferreira
  • Benar Fux Svaiter
    • Categories Infinite Dimensional Optimization, Nonlinear Optimization

    We prove that under semi-local assumptions, the inexact Newton method with a fixed relative residual error tolerance converges Q-linearly to a zero of the non-linear operator under consideration. Using this result we show that Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a … Read more

    Approximate spectral factorization for design of efficient sub-filter sequences

  • Mats Andersson
  • Oleg Burdakov
  • Hans Knutsson
  • Björn Norell
    • Categories Constrained Nonlinear Optimization, Global Optimization Applications, Nonlinear Systems and Least-Squares

    A well-known approach to the design of computationally efficient filters is to use spectral factorization, i.e. a decomposition of a filter into a sequence of sub-filters. Due to the sparsity of the sub-filters, the typical processing speedup factor is within the range 1-10 in 2D, and for 3D it achieves 10-100. The design of such … Read more

    Optimization over the Efficient Set of a Bicriteria Convex Programming Problem

  • Thi Bach Kim Nguyen
  • Ngoc Thang Tran
    • Categories Global Optimization, Nonlinear Optimization Tags , , , , ,

    The problem of optimizing a real function over the efficient set of a multiple objective programming problem arises in a variety of applications. In this article, we propose an outer approximation algorithm for maximizing a function $h(x) = \varphi(f(x))$ over the efficient set $X_E$ of the bi-criteria convex programming problem $ {\rm Vmin} \{f(x)=(f_1(x), f_2(x))^T … Read more