numerical experiments

On Stationary Conditions and the Convergence of Augmented Lagrangian methods for Generalized Nash Equilibrium Problems

  • Lennin Mallma Ramirez
  • Frank Navarro Rojas
  • Alberto Ramos
    • Categories Constrained Nonlinear Optimization, Game Theory, Nonlinear Optimization Tags , , , ,

    In this work, we study stationarity conditions and constraint qualifications (CQs) tailored to Generalized Nash Equilibrium Problems (GNEPs) and analyze their relationships and implications for the global convergence of algorithms. We recall that GNEPs generalize Nash Equilibrium Problems (NEPs) in that the feasible strategy set of each player depends on the strategies chosen by the … Read more

    Active-set Newton-MR methods for nonconvex optimization problems with bound constraints

  • Ernesto G. Birgin
  • Geovani Nunes Grapiglia
  • Diaulas S. Marcondes
    • Categories Bound-constrained Optimization, Nonlinear Optimization, Unconstrained Optimization Tags , , , ,

    This paper presents active-set methods for minimizing nonconvex twice-continuously differentiable functions subject to bound constraints. Within the faces of the feasible set, we employ descent methods with Armijo line search, utilizing approximated Newton directions obtained through the Minimum Residual (MINRES) method. To escape the faces, we investigate the use of the Spectral Projected Gradient (SPG) … Read more

    A general merit function-based global convergent framework for nonlinear optimization

  • Luís Felipe Bueno
  • Ernesto G. Birgin
  • Tiara Martini
  • Dimary Moreno López
  • Thadeu Senne
  • Thiago Siqueira Santos
    • Categories Nonlinear Optimization Tags , , , , ,

    In this paper, we revisit the convergence theory of the inexact restoration paradigm for non-linear optimization. The paper first identifies the basic elements of a globally convergent method based on merit functions. Then, the inexact restoration method that employs a two-phase iteration is introduced as a special case. A specific implementation is presented that is … Read more

    The Role of Level-Set Geometry on the Performance of PDHG for Conic Linear Optimization

  • Zikai Xiong
  • Robert M. Freund
    • Categories Convex Optimization, Linear Programming, Linear, Cone and Semidefinite Programming Tags , , , , ,

    We consider solving huge-scale instances of (convex) conic linear optimization problems, at the scale where matrix-factorization-free methods are attractive or necessary. The restarted primal-dual hybrid gradient method (rPDHG) — with heuristic enhancements and GPU implementation — has been very successful in solving huge-scale linear programming (LP) problems; however its application to more general conic convex … Read more

    On the global convergence of a general class of augmented Lagrangian methods

  • Ernesto G. Birgin
  • Gabriel Haeser
  • Nelson Maculan
  • Lennin Mallma Ramirez
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

    In [E. G. Birgin, R. Castillo and J. M. Martínez, Computational Optimization and Applications 31, pp. 31-55, 2005], a general class of safeguarded augmented Lagrangian methods is introduced which includes a large number of different methods from the literature. Besides a numerical comparison including 65 different methods, primal-dual global convergence to a KKT point is … Read more

    Accelerated derivative-free spectral residual method for nonlinear systems of equations

  • Ernesto G. Birgin
  • José Mario Martínez
  • John L. Gardenghi
  • Diaulas S. Marcondes
    • Categories Nonlinear Systems and Least-Squares Tags , , , , ,

    Spectral residual methods are powerful tools for solving nonlinear systems of equations without derivatives. In a recent paper, it was shown that an acceleration technique based on the Sequential Secant Method can greatly improve its efficiency and robustness. In the present work, an R implementation of the method is presented. Numerical experiments with a widely … Read more

    Globally convergent Newton-type methods for multiobjective optimization

  • Max L. N. Gonçalves
  • Leandro Fonseca Prudente
  • F. S. Lima
    • Categories Multi-Criteria Optimization, Unconstrained Optimization Tags , , ,

    We propose two Newton-type methods for solving (possibly) nonconvex unconstrained multiobjective optimization problems. The first is directly inspired by the Newton method designed to solve convex problems, whereas  the second uses  second-order information of the objective functions with ingredients of the steepest descent method.  One of the key points of our approaches  is to impose … Read more

    Penalized stochastic gradient methods for stochastic convex optimization with expectation constraints

  • Xiantao Xiao
    • Categories Stochastic Programming Tags , , , ,

    Stochastic gradient method and its variants are simple yet effective for minimizing an expectation function over a closed convex set. However, none of these methods are applicable to solve stochastic programs with expectation constraints, since the projection onto the feasible set is prohibitive. To deal with the expectation constrained stochastic convex optimization problems, we propose … Read more

    Complexity and performance of an Augmented Lagrangian algorithm

  • Ernesto G. Birgin
  • José Mario Martínez
    • Categories Constrained Nonlinear Optimization Tags , , ,

    Algencan is a well established safeguarded Augmented Lagrangian algorithm introduced in [R. Andreani, E. G. Birgin, J. M. Martínez and M. L. Schuverdt, On Augmented Lagrangian methods with general lower-level constraints, SIAM Journal on Optimization 18, pp. 1286-1309, 2008]. Complexity results that report its worst-case behavior in terms of iterations and evaluations of functions and … Read more

    Hybrid methods for nonlinear least squares problems

  • Ladislav Lukšan
  • Ctirad Matonoha
  • Jan Vlček
    • Categories Nonlinear Systems and Least-Squares, Unconstrained Optimization Tags , , , , , ,

    This contribution contains a description and analysis of effective methods for minimization of the nonlinear least squares function $F(x) = (1/2) f^T(x) f(x)$, where $x \in R^n$ and $f \in R^m$, together with extensive computational tests and comparisons of the introduced methods. All hybrid methods are described in detail and their global convergence is proved … Read more