Ernesto G. Birgin – Optimization Online

Safeguarded augmented Lagrangian algorithms with scaled stopping criterion for the subproblems

Published: 2023/10/27

Nonlinear Optimization augmented lagrangian method, Convergence, nonlinear optimization, scaled stopping criteria

At each iteration of the Safeguarded Augmented Lagrangian algorithm Algencan, a bound-constrained subproblem consisting of the minimization of the Powell-Hestenes-Rockafellar augmented Lagrangian function is considered, for which a minimizer with tolerance tending to zero is sought. More precisely, a point that satisfies a subproblem first-order necessary optimality condition with tolerance tending to zero is required. … Read more

Optimization of the first Dirichlet Laplacian eigenvalue with respect to a union of balls

Published: 2022/12/05

Nonlinear Optimization, Optimization of Systems modeled by PDEs Dirichlet Laplacian eigenvalues, numerical optimization, shape optimization

The problem of minimizing the first eigenvalue of the Dirichlet Laplacian with respect to a union of m balls with fixed identical radii and variable centers in the plane is investigated in the present work. The existence of a minimizer is shown and the shape sensitivity analysis of the eigenvalue with respect to the centers’ … Read more

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

Published: 2021/04/27

Nonlinear Systems and Least-Squares acceleration, derivative-free, nonlinear systems, numerical experiments, sequential residual methods, sequential secant approach

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

Accelerated derivative-free nonlinear least-squares applied to the estimation of Manning coefficients

Published: 2021/04/06

Ernesto G. Birgin

José Mario Martínez

Nonlinear Systems and Least-Squares acceleration, derivative-free methods, manning coefficients, nonlinear least squares

A general framework for solving nonlinear least squares problems without the employment of derivatives is proposed in the present paper together with a new general global convergence theory. With the aim to cope with the case in which the number of variables is big (for the standards of derivative-free optimization), two dimension-reduction procedures are introduced. … Read more

Secant acceleration of sequential residual methods for solving large-scale nonlinear systems of equations

Published: 2020/12/24

Ernesto G. Birgin

José Mario Martínez

Nonlinear Systems and Least-Squares acceleration, large-scale optimization, nonlinear systems of equations, sequential residual methods

Sequential Residual Methods try to solve nonlinear systems of equations $F(x)=0$ by iteratively updating the current approximate solution along a residual-related direction. Therefore, memory requirements are minimal and, consequently, these methods are attractive for solving large-scale nonlinear systems. However, the convergence of these algorithms may be slow in critical cases; therefore, acceleration procedures are welcome. … Read more

On complexity and convergence of high-order coordinate descent algorithms

Published: 2020/12/16

Nonlinear Optimization bound-constrained minimization, coordinate descent methods, worst-case evaluation complexity

Coordinate descent methods with high-order regularized models for box-constrained minimization are introduced. High-order stationarity asymptotic convergence and first-order stationarity worst-case evaluation complexity bounds are established. The computer work that is necessary for obtaining first-order $\varepsilon$-stationarity with respect to the variables of each coordinate-descent block is $O(\varepsilon^{-(p+1)/p})$ whereas the computer work for getting first-order $\varepsilon$-stationarity with … Read more

Economic inexact restoration for derivative-free expensive function minimization and applications

Published: 2020/09/18, Updated: 2021/06/03

Ernesto G. Birgin

Nataša Krejić

José Mario Martínez

Nonlinear Optimization derivative-free, expensive function, global convergence, inexact evaluation, inexact restoration

The Inexact Restoration approach has proved to be an adequate tool for handling the problem of minimizing an expensive function within an arbitrary feasible set by using different degrees of precision in the objective function. The Inexact Restoration framework allows one to obtain suitable convergence and complexity results for an approach that rationally combines low- … Read more

On complexity and convergence of high-order coordinate descent algorithms

Published: 2020/09/03

Bound-constrained Optimization, Nonlinear Optimization bound-constrained minimization, coordinate descent methods, worst-case evaluation complexity

Complexity and performance of an Augmented Lagrangian algorithm

Published: 2019/07/01, Updated: 2019/07/04

Ernesto G. Birgin

José Mario Martínez

Constrained Nonlinear Optimization augmented lagrangian method, complexity, nonlinear programming, numerical experiments

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

An Augmented Lagrangian algorithm for nonlinear semidefinite programming applied to the covering problem

Published: 2019/03/26, Updated: 2019/10/16

Constrained Nonlinear Optimization, Convex Optimization, Semi-definite Programming augmented lagrangian method, convex algebraic geometry, covering problem, nonlinear semidefinite programming

In this work we present an Augmented Lagrangian algorithm for nonlinear semidefinite problems (NLSDPs), which is a natural extension of its consolidated counterpart in nonlinear programming. This method works with two levels of constraints; one that is penalized and other that is kept within the subproblems. This is done in order to allow exploiting the … Read more