We propose a new variational problem which we call the Split Variational Inequality Problem (SVIP). It entails finding a solution of one Variational Inequality Problem (VIP), the image of which under a given bounded linear transformation is a solution of another VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert … Read more
This paper proposes a framework for trade-off analyses of blackbox constrained optimization problems. Two strategies are developed to show the trade-off of the optimal objective function value with tightening or loosening general constraints. These are a simple method which may be performed immediately after a single optimization and a detailed method performing biobjective optimization on … Read more
The goal of this paper is an easy and self-contained presentation of optimality conditions for nonlinear semidefinite programs concentrating on the differences between nonlinear semidefinite programs and nonlinear programs. Citation Technical Report, Department of Mathematics, Universit\”at D\”usseldorf. Article Download View Elementary optimality conditions for nonlinear SDPs
This paper develops a new error criterion for the approximate minimization of augmented Lagrangian subproblems. This criterion is practical in the sense that it requires only information that is ordinarily readily available, such as the gradient (or a subgradient) of the augmented Lagrangian. It is also “relative” in the sense of relative error criteria for … Read more
Augmented Lagrangian methods for derivative-free continuous optimization with constraints are introduced in this paper. The algorithms inherit the convergence results obtained by Andreani, Birgin, Martínez and Schuverdt for the case in which analytic derivatives exist and are available. In particular, feasible limit points satisfy KKT conditions under the Constant Positive Linear Dependence (CPLD) constraint qualification. … Read more
Penalty and interior-point methods for nonlinear optimization problems have enjoyed great successes for decades. Penalty methods have proved to be effective for a variety of problem classes due to their regularization effects on the constraints. They have also been shown to allow for rapid infeasibility detection. Interior-point methods have become the workhorse in large-scale optimization … Read more
At each outer iteration of standard Augmented Lagrangian methods one tries to solve a box-constrained optimization problem with some prescribed tolerance. In the continuous world, using exact arithmetic, this subproblem is always solvable. Therefore, the possibility of finishing the subproblem resolution without satisfying the theoretical stopping conditions is not contemplated in usual convergence theories. However, … Read more
In this work we introduce a relaxed version of the constant positive linear dependence constraint qualification (CPLD) that we call RCPLD. This development is inspired by a recent generalization of the constant rank constraint qualification from Minchenko and Stakhovski that was called RCR. We show that RCPLD is enough to ensure the convergence of an … Read more
We provide Completely Positive and Copositive Programming formulations for the Constrained Fractional Quadratic Problem (CFQP) and Standard Fractional Quadratic Problem (StFQP). Based on these formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which are used in a global optimization branch-and-bound approach. Applications of the CFQP and StFQP, … Read more
We study the local behavior of a primal-dual inexact interior point methods for solving nonlinear systems arising from the solution of nonlinear optimization problems or more generally from nonlinear complementarity problems. The algorithm is based on the Newton method applied to a sequence of perturbed systems that follows by perturbation of the complementarity equations of … Read more