Ana Friedlander

A new trust region algorithm for bound constrained minimization

We introduce a new algorithm of trust-region type for minimizing a differentiable function of many variables with box constraints. At each step of the algorithm we use an approximation to the minimizer of a quadratic in a box. We introduce a new method for solving this subproblem, that has finite termination without dual nondegeneracy assumptions. We prove the global convergence of the main algorithm and a result concerning the identification of the active constraints in finite time. We describe an implementation of the method and we present numerical experiments showing the effect of solving the subproblem with different degrees of accuracy.

On the Maximization of a Concave Quadratic Function with Box Constraints

A new method for maximizing a concave quadratic function with bounds on the variables is introduced. The new algorithm combines conjugate gradients with gradient projection techniques, as the algorithm of More and Toraldo [SIAM J. Optimization, 1 (1991), pp. 93–113] and other well-known methods do. A new strategy for the decision of leaving the current face is introduced that makes it possible to obtain finite convergence even for a singular Hessian and in the presence of dual degeneracy. Numerical experiments are presented.

Augmented Lagrangians with Adaptive Precision Control for Quadratic Programming with Simple Bounds and Equality Constraints

In this paper we discuss a specialization of the augmented Lagrangian-type algorithm of Conn, Gould, and Toint to the solution of strictly convex quadratic programming problems with simple bounds and equality constraints. The new feature of the presented algorithm is the adaptive precision control of the solution of auxiliary problems in the inner loop of the basic algorithm which yields a rate of convergence that does not have any term that accounts for inexact solution of auxiliary problems. Moreover, boundedness of the penalty parameter is achieved for the precision control used. Numerical experiments illustrate the efficiency of the presented algorithm and encourage its usage.

On the Resolution of the Generalized Nonlinear Complementarity Problem

Minimization of a differentiable function subject to box constraints is proposed as a strategy to solve the generalized nonlinear complementarity problem (GNCP) defined on a polyhedral cone. It is not necessary to calculate projections that complicate and sometimes even disable the implementation of algorithms for solving these kinds of problems. Theoretical results that relate stationary points of the function that is minimized to the solutions of the GNCP are presented. Perturbations of the GNCP are also considered, and results are obtained related to the resolution of GNCPs with very general assumptions on the data. These theoretical results show that local methods for box-constrained optimization applied to the associated problem are efficient tools for solving the GNCP. Numerical experiments are presented that encourage the use of this approach.

Solution of linear complementarity problems using minimization with simple bounds

We define a minimization problem with simple bounds associated to the horizontal linear complementarity problem (HLCP). When the HLCP is solvable, its solutions are the global minimizers of the associated problem. When the HLCP is feasible, we are able to prove a number of properties of the stationary points of the associated problem. In many cases, the stationary points are solutions of the HLCP. The theoretical results allow us to conjecture that local methods for box constrained optimization applied to the associated problem are efficient tools for solving linear complementarity problems. Numerical experiments seem to confirm this conjecture.

Augmented Lagrangians with Adaptive Precision Control forQuadratic Programming with Equality Constraints

In this paper we introduce an augmented Lagrangian type algorithm for strictly convex quadratic programming problems with equality constraints. The new feature of the proposed algorithm is the adaptive precision control of the solution of auxiliary problems in the inner loop of the basic algorithm. Global convergence and boundedness of the penalty parameter are proved and an error estimate is given that does not have any term that accounts for the inexact solution of the auxiliary problems. Numerical experiments illustrate efficiency of the algorithm presented

A new line search inexact restoration approach for nonlinear programming

A new general scheme for Inexact Restoration methods for Nonlinear Programming is introduced. After computing an inexactly restored point, the new iterate is determined in an approximate tangent affine subspace by means of a simple line search on a penalty function. This differs from previous methods, in which the tangent phase needs both a line search based on the objective function (or its Lagrangian) and a confirmation based on a penalty function or a filter decision scheme. Besides its simplicity the new scheme enjoys some nice theoretical properties. In particular, a key condition for the inexact restoration step could be weakened. To some extent this also enables the application of the new scheme to mathematical programs with complementarity constraints.

On the Resolution of Linearly Constrained Convex Minimization Problems

The problem of minimizing a twice differentiable convex function f is considered, subject to

INEXACT RESTORATION METHOD FOR DERIVATIVE-FREE OPTIMIZATION WITH SMOOTH CONSTRAINTS ∗

Ax = b, x \geq 0

Analysis of semicoercive contact problems using symmetric BEM and augmented Lagrangians

, where