Robustness of the Hybrid Extragradient Proximal-Point Algorithm

The hybrid extragradient proximal-point method recently proposed by Solodov and Svaiter has the distinctive feature of allowing a relative error tolerance. We extend the error tolerance of this method, proving that it converges even if a summable error is added to the relative error. Furthermore, the extragradient step may be performed inexactly with a summable error. We present a convergence analysis, which encompasses other well-known variations of the proximal-point method, previously unrelated. We establish weak global convergence under mild assumptions.     

A note on the existence of zeroes of convexly regularized sums of maximal monotone operators

Many algorithms for solving the problem of finding zeroes of a sum of two maximal monotone operators T₁ and T₂, have regularized subproblems of the kind 0T₁(x)+T₂(x)+∂D(x), where D is a convex function. We develop an unified analysis for existence of solutions of these subproblems, through the introduction of the concept of convex regularization, which includes several well-known cases in the literature. Finally, we establish conditions, either on D or on the operators, which assure solvability of the subproblems.     

On some properties and an application of the logarithmic barrier method

We analyze the logarithmic barrier method for nonsmooth convex optimization in the setting of point-to-set theory. This general framework allows us to both extend and include classical results. We also propose an application for finding efficient points of nonsmooth constrained convex vector-valued problems. Susana Scheimberg was partially supported CNPq grant 302393/85-4 (RN).     

A Computational Study of Global Algorithms for Linear Bilevel Programming

We analyze two global algorithms for solving the linear bilevel program (LBP) problem. The first one is a recent algorithm built on a new concept of equilibrium point and a modified version of the outer approximation method. The second one is an efficient branch-and-bound algorithm known in the literature. Based on computational results we propose some modifications in both algorithms to improve their computational performance. A significant number of experiments is carried out and a comparative study with the algorithms is presented. The modified procedures has better performance than the original versions.     

Note on a sufficient condition for a local minimum of a class of nonconvex quadratic programs

A counterexample is given to show that a previously proposed sufficient condition for a local minimum of a class of nonconvex quadratic programs is not correct. This class of problems arises in combinatorial optimization. The problem with the original proof is pointed out. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

A Simplex Approach for Finding Local Solutions of a Linear Bilevel Program by Equilibrium Points

In this paper, a linear bilevel programming problem (LBP) is considered. Local optimality conditions are derived. They are based on the notion of equilibrium point of an exact penalization for LBP. It is described how an equilibrium point can be obtained with the simplex method. It is shown that the information in the simplex tableaux can be used to get necessary and sufficient local optimality conditions for LBP. Based on these conditions, a simplex type algorithm is proposed, which attains a local solution of LBP by moving in equilibrium points. A numerical example illustrates how the algorithm works. Some computational results are reported.     

A Two-Phase Algorithm for a Variational Inequality Formulation of Equilibrium Problems

We introduce an explicit algorithm for solving nonsmooth equilibrium problems in finite-dimensional spaces. A particular iteration proceeds in two phases. In the first phase, an orthogonal projection onto the feasible set is replaced by projections onto suitable hyperplanes. In the second phase, a projected subgradient type iteration is replaced by a specific projection onto a halfspace. We prove, under suitable assumptions, convergence of the whole generated sequence to a solution of the problem. The proposed algorithm has a low computational cost per iteration and, some numerical results are reported.     

Descent algorithm for a class of convex nondifferentiable functions

We present first an -descent basic method for minimizing a convex minmax problem. We consider first- and second-order information in order to generate the search direction. Preliminarily, we introduce some properties for the second-order information, the subhessian, and its characterization for max functions. The algorithm has -global convergence. Finally, we give a generalization of this algorithm for an unconstrained convex problem having second-order information. In this case, we obtain global -convergence.This research was supported in part by CAPES (Coordinação de Aperfeiçoamento de Pessoal de Nivel Superior) and in part by the IM-COPPE/UFRJ, Brazil. The authors wish to thank Nguyen Van Hien and Jean-Jacques Strodiot for their constructive remarks on an earlier draft of the paper. This work was completed while the first author was with the Department of Mathematics of the Facultés Universitaires de Namur. The authors are also grateful for the helpful comments of two anonymous referees.