Susana Scheimberg

Inexact Bundle Methods for Two-Stage Stochastic Programming

Stochastic programming problems arise in many practical situations. In general, the deterministic equivalents of these problems can be very large and may not be solvable directly by general-purpose optimization approaches. For the particular case of two-stage stochastic programs, we consider decomposition approaches akin to a regularized L-shaped method that can handle inexactness in the subproblem solution. From a nonsmooth optimization perspective, these variants amount to applying a proximal bundle method to an oracle that gives inaccurate values for the objective function and a subgradient. Rather than forcing early termination of the subproblem optimization to define inexact oracles, we select a small subset of scenarios for which the subproblem solution is exact, and we replace the information for the remaining scenarios by a fast procedure that does not involve solving an optimization problem. The inaccurate oracle information creates inexact cuts in the master program, which are well handled by th...

A Proximal Point Method for the Variational Inequality Problem in Banach Spaces

In this paper we prove well-definedness and weak convergence of the generalized proximal point method when applied to the variational inequality problem in reflexive Banach spaces. The proximal version we consider makes use of Bregman functions, whose original definition for finite dimensional spaces has here been properly extended to our more general framework.

An inexact subgradient algorithm for Equilibrium Problems

We present an inexact subgradient projection type method for solving a nonsmooth Equilibrium Problem in a finite-dimensional space. The proposed algorithm has a low computational cost per iteration. Some numerical results are reported.

A projection algorithm for general variational inequalities with perturbed constraint sets

We consider the problem of general variational inequalities, GVI, with nonmonotone operator, in a finite dimensional space. We propose a method to solve GVI that at each iteration considers only one projection on an easy approximation of the constraint set, which is important from a practical point of view. We analyse the convergence of the algorithm under a weak cocoercivity condition, using variational metric analysis. Computational experience is reported and comparative analysis with other two algorithms is also given for the monotone case.

A note on a modified simplex approach for solving bilevel linear programming problems

Abstract We analyze the article “A modified simplex approach for solving bilevel linear programming problems” (EJOR, 67, 126–135). We point out some problems in its theoretical analysis. Moreover, the algorithm proposed may not find a global solution as it is claimed. We give some examples in order to illustrate these remarks.

A relaxed projection method for finite-dimensional equilibrium problems

In this article we consider an equilibrium problem for a differentiable bifunction which is not necessarily monotone. We present an implementable projection method. At each iteration, only one inexact projection onto a simple approximation of the constraint set is performed, such as a polyhedron, which renders it numerically attractive. The algorithm can identify, in practice, a subsequence that converges to a solution under reasonable assumptions. Some numerical results are reported showing the performance of our algorithm.

Duality for generalized equilibrium problem

We introduce a generalized equilibrium problem (GEP) that allow us to develop a robust dual scheme for this problem, based on the theory of conjugate functions. We obtain a unified dual analysis for interesting problems. Indeed, the Lagrangian duality for convex optimization is a particular case of our dual problem. We establish necessary and sufficient optimality conditions for GEP that become a well-known theorem given by Mosco and the dual results obtained by Morgan and Romaniello, which extend those introduced by Auslender and Teboulle for a variational inequality problem.

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.

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.

An inexact method of partial inverses and a parallel bundle method

For a maximal monotone operator T on a Hilbert space H and a closed subspace A of H, we consider the problem of finding (x, y∈T(x)) satisfying x∈A and y∈A ⊥. An equivalent formulation of this problem makes use of the partial inverse operator of Spingarn. The resulting generalized equation can be solved by using the proximal point algorithm. We consider instead the use of hybrid proximal methods. Hybrid methods use enlargements of operators, close in spirit to the concept of ϵ-subdifferentials. We characterize the enlargement of the partial inverse operator in terms of the enlargement of T itself. We present a new algorithm of resolution that combines Spingarn and hybrid methods, we prove for this method global convergence only assuming existence of solutions and maximal monotonicity of T. We also show that, under standard assumptions, the method has a linear rate of convergence. For the important problem of finding a zero of a sum of maximal monotone operators T 1, …, T m , we present a highly parallelizable scheme. Finally, we derive a parallel bundle method for minimizing the sum of polyhedral functions.