An interior point method for constrained saddle point problems

Abstract

Abstract. We present an algorithm for the constrained saddle point problem with a convexconcave function L and convex sets with nonempty interior. The method consists of moving away from the current iterate by choosing certain perturbed vectors. The values of gradients of L at these vectors provide an appropriate direction. Bregman functions allow us to define a curve which starts at the current iterate with this direction, and is fully contained in the interior of the feasible set. The next iterate is obtained by moving along such a curve with a certain step size.We establish convergence to a solution with minimal conditions upon the function L, weaker than Lipschitz continuity of the gradient of L, for instance, and including cases where the solution needs not be unique. We also consider the case in which the perturbed vectors are on certain specific curves starting at the current iterate, in which case another convergence proof is provided.In the case of linear programming, we obtain a family of interior point methods where all the iterates and perturbed vectors are computed with very simple formulae, without factorization of matrices or solution of linear systems, which makes the method attractive for very large and sparse matrices. The method may be of interest for massively parallel computing. Numerical examples for the linear programming case are given.Mathematical subject classification: 90C25, 90C30.

Interior Proximal Method for Variational Inequalities: Case of Nonparamonotone Operators

“…The following statement is a generalization of Proposition 4 in [12]. Note that in [12], in distinction to our consideration, the operator T is supposed to be singlevalued, monotone and continuous.…”

“…Lemata 3.1 and 3.2 are proved rather similar to Lemma 2 in [16] and Proposition 4 in [12], respectively, and their proofs can be found in Appendix. …”

Some Results About Proximal-Like Methods

“…To avoid needless formalization, let us turn to the optimization problem (15). Suppose that mm{J2{u) : UGK} (16) splits up into independent problems on the sets Ki, ...^Kg, K -nf^^Ki is the Cartesian product. Then the auxiliary problems with C^ = 0 can be rewritten as…”

“…and obviously, they can be decomposed in the same manner as (16). For VI (1) and scheme (2) such decomposition is described in [20].…”

“…The convergence analysis uses essentially the following modification of Proposition 4 in lusem/Kallio [16] (which was proved for a VI with singlevalued, monotone and continuous operator): The second class of Vis considered in [28] is the complementarity problem hnd u* eW^, qeQ{u*):…”