On the Convergence of a Modiied Barrier Function Method for Convex Quadratic and Linear Programming

We analyze convergence properties of the Modiied Barrier Function Method (MBFM), which operates like the usual Augmented Lagrangian method, except that it uses a shifted logarithmic barrier function in place of the usual quadratic penalty function. The MBFM avoids some of the drawbacks of classical barrier methods. The convergence of the MBFM had previously been established under strict regularity assumptions for general convex programs. We focus on Convex Quadratic and Linear Programs and strengthen substantially the existing convergence results. For Linear Programs, we also analyze the speed of convergence of the dual sequence. 1. Foundations In the following we are going to consider the solution of the convex quadratic program (QP) min q(x) 1 2 x T Qx + c T x x 2 S := fx 2 IR n : Ax bg = fx 2 IR n : a T j x ? b j 0; j = 1; : : :; mg; using the Modiied Barrier Function Method (MBFM) introduced by R. Polyak in 3]. Here Q is a symmetric positive semideenite n n matrix, A is a m n matrix with the rows a T 1 ; : : :; a T m. Thus (QP) is a convex program. We denote by q := inffq(x) : x 2 Sg the optimal function value and by S := fx 2 S : q(x) = q g the set of optimal solutions for (QP). Throughout the paper we will make the following assumption: Assumption A: The optimal solution set S for (QP) is nonempty and bounded. The Lagrangian L associated with (QP) is given by L(x;) := 1 2 x T Qx+c T x+ T (b?Ax) if 0 and L(x;) := ?1 otherwise. By the Kuhn{Tucker Theorem from a feasible vector x 2 S is an optimal solution of (QP) if and only if there is a vector 0 such that r x L(x ;) = Qx + c ? A T = 0 and the complementary slackness condition () T (Ax ? b) T = 0 is satissed. The (Lagrange) dual program (D) associated with (QP) is given by (D) max 0 d(); (1) where d() := inf x2IR n L(x;) is the dual functional for (QP). It is easy to see that under our assumptions d is concave, proper and closed. The weak duality theorem asserts that the value d() of any vector 2 IR m is …