A variant of the Topkis-Veinott method for solving inequality constrained optimization problems

In this paper we give a variant of the Topkis-Veinott method for solving in equality constrained optimization problems. This method uses a linearly con strained positive semidefinite quadratic problem to generate a feasible des cent direction at each iteration. Under mild assumptions, the algorithm is shown to be globally convergent in the sense that every accumulation point of the sequence generated by the algorithm is a Fritz-John point of the pro blem. We introduce a Fritz-John (FJ) function, an FJ1 strong second-order s ufficiency condition (FJ1-SSOSC), and an FJ2 strong second-order sufficienc y condition (FJ2-SSOSC), and then show, without any constraint qualificatio n (CQ), that (i) if an FJ point z satisfies the FJ1-SSOSC, then there exist s a neighborhood N(z) of z such that, for any FJ point y is an element of N (z)\{z}, f(0)(y) not equal f(0)(z), where f(0) is the objective function of the problem; (ii) if an FJ point z satisfies the FJZ-SSOSC, then z is a st rict local minimum of the problem. The result (i) implies that the entire i teration point sequence generated by the method converges to an FJ point. W e also show that if the parameters are chosen large enough, a unit step len gth can be accepted by the proposed algorithm.