A Superlinearly Convergent Strongly Sub-Feasible SQP Algorithm For Constrained Optimization Without Strict Complementarity

Sequential quadratic programming (SQP) algorithms are widely acknowledged to be among the most successful algorithms for solving nonlinear optimization problems. In recent decades, feasible SQP algorithms have been studied widely, since they have fast rate of convergence, generate feasible iterates, and do not require any penalty functions. However, these feasible algorithms usually require initial feasible points, and it is often a time-consuming procedure for computing such points. On the other hand, feasible SQP algorithms usually require to solve two or three quadratic subprogram at each iteration, so the computational cost is relative high. And the strict complementarity condition is necessary, which is strong and difficult for testing.In this paper, combined the generalized projection technique and the idea of strongly sub-feasible direction method, a new sequential quadratic programming algorithm for inequality constrained optimization problems is presented. The algorithm, in which a new Armijo type step-length search is introduced, starts with an arbitrary initial point, and it must generate a feasible point in a finite number of iterations, then automatically becomes a feasible descent SQP algorithm. At each iteration, only one quadratic program (QP) needs solving, and two correction directions are obtained only by explicit formulas. As a result, the computational cost per iteration is relatively low. Furthermore, the global convergence and superlinear convergence rate are proved under mild assumptions without the strict complementarity. Finally, some preliminary numerical results show that the proposed algorithm is stable and effective.