A method combining norm-relaxed QP subproblems with systems of linear equations for constrained optimization

Based on the ideas of norm-relaxed sequential quadratic programming (SQP) method and the strongly sub-feasible direction method, we propose a new SQP algorithm for the solution of nonlinear inequality constrained optimization. Unlike the previous work, at each iteration, the norm-relaxed quadratic programming subproblem (NRQPS) in our algorithm only consists of the constraints corresponding to an estimate of the active set, and the high-order correction direction (used to avoid the Maratos effect) is obtained by solving a system of linear equations (SLE) which also only consists of such a subset of constraints and gradients. Moreover, the line search technique can effectively combine the initialization process with the optimization process, and therefore (if the starting point is not feasible) the iteration points always get into the feasible set after a finite number of iterations. The global convergence is proved under the Mangasarian–Fromovitz constraint qualification (MFCQ), and the superlinear convergence is obtained without assuming the strict complementarity. Finally, the numerical experiments show that the proposed algorithm is effective and promising for the test problems.