A new treatment of inconsistent quadratic programs in a SQP-based algorithm

Abstract

Sequential quadratic programming (SQP) algorithms are often considered to be the best choice for solving nonlinear programming problems (NLP). The interest in solving NLPs has increased recently. It has become advantageous to perform on-line economic optimization and non-linear process control in the chemical industries. SQP methods may not be always robust and efficient. They depend on the solution of an approximating quadratic programming problem (QP), whose feasible region is based on a linearization of the constraints of the NLP. The linearized constraints may become infeasible. Moreover, regularity conditions are often not met and the Karush–Kuhn–Tucker (KKT) system of the QP may be singular when some constraints are activated. Difficulties become worse when the SQP method makes use of non-convex QPs. In this case, cycling or convergence to a non-stationary point can be obtained. In this paper we show when singularities in the KKT system appear and inconsistencies in the QPs are carefully defined and analyzed. Discussion is not limited to the case of singular linearized constraints. Strongly inconsistent and critical QPs are defined. Examples are presented showing how each type of inconsistency leads to convergence problems in existing SQP approaches. We also discuss some other numerical difficulties related to constraints with strong curvatures or NLPs whose functions are not of class C 2. We further present a procedure for detecting inconsistencies in the QPs. When the KKT system is not of full rank, we solve an extended KKT system (EKKT). Some modifications in the SQP method are proposed. They make use of the EKKT system and include a separable term in the objective function of the NLP. The resulting SQP algorithm does not need any preprocessing phase and it is shown to work for problems where other codes may fail to converge.