Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems

Abstract

In this paper a family of trust{region interior{point SQP algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise e.g. from the discretization of optimal control problems. The algorithms treat states and controls as independent v ariables. They are designed to take a d v antage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints, but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial di erential equations. The algorithms keep strict feasibility with respect to the bound constraints by using an a ne scaling method proposed for a di erent class of problems by Coleman and Li and they exploit trust{region techniques for equality{ constrained optimization. Thus, they allow the computation of the steps using a variety of methods, including many iterative techniques. Global convergence of these algorithms to a rst{order KKT limit point i s p r o ved under very mild conditions on the trial steps. Under reasonable, but more stringent conditions on the quadratic model and on the trial steps, the sequence of iterates generated by the algorithms is shown to have a limit point satisfying the second{order necessary KKT conditions. The local rate of convergence to a nondegenerate strict local minimizer is q{quadratic. The results given here include as special cases current results for only equality constraints and for only simple bounds. Numerical results for the solution of an optimal control problem governed by a nonlinear heat equation are reported.

A primal-dual trust region algorithm for nonlinear optimization

“…with Lagrange multipliers for the inequalities being defined as y i = µ/c i (x) for i ∈ I. Barrier-SQP methods exploit this interpretation by replacing the general mixed-constraint problem (NP) by a sequence of equality constraint problems in which the inequalities are eliminated using a logarithmic barrier transformation (see, e.g., [2,9,12,14,31,57]). …”

“…Trust-region barrier-SQP methods are largely based on the Byrd-Omojokun algorithm [8,47], which uses different trust-regions to control the normal and tangential components of the SQP search direction (see, e.g., [9,10,14,59]). …”

“…Some merit functions exploit the fact that equations (1.6) are the optimality conditions for the subproblems generated by an SQP method for the problem (1.2). This provides the motivation for many algorithms based on standard line-search or trust-region SQP strategies (see, e.g., [2,9,10,14,31]). These algorithms usually converge to a second-order point, although this can be guaranteed for only some of the methods.…”

Second-order negative-curvature methods for box-constrained and general constrained optimization

“…Conn, Gould, Orban and Toint [14] employ a logarithmic barrier method for inequality constrained optimization with linear equality constraints. Dennis, Heinkenschloss and Vicente [17] use affine scaling directions and, also, the SQP approach for optimization with equality constraints and simple bounds (see, also, [18]). Di Pillo, Lucidi and Palagi [19] define a primal-dual algorithm model for inequality constrained optimization problems that exploits the equivalence between the original constrained problem and the unconstrained minimization of an exact Augmented Lagrangian function.…”

Examples of ill-behaved central paths in convex optimization

“…Internal penalty methods, also known as barrier methods, had an explosive development in the last fifteen years, due to the success of interior point methods for linear programming and for linear complementarity problems (see the monographs [10,18,19,25,27,28,33]; see also the extensions to nonlinear programming in [4,9,11,14,15,29]). The first deep study of the path of optimizers, now known as central path, is due to McLinden [21], followed by Bayer and Lagarias [3] and by Megiddo [22], who gave a definitive characterization of the primal-dual central path.…”