Sequential Quadratic Programming

Sequential quadratic programming (SQP) methods are very effective for solving optimization problems with significant nonlinearities in constraints. These are active-set methods and generate steps by solving quadratic programming subproblems at every iteration. These methods are used both in the line-search and in the trust-region paradigm. In this chapter, we consider both the equality quadratic programming and the inequality quadratic programming. There are some differences between these approaches. In the inequality quadratic programming, at every iteration, a general inequality-constrained quadratic programming problem is solved for computing a step and for generating an estimate of the optimal active-set. On the other hand, in the equality quadratic programming, these are separated. Firstly, they compute an estimate of the optimal active-set, then they solve an equality-constrained quadratic programming problem to find the step. Our presentation proceeds to develop the theory of the sequential quadratic programming approach for the step computation, followed by practical line-search and trust-region methods that achieve the convergence from remote starting points. At the same time, a number of three implementations of these methods are discussed, which are representative of the following: an SQP algorithm for large-scale-constrained optimization (SNOPT), an SQP algorithm with successive error restoration (NLPQLP), and the active-set sequential linear-quadratic programming (KNITRO/ACTIVE). Their performances are illustrated in solving large-scale problems from the LACOP collection.