Convexification Schemes for Sqp Methods

Sequential quadratic programming (SQP) methods solve nonlinear optimization problems by finding an approximate solution of a sequence of quadratic programming (QP) subproblems. Each subproblem involves the minimization of a quadratic model of the objective function subject to the linearized constraints. Depending on the definition of the quadratic model, the QP subproblem may be nonconvex, leading to difficulties in the formulation and analysis of a conventional SQP method. Convexification is a process for defining a local convex approximation of a nonconvex problem. We describe three forms of convexification: preconvexification, concurrent convexification, and post-convexification. The methods require only minor changes to the algorithms used to solve the QP subproblem, and are designed so that modifications to the original problem are minimized and applied only when necessary.