Regularized Sequential Quadratic Programming Methods

Sequential quadratic programming (SQP) methods are a popular class of methods for nonlinearly constrained optimization. They are particularly effective for solving a sequence of related problems, such as those arising in mixed-integer nonlinear programming and the optimization of functions subject to differential equation constraints.

Recently, there has been considerable interest in the formulation of stabilized SQP methods, which are specifically designed to handle degenerate optimization problems. Existing stabilized SQP methods are essentially local, in the sense that both the formulation and analysis focus on a neighborhood of a solution. We present the formulation and analysis of a new SQP method that has favorable global convergence properties yet is equivalent to a variant of the conventional stabilized SQP method in the neighborhood of a solution. The method is based on the combination of a primal-dual generalized augmented Lagrangian merit function with a flexible line search to obtain a sequence of improving estimates of the solution. An important feature of the method is that the quadratic programming (QP) subproblem is defined using the exact Hessian of the Lagrangian, yet has a unique bounded solution. This gives the potential for fast convergence in the neighborhood of a solution. Additional benefits of the method include: (i) each QP subproblem is regularized; (ii) the QP subproblem always has a known feasible point; and (iii) a projected gradient method may be used to identify the QP active set when far from the solution.

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