Second-derivative SQP methods for large-scale nonconvex nonlinear optimization
Interior Point methods and Sequential Quadratic Programming (SQP) methods have become two of the most crucial methods for solving large-scale nonlinear optimization problems. The two methods take very different approaches to solving the same problem. SQP methods find approximate solutions to a sequence of linearly constrained quadratic subproblems in which a quadratic model of the Lagrangian is minimized subject to a linear model of the constraints. Typically, the QP subproblems are solved using an active-set method, giving the problem a major-minor iteration pattern in which each iteration of the active-set method solves an indefinite system. In contrast, interior point methods follow a continuous path towards the optimal solution by perturbing the first-order optimality conditions of the problem. In this talk, we discuss a shifted primal dual interior point method and its potential applicability in solving the QP subproblem of an SQP method. We also discuss some of the potential issues with this approach that we hope to overcome.