Infeasibility and negative curvature in optimization

We consider the constrained nonlinear optimization problem where the feasible set is possibly empty, in which case no solution exists. However, in many real-life situations some decision still has to be made. We propose a method where the constraints are allowed to be violated, a so-called elastic approach. Our new goal is to find a near-feasible point with a “good” objective value. The approach we advocate is also useful for problems that have a solution but the solution is determined by solving subproblems. A subproblem may be infeasible even if the original problem is feasible. We focus on SQP (Sequential Quadratic Programming) methods, a popular class of methods for solving constrained nonlinear optimization problems. A characteristic feature of SQP methods is that they solve a sequence of quadratic subproblems with linear constraints. It is often assumed that all the subproblems are feasible. This does not always hold, even when the original nonlinear problem is feasible. In the second part of this thesis, we turn to the problem of how to compute a direction of negative curvature for a symmetric matrix H=12Fx . When second derivatives are known we wish to be able to determine a point that satisfies the second-order (necessary) conditions for a minimizer. It can be shown that an essential feature of such algorithms is to be able to compute a direction of negative curvature for a symmetric matrix. We describe some new work on how to compute such directions. In particular, we show how, with little effort, some iterative methods can be used to obtain a good direction of negative curvature from a poor direction.