Direct Search Methods for Nonlinearly Constrained Optimization Using Filters and Frames

A direct search method for nonlinear optimization problems with nonlinear inequality constraints is presented. A filter based approach is used, which allows infeasible starting points. The constraints are assumed to be continuously differentiable, and approximations to the constraint gradients are used. For simplicity it is assumed that the active constraint normals are linearly independent at all points of interest on the boundary of the feasible region. An infinite sequence of iterates is generated, some of which are surrounded by sets of points called bent frames. An infinite subsequence of these iterates is identified, and its convergence properties are studied by applying Clarke's non-smooth calculus to the bent frames. It is shown that each cluster point of this subsequence is a Karush-Kuhn-Tucker point of the optimization problem under mild conditions which include strict differentiability of the objective function at each cluster point. This permits the objective function to be non-smooth, infinite, or undefined away from these cluster points. When the objective function is only locally Lipschitz at these cluster points it is shown that certain directions still have interesting properties at these cluster points.