Numerical methods for optimizing discontinuous functions (Q1904790)

Problems of minimizing nonsmooth and discontinuous functions cause serious complications for numerical methods. Methods based on the approximational gradient allow one to handle many nonsmooth and discontinuous problems. Analogs of the steepest descent and of the conjugate gradient methods based on the approximational gradient (which really consists of evaluating an integral over certain region) are studied. The analogs require that formulas be specified only for the function being minimized and they are based on the search for generalized stationary points (the notion of which is given in this paper). Under assumptions satisfied for a wide class of functions, including discontinuous ones, several convergence theorems for the approximational gradient methods are proved. One possible approach to the solution of constrained optimization problems via the approximational gradient methods by reducing the constrained problem to a problem of minimizing some function \(F(x)\) on the entire space \(\mathbb{R}^n\) is analyzed and this approach is compared with that of penalty function methods. The methods under discussion are tested on a broad collection of benchmark problems.