Numerical algorithm for solving a class of optimization problems with a constraint in the form of a subset of points of a smooth surface

A numerical algorithm is proposed for minimizing a convex function on the set-theoretic difference between a set of points of a smooth surface and the union of finitely many convex open sets in $n$-dimensional Euclidean space. The idea behind the algorithm is to reduce the original problem to a sequence of convex programming problems. Necessary extremum conditions are examined, and the convergence of the algorithm is analyzed.