Solving Convex Programs by Means of Ordinary Differential Equations

Within a Hilbert space we consider nonsmooth convex programs with sharp constraints. Examples include all problem instances that are bounded and strictly feasible. To solve such programs we pursue an absolutely continuous trajectory generated by a differential inclusion of subgradient type. Whenever this inclusion offers some freedom of choice, we select a steepest descent direction. It is shown that the proposed algorithm converges to an optimal solution in finite time.