A hybrid direct search and projected simplex gradient method for convex constrained minimization

We propose a new Derivative-free Optimization (DFO) approach for solving convex constrained minimization problems. The feasible set is assumed to be the non-empty intersection of a finite collection of closed convex sets, such that the projection onto each of these individual convex sets is simple and inexpensive to compute. Our iterative approach alternates between steps that use Directional Direct Search (DDS), considering adequate poll directions, and a Spectral Projected Gradient (SPG) method, replacing the real gradient by a simplex gradient, under a DFO approach. In the SPG steps, if the convex feasible set is simple, then a direct projection is computed. If the feasible set is the intersection of finitely many convex simple sets, then Dykstra's alternating projection method is applied. Convergence properties are established under standard assumptions usually associated to DFO methods. Some preliminary numerical experiments are reported to illustrate the performance of the proposed algorithm, in particular by comparing it with a classical DDS method. Our results indicate that the hybrid algorithm is a robust and effective approach for derivative-free convex constrained optimization.