A Feasible Directions Method for Nonsmooth Convex Optimization

We propose a new technique for minimization of convex functions not necessarily smooth. Our approach employs an equivalent constrained optimization problem and approximated linear programs obtained with cutting planes. At each iteration a search direction and a step length are computed. If the step length is considered “non serious”, a cutting plane is added and a new search direction is computed. This procedure is repeated until a “serious” step is obtained. When this happens, the search direction is a feasible descent direction of the constrained equivalent problem. The search directions are computed with FDIPA, the Feasible Directions Interior Point Algorithm. We prove global convergence and solve several test problems very efficiently.