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. To compute the search directions we employ the same formulation as in FDIPA, the Feasible Directions Interior Point Algorithm for constrained optimization. We prove global convergence of the present method. A set of numerical tests is described. The present technique was also successfully applied to the topology optimization of robust trusses. Our results are comparable to those obtained with other well known established methods.