The Proximal Chebychev Center Cutting Plane Methods For Constrained Nonsmooth Optimization

Nonsmooth optimization,also known as non-differentiable optimization,has wide application value in industry,agriculture and military.Due to the high cost of traditional non-smooth optimization methods to solve the non-smooth problems with complex constraints,it is a meaningful work to design an effi-cient and feasible algorithm to solve the problem.In this paper,the proximal Chebychev center cutting plane method is ex-tended to solve nonsmooth constrained optimization problems,and two new proximal Chebychev center cutting plane methods are proposed.Firstly,based on the improved function method,a new proximal Cheby-chev center cutting plane algorithm(CPC~3PA)is proposed.In this method,the constrained function is treated by the improved function method to make it into an unconstrained optimization problem,and then solved by the proxi-mal Chebychev center cutting plane method.The idea this algorithm is that a bounded,non-empty polyhedron is defined by a set of linear inequalities,and each iteration point is the center of the polyhedron’s maximum inner catch.During the iteration process,the non-optimal region is gradually cut,and the maximum inner radius of the ball tends to 0,so as to obtain the optimal solution of the problem.Secondly,in order to limit the increase of subproblem constraints,the new proximal Chebychev center cutting plane method is improved,and the aggregat-ing technique is introduced to obtain the second algorithm(ACPC~3PA),which inherits the advantages of the first algorithm and saves storage space.Finally,the strong convergence of the two algorithms is demonstrated and analyzed.The preliminary numerical experiments show that the two algorithms in this paper have significant advantages over the existing algorithms,especially for problems with large dimensions,the proximal Chebychev center cutting plane algorithm with aggregation is more efficient.