A new proximal chebychev center cutting plane algorithm for nonsmooth optimization and its convergence

Abstract

Motivated by the proximal-like bundle method [K. C. Kiwiel, Journal of Optimization Theory and Applications, 104(3) (2000), 589-603.], we establish a new proximal Chebychev center cutting plane algorithm for a type of nonsmooth optimization problems. At each step of the algorithm, a new optimality measure is investigated instead of the classical optimality measure. The convergence analysis shows that an ε-optimal solution can be obtained within O(1/ε 3) iterations. The numerical result is presented to show the validity of the conclusion and it shows that the method is competitive to the classical proximal-like bundle method.

An Infeasible Incremental Bundle Method for Nonsmooth Optimization Problem Based on CVaR Portfolio

“…Furthermore, g i H k+1 ∈ z e k+1 i H k+1 (x k+1 , α k+1 ), where e k+1 i ≥ 0 and g i H k+1 are defined in (8) written with k replaced by k + 1.…”

“…Nonsmooth optimization (NSO) problems are in general difficult to solve. Lots of approaches are proposed to solve these problems [4][5][6][7][8]. Among others, bundle methods are considered as one of the most efficient and promising methods.…”