A Proximal Bundle Method For Nonsmooth DC Optimization With Inexact Oracle

In real applications,we encounter optimization problems with nonsmooth nonconvex functions,and convex difference optimization is one important part of nonconvex nonsmooth optimization.In addition,for some problem we cannot obtain the exact function values and subgradients,or although the exact function values and the subgradients can be obtained but cause heavy computational burden.For these reasons,we propose a proximal bundle method with informative and controllable lower Oracle for convex difference optimization.The novelty of this thesis lies in that we can control the accuracy between the exact objective values and subgradients and the inexact ones in Oracle,by controlling the update formula of the upper bound of the error.Then we use the special structure of the convex difference function,at each iteration we maintain two separate bundles containing available information including the subgradients and linearization errors obtained from the iterative process.Using these information stored in the double bundles,we construct a piecewise linear approximation of the objective function.By solving an approximated model by adding a stabilizing term,the search direction is obtained,and a new iteration point is obtained.The information at this point is used to improve the approximated model.This process is repeated until the stop condition is met.Show that the proposed method converges to the ?-critical point.Our method is divided into two parts: 'outer loop' and 'inner loop'.First,with the aid of ?-subgradient,we prove that the 'inner loop' terminates after a finite number of steps.Based on which,we show the finite termination of the overall proximal bundle method with informative and controllable lower Oracle.