An Approximate Sequential Bundle Method for Solving a Convex Nondifferentiable Bilevel Programming Problem
Jie Shen *
School of Mathematics, Liaoning Normal University, Dalian 116029, 0086-0411-84258356, China.
Li-Ping Pang
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China.
Xi-Jun Liang
College of Science, China University of Petroleum, Qingdao 266555, China.
Zun-Quan Xia
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China.
*Author to whom correspondence should be addressed.
Abstract
By combining two bundle methods, PBMASL (proximal bundle method with approximate subgradient linearizations) and DPLBM (descent proximal level bundle method), we present an approximate sequential bundle algorithm for solving a bilevel programming problem with a nondifferentiable convex objective function and two separable constraints. In the proposed algorithm, the values of the objective function in the constraints and its subgradients are computed approximately, the estimates of the tolerances are not required for convergence proof. The presented results improve and extend the earlier work.
Keywords: Nonsmooth optimization, Bilevel programming problem, Bundle method, Subgradient, Proximal bundle method.