Affiliation: | (1) Institute of Antennas and EM Scattering, Xidian University, Xian, Shaanxi, 710071, Peoples Republic of China;(2) Department of Geography and Resource Management, Centre for Environmental Policy and Resource Management, and Joint Laboratory for GeoInformation Science, The Chinese University of Hong Kong, New Territories, Shatin, Hong Kong;(3) Institute for Information and System Sciences, Faculty of Sciences, Xian Jiaotong University, Xian, Shaanxi, 710049, Peoples Republic of China |
Abstract: | In this part of the two-part series of papers, algorithms for solving some variable programming (VP) problems proposed in Part I are investigated. It is demonstrated that the non-differentiability and the discontinuity of the maximum objective function, as well as the summation objective function in the VP problems constitute difficulty in finding their solutions. Based on the principle of statistical mechanics, we derive smooth functions to approximate these non-smooth objective functions with specific activated feasible sets. By transforming the minimax problem and the corresponding variable programming problems into their smooth versions we can solve the resulting problems by some efficient algorithms for smooth functions. Relevant theoretical underpinnings about the smoothing techniques are established. The algorithms, in which the minimization of the smooth functions is carried out by the standard quasi-Newton method with BFGS formula, are tested on some standard minimax and variable programming problems. The numerical results show that the smoothing techniques yield accurate optimal solutions and that the algorithms proposed are feasible and efficient.This work was supported by the RGC grant CUHK 152/96H of the Hong Kong Research Grant Council. |