A Cone Constrained Convex Program: Structure and Algorithms

In this paper, we consider the positive semi-definite space tensor cone constrained convex program, its structure and algorithms. We study defining functions, defining sequences and polyhedral outer approximations for this positive semidefinite space tensor cone, give an error bound for the polyhedral outer approximation approach, and thus establish convergence of three polyhedral outer approximation algorithms for solving this problem. We then study some other approaches for solving this structured convex program. These include the conic linear programming approach, the nonsmooth convex program approach and the bi-level program approach. Some numerical examples are presented. L. Qi was supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 501909, 502510, 502111 and 501212). Y.-X. Yuan was supported by National Natural Science Foundation of China (Grant Nos. 10831006 and 11021101) and by CAS grant (No. kjcx-yw-s7). X. Zhang was supported by the National Natural Science Foundation of China (Grant Nos. 11101303 and 11171180). L. Qi ( ) · Y. Xu Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China e-mail: [email protected] Y. Xu e-mail: [email protected] Y.-X. Yuan State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, Zhong Guan Cun Donglu 55, Beijing, 100190, P.R. China e-mail: [email protected] X. Zhang Department of Mathematics, School of Science, Tianjin University, Tianjin, 300072, China e-mail: [email protected]