The nonconvex program plays an important role in the field of optimization and has a lot of applications in practice. However, for general nonconvex programming problems, the lack of verifiable global optimal conditions and the multiple local minimizers make global optimization hard in computation. In this paper, a convex proximal point algorithm (CPPA) is considered for globally solving nonconvex programming problems. We prove that every accumulation point of CPPA is a stationary point and the initial point of CPPA is key to get the global minimum. Serval sufficient conditions for the initial point selection are provided for CPPA getting the global minimum. Motivated by these sufficient conditions, CPPA is applied for the nonconvex quadratic programming problem with convex quadratic constraints with the initial point getting from its Lagrangian dual problem. Numerical results show that the possibility to get the global minimum is much higher than that of randomly selecting initial points.
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China. July/2021