Solving Complex Semidefinite Programming With Inequality Constraints And Application

Semi-definite programming(SDP), as a powerful convex technique, has been wildly applied to solve optimization problems. In communication field, SDP attracts researcher's attention in two aspects:(1) reformulate or relax the communication problems which are difficult to solve directly, into SDP problems;(2) propose an efficient and practical algorithm to solve the above problems. Interior point methods, such as SeDuMi(Self dual minimization), are mature and efficient to solve small SDP problems. However, for large scale problems, interior point methods are inefficient or even infeasible. It is because that interior point methods require more storages and execution time.In this paper, we propose a practical low rank factorization(LRF) algorithm to solve SDP problems with inequality constraints(SDPI). The LRF algorithm overcomes the two serious disadvantages of semi-definite programming low rank(SDPLR) algorithm. One is that SDPLR can solve the real SDP problems only, another one is that SDPLR need an enormous amount of auxiliary variables to solve SDPI problems that increase the problem's scale rapidly. Following the idea of low rank factorization, we replace the positive semi-definite matrix X?(N×N) by a low dimension matrix N ×rR?(N×r) through the factorization X=RR~H, where r is far less than N. Then, the SDPI problem can be reformulated as a nonlinear problem which can be easily solved through an iteration algorithm based on a simplified augmented Lagrangian function(ALF).As important as a kind of technique in communication, Eddy formulates the precoder-decorrelator design with imperfect channel state information(CSI) as an optimization problem to maximize the worst-case SINR among all users' data streams, subject to the maximum transmit power per source node in the K-pair quasi-static MIMO interference channel. The above optimal problem can be solved by alternating optimization Q_v and Q_u subproblem. LRF and SeDuMi are used to solve the SDPI subproblem Q_v, respectively. The numerical results show that the proposed LRF algorithm has the same performance as SeDuMi. With the problem's scale increasing, the LRF spends much less execution time than SeDuMi.