An Explicit Exchange Algorithm For Linear Semi-Infinite Programming Problems With Second-Order Cone Constraints

Abstract

In this paper, we propose an explicit exchange algorithm for solving the semi-infinite programming problem (SIP) with second-order cone (SOC) constraints. We prove, by using the complementarity slackness conditions, that the algorithm terminates in a finite number of iterations and the obtained solution sufficiently approximates the original SIP solution. In existing studies on SIPs, only the nonnegative constraints were considered, and hence, the complementarity slackness conditions were separable to each scalar component. However, in our study, the existing componentwise analyses are no longer applicable since the complementarity slackness conditions are associated with SOCs. In order to overcome such a difficulty, we introduce a certain coordinate system based on the spectral factorization. In the numerical experiments, we solve some test problems to see the effectiveness of the proposed algorithm. First, we compare our algorithm with two other algorithms based on the linear SIP (LSIP) reformulation and observe that, by exploiting the SDPT3 solver for subproblems, our algorithm finds the solution much faster than LSIP reformulation approaches when the size of the problem is large. Also, we apply the algorithm to some filter design problems and see that those problems can be solved efficiently.