Simplex-type algorithm for second-order cone programmes via semi-infinite programming reformulation
Optimization Methods and Software
To solve the (linear) second-order cone programs (SOCPs), the primal-dual interior-point method has been studied extensively so far and said to be the most efficient method by many researchers. On the other hand, the simplex type method for SOCP is much less spotlighted, while it still keeps an important position for linear programs. In this paper, we apply the dual-simplex primalexchange (DSPE) method, which was originally developed for solving linear semi-infinite programs (LSIP), to the SOCP
... by reformulating the second-order cone constraint as an infinite number of linear inequality constraints. Then, we show that the sequence generated by the DSPE method converges to the SOCP optimum under mild assumptions. By means of numerical experiments, we observe that such a simplex type method can be more efficient than the existing interior-point based method, when we solve multiple SOCPs having similar data structures successively applying the so-called "hot start" technique. the magnetic shield design problem for maglev trains  , and so on. Since the nonnegative orthant is identical with the Cartesian product of one-dimensional SOCs, i.e., R n + = K 1 × K 1 × · · · × K 1 , the linear program (LP) can be regarded as a subclass of SOCP. Moreover, the quadratic program (QP) and a certain class of robust optimization problems can be reformulated as an SOCP. On the contrary, the SOCP is involved in the semidefinite program (SDP) as a subclass. However, it is not reasonable to solve the SOCP as an SDP  since the SDP has matrix variables and therefore the computational cost tends to be much higher than that of SOCP. Currently, the most popular method for solving the SOCP is the primal-dual interior-point method [8, 9, 17] , which was originally developed for solving LPs and has been extended to SOCP and SDP. This method is known to be quite efficient both theoretically and practically, and several software packages [16, 15, 19] have been developed. On the other hand, there are only a few studies on the simplex method for SOCP, though it is still important and popular for LP. For example, Muramatsu [10, 11] defined a "dictionary" with respect to the simplex method and proposed an implementable simplex type algorithm for SOCP with some restricted structure. Pataki  extended the simplex method for LP to SDP in a theoretical manner. However, his approach has not been implemented to practical problems. The main reason why the simplex method for SOCP has not been studied so much is that the feasible region of an SOCP has infinitely many extreme points unlike LP. However, the simplex type method has an advantage when we need to solve multiple problems whose structures are similar to each other. In such a case, the simplex type method can be accelerated by using the "hot start" technique, which inherits the information of the bases of the problem solved in the previous step. In this paper, we first reformulate the SOCP as a linear semi-infinite program (LSIP), and then apply to it the dual-simplex primal-exchange (DSPE) method [5, Chap. 12]. Since any (linear) SOCP can be reformulated as an LSIP, we do not need to restrict the structure of SOCPs. Moreover, in the step of finding the "most violated index" in the DSPE method, we have only to substitute an obtained vector into an explicit formula. This advantage is due to the special structure of the SOC. Indeed, such an index is usually obtained by solving a nonconvex optimization problem when we apply the DSPE method to general LSIPs. We also show the global convergence property of the algorithm under mild assumptions. In the proof, we apply some mathematical techniques used in the convergence analyses for the exchange method [7, 6, 12, 20] . The paper is organized as follows. In Section 2, we give some fundamental background on SOCP and LSIP, and reformulate the SOCP as an LSIP. In Section 3, we introduce the DSPE method for solving such an LSIP, and mention some properties important for implementation. In Section 4, we establish global convergence for the DSPE method under some mild assumptions. In Section 5, we report some numerical results. Especially, we compare the simplex type algorithm with the existing primal-dual interior-point method, and observe that the simplex type algorithm is often more efficient in solving multiple problems with similar structures successively. In Section 6, we conclude the paper with some remarks.