Takayuki Okuno

A Regularized Explicit Exchange Method for Semi-Infinite Programs with an Infinite Number of Conic Constraints

The semi-infinite program (SIP) is normally represented with infinitely many inequality constraints and has been studied extensively so far. However, there have been very few studies on the SIP involving conic constraints, even though it has important applications such as Chebyshev-like approximation, filter design, and so on. In this paper, we focus on the SIP with infinitely many conic constraints, called an SICP for short. We show that under the Robinson constraint qualification a local optimum of the SICP satisfies the KKT conditions that can be represented only with a finite subset of the conic constraints. We also introduce two exchange type algorithms for solving the convex SICP. We first provide an explicit exchange method and show that it has global convergence under the strict convexity assumption on the objective function. We then propose an algorithm combining a regularization method with the explicit exchange method and establish global convergence of the hybrid algorithm without the strict c...

Simplex-type algorithm for second-order cone programmes via semi-infinite programming reformulation

To solve the (linear) second-order cone programmes (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 programmes. In this paper, we apply the dual–simplex primal-exchange (DSPE) method, which was originally developed for solving linear semi-infinite programmes, 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 certain assumptions. In the numerical experiments, we consider the situation to solve multiple SOCPs with similar structures successively. Then we observe that our simplex-type method can be more efficient than the existing interior-point method when we apply the so-called ‘hot start’ technique.

Local reduction based SQP-type method for semi-infinite programs with an infinite number of second-order cone constraints

The second-order cone program (SOCP) is an optimization problem with second-order cone (SOC) constraints and has achieved notable developments in the last decade. The classical semi-infinite program (SIP) is represented with infinitely many inequality constraints, and has been studied extensively so far. In this paper, we consider the SIP with infinitely many SOC constraints, called the SISOCP for short. Compared with the standard SIP and SOCP, the studies on the SISOCP are scarce, even though it has important applications such as Chebychev approximation for vector-valued functions. For solving the SISOCP, we develop an algorithm that combines a local reduction method with an SQP-type method. In this method, we reduce the SISOCP to an SOCP with finitely many SOC constraints by means of implicit functions and apply an SQP-type method to the latter problem. We study the global and local convergence properties of the proposed algorithm. Finally, we observe the effectiveness of the algorithm through some numerical experiments.

An exchange method with refined subproblems for convex semi-infinite programming problems

In this paper, we propose a new exchange method for solving convex semi-infinite programming problems (SIPs). The traditional exchange method solves a sequence of finitely relaxed subproblems, that is, subproblems with finitely many constraints chosen from the original constraints. On the other hand, our exchange method solves a sequence of new subproblems, in which the traditional finite subproblems are refined by the quadratic approximation. Under mild assumptions, the refined subproblems approximate the original SIP more precisely than the traditional subproblems. Moreover, although those subproblems are still SIPs, they can be solved efficiently by reformulating them as certain optimization problems with finitely many constraints. We establish the global convergence property of the proposed algorithm. Finally, we examine the efficiency of the algorithm by some numerical experiments.