Shunsuke Hayashi

A Combined Smoothing and Regularization Method for Monotone Second-Order Cone Complementarity Problems

The second-order cone complementarity problem (SOCCP) is a wide class of problems containing the nonlinear complementarity problem (NCP) and the second-order cone programming problem (SOCP). Recently, Fukushima, Luo, and Tseng [SIAM J. Optim., 12 (2001), pp. 436--460] extended some merit functions and their smoothing functions for NCP to SOCCP. Moreover, they derived computable formulas for the Jacobians of the smoothing functions and gave conditions for the Jacobians to be invertible. In this paper, we propose a globally and quadratically convergent algorithm, which is based on smoothing and regularization methods, for solving monotone SOCCP. In particular, we study strong semismoothness and Jacobian consistency, which play an important role in establishing quadratic convergence of the algorithm. Furthermore, we examine the effectiveness of the algorithm by means of numerical experiments.

Dynamic Spectrum Management: When is FDMA Sum-Rate Optimal?

Consider a multiuser communication system in a frequency selective environment whereby users share a common spectrum and can interfere with each other. Assuming Gaussian signaling and treating interference as noise, we study optimal spectrum sharing strategies for the maximization of weighted sum-rate. In this work, we show that, if the normalized crosstalk gains are larger than a given threshold (roughly equal to 1/2), then the optimal spectrum sharing strategy is frequency division multiple access (FDMA). We also propose several simple distributed spectrum allocation algorithms that can approximately maximize weighted sum-rates. Numerical simulation of DSL applications shows that these algorithms are efficient and can achieve substantially larger weighted sumrates than those obtained by the existing iterative waterfilling algorithm.

The Corridor Problem with Discrete Multiple Bottlenecks

This paper presents a transparent approach to the analysis of dynamic user equilibrium and clarifies the properties of a departuretime choice equilibrium of a corridor problem where discrete multiple bottlenecks exist along a freeway. The basis of our approach is the transformation of the formulation of equilibrium conditions in a conventional “Eulerian coordinate system” into one in a “Lagrangian-like coordinate system.” This enables us to evaluate dynamic travel times easily, and to achieve a deep understanding of the mathematical structure of the problem, in particular, about the properties of the demand and supply (queuing) sub-models, relations with dynamic system optimal assignment, and differences between the morning and evening rush problems. Building on these foundations, we establish rigorous results on the existence and uniqueness of equilibria. c

Semidefinite complementarity reformulation for robust Nash equilibrium problems with Euclidean uncertainty sets

Consider the N-person non-cooperative game in which each player’s cost function and the opponents’ strategies are uncertain. For such an incomplete information game, the new solution concept called a robust Nash equilibrium has attracted much attention over the past several years. The robust Nash equilibrium results from each player’s decision-making based on the robust optimization policy. In this paper, we focus on the robust Nash equilibrium problem in which each player’s cost function is quadratic, and the uncertainty sets for the opponents’ strategies and the cost matrices are represented by means of Euclidean and Frobenius norms, respectively. Then, we show that the robust Nash equilibrium problem can be reformulated as a semidefinite complementarity problem (SDCP), by utilizing the semidefinite programming (SDP) reformulation technique in robust optimization. We also give some numerical example to illustrate the behavior of robust Nash equilibria.

SDP reformulation for robust optimization problems based on nonconvex QP duality

In a real situation, optimization problems often involve uncertain parameters. Robust optimization is one of distribution-free methodologies based on worst-case analyses for handling such problems. In this paper, we first focus on a special class of uncertain linear programs (LPs). Applying the duality theory for nonconvex quadratic programs (QPs), we reformulate the robust counterpart as a semidefinite program (SDP) and show the equivalence property under mild assumptions. We also apply the same technique to the uncertain second-order cone programs (SOCPs) with “single” (not side-wise) ellipsoidal uncertainty. Then we derive similar results on the reformulation and the equivalence property. In the numerical experiments, we solve some test problems to demonstrate the efficiency of our reformulation approach. Especially, we compare our approach with another recent method based on Hildebrand’s Lorentz positivity.

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...

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

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.

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.

A Smoothing Method with Appropriate Parameter Control Based on Fischer-Burmeister Function for Second-Order Cone Complementarity Problems

We deal with complementarity problems over second-order cones. The complementarity problem is an important class of problems in the real world and involves many optimization problems. The complementarity problem can be reformulated as a nonsmooth system of equations. Based on the smoothed Fischer-Burmeister function, we construct a smoothing Newton method for solving such a nonsmooth system. The proposed method controls a smoothing parameter appropriately. We show the global and quadratic convergence of the method. Finally, some numerical results are given.

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.