Papers by Masakazu Kojima
Journal of Global Optimization, Jan 30, 2020
We study the equivalence among a nonconvex QOP, its CPP and DNN relaxations under the assumption ... more We study the equivalence among a nonconvex QOP, its CPP and DNN relaxations under the assumption that the aggregated and correlative sparsity of the data matrices of the CPP relaxation is represented by a block-clique graph G. By exploiting the correlative sparsity, we decompose the CPP relaxation problem into a cliquetree structured family of smaller subproblems. Each subproblem is associated with a node of a clique tree of G. The optimal value can be obtained by applying an algorithm that we propose for solving the subproblems recursively from leaf nodes to the root node of the clique-tree. We establish the equivalence between the QOP and its DNN relaxation from the equivalence between the reduced family of subproblems and their DNN relaxations by applying the known results on: (i) CPP and DNN reformulation of a class of QOPs with linear equality, complementarity and binary constraints in 4 nonnegative variables. (ii) DNN reformulation of a class of quadratically constrained convex QOPs with any size. (iii) DNN reformulation of LPs with any size. As a result, we show that a QOP whose subproblems are the QOPs mentioned in (i), (ii) and (iii) is equivalent to its DNN relaxation, if the subproblems form a clique-tree structured family induced from a block-clique graph.
arXiv (Cornell University), May 29, 2019
For the Lagrangian-DNN relaxation of quadratic optimization problems (QOPs), we propose a Newton-... more For the Lagrangian-DNN relaxation of quadratic optimization problems (QOPs), we propose a Newton-bracketing method to improve the performance of the bisectionprojection method implemented in BBCPOP [to appear in ACM Tran. Softw., 2019]. The relaxation problem is converted into the problem of finding the largest zero y * of a continuously differentiable (except at y *) convex function g : R → R such that g(y) = 0 if y ≤ y * and g(y) > 0 otherwise. In theory, the method generates lower and upper bounds of y * both converging to y *. Their convergence is quadratic if the right derivative of g at y * is positive. Accurate computation of g (y) is necessary for the robustness of the method, but it is difficult to achieve in practice. As an alternative, we present a secant-bracketing method. We demonstrate that the method improves the quality of the lower bounds obtained by BBCPOP and SDPNAL+ for binary QOP instances from BIQMAC. Moreover, new lower bounds for the unknown optimal values of large scale QAP instances from QAPLIB are reported.
数理解析研究所講究録, 1994
Abstract. We present a variational condition for a globaloptimality to a $ cIass $ of quasiconvex... more Abstract. We present a variational condition for a globaloptimality to a $ cIass $ of quasiconvex mininization problemswhere vanishing gradients are not enough to the optimality. Using this condition we show that a quasiconvex minimization problem, whose ...
arXiv (Cornell University), Oct 28, 2022
Tai256c is the largest unsolved quadratic assignment problem (QAP) instance in QAPLIB; a 1.48% ga... more Tai256c is the largest unsolved quadratic assignment problem (QAP) instance in QAPLIB; a 1.48% gap remains between the best known feasible objective value and lower bound of the unknown optimal value. This paper shows that the instance can be converted into a 256 dimensional binary quadratic optimization problem (BQOP) with a single cardinality constraint which requires the sum of the binary variables to be 92. The converted BQOP is much simpler than the original QAP tai256c and it also inherits some of the symmetry properties. However, it is still very difficult to solve. We present an efficient branch and bound method for improving the lower bound effectively. A new lower bound with 1.36% gap is also provided.
To solve a partial differential equation (PDE) numerically, we formulate it as a polynomial optim... more To solve a partial differential equation (PDE) numerically, we formulate it as a polynomial optimization problem (POP) by discretizing it via a finite difference approximation. The resulting POP satisfies a structured sparsity, which we can exploit to apply the sparse SDP relaxation of Waki, Kim, Kojima and Muramatsu [20] to the POP to obtain a roughly approximate solution of the PDE. To compute a more accurate solution, we incorporate a grid-refining method with repeated applications of the sparse SDP relaxation or Newton’s method. The main features of this approach are: (a) we can choose an appropriate objective function, and (b) we can add inequality constraints on the unknown variables and their derivatives. These features make it possible for us to compute a specific solution when the PDE has multiple solutions. Some numerical results on the proposed method applied to ordinary differential equations, PDEs, differential algebraic equations and an optimal control problem are repo...
Journal of the Operations Research Society of Japan, 2008
The aim of this paper is to study how eMciently we evaluate a system of mu]tivariate polynomials ... more The aim of this paper is to study how eMciently we evaluate a system of mu]tivariate polynomials and their partial derivatives in homotopy continuation methods, Our major tool is an extension of the Hor ner scheme, which is popular in evaluating a univariate polynomial, to a multivariate polynomial. But the extension is not unique, and there are many Horner factorizations of a given multivariate polynomial which require different numbers of mukiplications. Wle present exact method for computing a minimum Horner factorization, which can process smaller size polynomials, as well as heuristic methods for a smaller number of multiplications, which can process larger $ize polynemials. Based on these Horner factorization methods, we then present method$ to evaluate a system of multivariate po]ynomials and their partia] derivatives. Numerical results are shown to verify the effectiveness and the eficieney of the proposed methods.
Journal of the Operations Research Society of Japan, 1988
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Journal of the Operations Research Society of Japan, 1984
Journal of the Operations Research Society of Japan, 1986
This paper extends Newton and quasi-Newton methods to systerns ofPCi equations and establishes th... more This paper extends Newton and quasi-Newton methods to systerns ofPCi equations and establishes the quadratic convergence property of the extended Newton method and the Q-superlinear convergence property of the extended quasi-Newton method.
~~~~~~~ ~~~~-~~~~~Ĩ n this paper, we consider several deterministic models for a stochastic linea... more ~~~~~~~ ~~~~-~~~~~Ĩ n this paper, we consider several deterministic models for a stochastic linear program with a constant feasible region and stochastic cost coefficients having multi-variate normal distribution. Relationships among the solutions of these models are examined and it is shown that solving a parametric quadratic program associated with Markowitz's mean-variance model yields solutions to all other models considered for all relevant values of parameters.
Page 1. Research Reports on Mathematical and Computing Sciences Series B : Operations Research De... more Page 1. Research Reports on Mathematical and Computing Sciences Series B : Operations Research Department of Mathematical and Computing Sciences Tokyo Institute of Technology 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-0033, Japan ...
Journal of the Operations Research Society of Japan
This short note discusses the so-called copositive program, a related problem to semidefinite pro... more This short note discusses the so-called copositive program, a related problem to semidefinite programs. Copositive programs are very hard problems. We review some known facts to approximately solve these problems by semidefinite programs or second-order cone programs, and provide hints for further developments.
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Papers by Masakazu Kojima