arXiv subjects

Masakazu Kojima

Publications and source records attributed to Masakazu Kojima.

18 recordsLinked to original sources

arXiv2026

Local-to-Global Exactness of SDP Relaxations for Sparse QCQPs

We study exact semidefinite programming (SDP) relaxation for a given sparse quadratically constrained quadratic program (QCQP). The SDP relaxation is exact if, whenever it has an optimal solution, it admits a rank-at-most-one optimal solution that corresponds to an optimal solution of the QCQP. Using the maximal cliques of a chordal extension of the aggregate sparsity pattern graph of the data matrices, we formulate the SDP relaxation in terms of clique-wise matrix variables and develop a local-to-global framework for certifying exactness. For each clique-wise matrix variable, we introduce a local sub-SDP with two parameters: a local right-hand-side vector and a consistency matrix specifying the values of entries shared by overlapping clique-wise matrix variables. In the main theorem, these parameters are determined by an optimal solution of the global clique-wise SDP. The theorem shows that if the resulting local sub-SDPs are exact, then the original SDP relaxation is exact. Under the additional assumption that any two distinct cliques intersect in at most one node, we present three classes of local QCQPs that can be incorporated into this framework: convex local QCQPs, local QCQPs characterized by sign-pattern conditions, and separable local QCQPs with a limited number of constraints. Examples illustrate how these different local QCQP classes can be combined in sparse QCQPs.

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arXiv2026

Exact SDP relaxations for a class of quadratic programs with finite and infinite quadratic constraints

We investigate exact semidefinite programming (SDP) relaxations for the problem of minimizing a nonconvex quadratic objective function over a feasible region defined by both finitely and infinitely many nonconvex quadratic inequality constraints (semi-infinite QCQPs). Sufficient conditions for the exactness of SDP relaxations for QCQPs with finitely many constraints have been extensively studied, notably by Argue et al. (MOR, 48:100-126, 2023), Arima et al. (SIOPT, 34:3194-3211, 2024), and Joyce and Yang (MP, 205:539-558, 2024). In this work, we present three new sufficient conditions that generalize the existing conditions in these works for both finite and semi-infinite QCQPs. Specifically, we establish relationships among the proposed and existing conditions, and prove that one of the proposed conditions is the weakest among them, since it is implied by all the others. Illustrative examples are also provided to demonstrate the effectiveness of the proposed conditions in comparison to the existing ones.

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arXiv2026

Separable QCQPs and Their Exact SDP Relaxations

This paper studies exact semidefinite programming relaxations (SDPRs) for separable quadratically constrained quadratic programs (QCQPs). We consider the construction of a larger separable QCQP from multiple QCQPs with exact SDPRs. We show that exactness is preserved when such QCQPs are combined through a separable horizontal connection, where the coupling is induced through the right-hand-side parameters of the constraints. The proposed framework provides a simple sufficient condition for exactness of the resulting SDPR. We then identify notable classes of QCQPs for which this condition holds, including convex QCQPs, QCQPs defined by sign-pattern and graph-structural conditions, and separable homogeneous QCQPs with a limited number of constraints. Two examples illustrate the constructive nature of the proposed framework, showing how heterogeneous QCQPs can be combined to yield new instances with exact SDP relaxations.

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arXiv2025

Extending Exact Convex Relaxations of Quadratically Constrained Quadratic Programs

A convex relaxation of a quadratically constrained quadratic program (QCQP) is called exact if it has a rank-$1$ optimal solution that corresponds to an optimal solution of the QCQP. Given a QCQP whose convex relaxation is exact, this paper investigates the incorporation of additional quadratic inequality constraints under a non-intersecting quadratic constraint condition while maintaining the exactness of the convex relaxation of the resulting QCQP. Specifically, we extend existing exact semidefinite programming relaxation, completely positive programming relaxation and doubly nonnegative programming relaxation of various classes of QCQPs in a unified manner. Illustrative examples are included to demonstrate the applicability of the established result.

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arXiv2025

Constructing QCQP Instances Equivalent to Their SDP Relaxations

General quadratically constrained quadratic programs (QCQPs) are challenging to solve as they are known to be NP-hard. A popular approach to approximating QCQP solutions is to use semidefinite programming (SDP) relaxations. It is well-known that the optimal value $η$ of the SDP relaxation problem bounds the optimal value $ζ$ of the QCQP from below, i.e., $η\leq ζ$. The two problems are considered equivalent if $η= ζ$. In the recent paper by Arima, Kim and Kojima [arXiv:2409.07213], a class of QCQPs that are equivalent to their SDP relaxations are proposed with no condition imposed on the quadratic objective function, which can be chosen arbitrarily. In this work, we explore the construction of various QCQP instances within this class to complement the results in [arXiv:2409.07213]. Specifically, we first construct QCQP instances with two variables and then extend them to higher dimensions. We also discuss how to compute an optimal QCQP solution from the SDP relaxation.

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arXiv2024

An Exceptionally Difficult Binary Quadratic Optimization Problem with Symmetry: a Challenge for The Largest Unsolved QAP Instance Tai256c

Tai256c is the largest unsolved quadratic assignment problem (QAP) instance in QAPLIB. It is known that QAP tai256c 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. As the BQOP is much simpler than the original QAP, the conversion increases the possibility to solve the QAP. Solving exactly the BQOP, however, is still very difficult. Indeed, a 1.48\% gap remains between the best known upper bound (UB) and lower bound (LB) of the unknown optimal value. This paper shows that the BQOP admits a nontrivial symmetry, a property that makes the BQOP very hard to solve. The symmetry induces equivalent subproblems in branch and bound (BB) methods. To effectively improve the LB, we propose an efficient BB method that incorporates a doubly nonnegative relaxation, the standard orbit branching and a technique to prune equivalent subproblems. With this BB method, a new LB with 1.25\% gap is successfully obtained, and computing an LB with $1.0\%$ gap is shown to be still quite difficult.

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arXiv2024

Further Development in Convex Conic Reformulation of Geometric Nonconvex Conic Optimization Problems

A geometric nonconvex conic optimization problem (COP) was recently proposed by Kim, Kojima and Toh as a unified framework for convex conic reformulation of a class of quadratic optimization problems and polynomial optimization problems. The nonconvex COP minimizes a linear function over the intersection of a nonconvex cone $\mathbb{K}$, a convex subcone $\mathbb{J}$ of the convex hull co$\mathbb{K}$ of $\mathbb{K}$, and an affine hyperplane with a normal vector $H$. Under the assumption co$(\mathbb{K} \cap \mathbb{J}) = \mathbb{J}$, the original nonconvex COP in their paper was shown to be equivalently formulated as a convex conic program by replacing the constraint set with the intersection of $\mathbb{J}$ and the affine hyperplane. This paper further studies some remaining issues, not fully investigated there, such as the key assumption co$(\mathbb{K} \cap \mathbb{J}) = \mathbb{J}$ in the framework. More specifically, we provide three sets of necessary-sufficient conditions for the assumption. As an application, we propose a new wide class of quadratically constrained quadratic programs with multiple nonconvex equality and inequality constraints that can be solved exactly by their semidefinite relaxation.

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arXiv2023

Equivalent Sufficient Conditions for Global Optimality of Quadratically Constrained Quadratic Program

We study the equivalence of several well-known sufficient optimality conditions for a general quadratically constrained quadratic program (QCQP). The conditions are classified in two categories. The first one is for determining an optimal solution and the second one is for finding an optimal value. The first category of conditions includes the existence of a saddle point of the Lagrangian function and the existence of a rank-1 optimal solution of the primal SDP relaxation of QCQP. The second category includes $η_p = ζ$, $η_d = ζ$, and $φ= ζ$, where $ζ$, $η_p$, $η_d$, and $φ$ denote the optimal values of QCQP, the dual SDP relaxation, the primal SDP relaxation and the Lagrangian dual, respectively. We show the equivalence of these conditions with or without the existence of an optimal solution of QCQP and/or the Slater constraint qualification for the primal SDP relaxation. The results on the conditions are also extended to the doubly nonnegative relaxation of equality constrained QCQP in nonnegative variables.

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arXiv2022

The Largest Unsolved QAP Instance Tai256c Can Be Converted into A 256-dimensional Simple BQOP with A Single Cardinality Constraint

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.

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arXiv2022

Strong duality of a conic optimization problem with a single hyperplane and two cone constraints

Strong (Lagrangian) duality of general conic optimization problems (COPs) has long been studied and its profound and complicated results appear in different forms in a wide range of literatures. As a result, characterizing the known and unknown results can sometimes be difficult. The aim of this article is to provide a unified and geometric view of strong duality of COPs for the known results. For our framework, we employ a COP minimizing a linear function in a vector variable $x$ subject to a single hyperplane constraint $x \in H$ and two cone constraints $x \in K_1$, $x \in K_2$. It can be identically reformulated as a simpler COP with the single hyperplane constraint $x \in H$ and the single cone constraint $x \in K_1 \cap K_2$. This simple COP and its dual as well as their duality relation can be represented geometrically, and they have no duality gap without any constraint qualification. The dual of the original target COP is equivalent to the dual of the reformulated COP if the Minkowski sum of the duals of the two cones $K_1$ and $K_2$ is closed or if the dual of the reformulated COP satisfies a certain Slater condition. Thus, these two conditions make it possible to transfer all duality results, including the existence and/or boundedness of optimal solutions, on the reformulated COP to the ones on the original target COP, and further to the ones on a standard primal-dual pair of COPs with symmetry.

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arXiv2021

Generating Cutting Inequalities Successively for Quadratic Optimization Problems in Binary Variables

We propose a successive generation of cutting inequalities for binary quadratic optimization problems. Multiple cutting inequalities are successively generated for the convex hull of the set of the optimal solutions $\subset \{0, 1\}^n$, while the standard cutting inequalities are used for the convex hull of the feasible region. An arbitrary linear inequality with integer coefficients and the right-hand side value in integer is considered as a candidate for a valid inequality. The validity of the linear inequality is determined by solving a conic relaxation of a subproblem such as the doubly nonnegative relaxation, under the assumption that an upper bound for the unknown optimal value of the problem is available. Moreover, the candidates generated for the multiple cutting inequalities are tested simultaneously for their validity in parallel. Preliminary numerical results on 60 quadratic unconstrained binary optimization problems with a simple implementation of the successive cutting inequalities using an 8- or 32-core machine show that the exact optimal values are obtained for 70\% of the tested problems, demonstrating the strong potential of the proposed technique.

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arXiv2021

Solving Challenging Large Scale QAPs

We report our progress on the project for solving larger scale quadratic assignment problems (QAPs). Our main approach to solve large scale NP-hard combinatorial optimization problems such as QAPs is a parallel branch-and-bound method efficiently implemented on a powerful computer system using the Ubiquity Generator (UG) framework that can utilize more than 100,000 cores. Lower bounding procedures incorporated in the branch-and-bound method play a crucial role in solving the problems. For a strong lower bounding procedure, we employ the Lagrangian doubly nonnegative (DNN) relaxation and the Newton-bracketing method developed by the authors' group. In this report, we describe some basic tools used in the project including the lower bounding procedure and branching rules, and present some preliminary numerical results. Our next target problem is QAPs with dimension at least 50, as we have succeeded to solve tai30a and sko42 from QAPLIB for the first time.

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arXiv2019

A Newton-bracketing method for a simple conic optimization problem

For the Lagrangian-DNN relaxation of quadratic optimization problems (QOPs), we propose a Newton-bracketing method to improve the performance of the bisection-projection 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 $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g(y) = 0$ if $y \leq y^*$ 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.

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arXiv2019

Doubly nonnegative relaxations are equivalent to completely positive reformulations of quadratic optimization problems with block-clique graph structures

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 clique-tree 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.

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arXiv2019

A Geometrical Analysis of a Class of Nonconvex Conic Programs for Convex Conic Reformulations of Quadratic and Polynomial Optimization Problems

We present a geometrical analysis on the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems with a class of geometrically defined nonconvex conic programs and their covexification. The class of nonconvex conic programs is described with a linear objective functionin a linear space $V$, and the constraint set is represented geometrically as the intersection of a nonconvex cone $K \subset V$, a face $J$ of the convex hull of $K$ and a parallel translation $L$ of a supporting hyperplane of the nonconvex cone $K$. We show that under a moderate assumption, the original nonconvex conic program can equivalently be reformulated as a convex conic program by replacing the constraint set with the intersection of $J$ and the hyperplane $L$. The replacement procedure is applied to derive the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems.

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arXiv2018

BBCPOP: A Sparse Doubly Nonnegative Relaxation of Polynomial Optimization Problems with Binary, Box and Complementarity Constraints

The software package BBCPOP is a MATLAB implementation of a hierarchy of sparse doubly nonnegative (DNN) relaxations of a class of polynomial optimization (minimization) problems (POPs) with binary, box and complementarity (BBC) constraints. Given a POP in the class and a relaxation order, BBCPOP constructs a simple conic optimization problem (COP), which serves as a DNN relaxation of the POP, and then solves the COP by applying the bisection and projection (BP) method. The COP is expressed with a linear objective function and constraints described as a single hyperplane and two cones, which are the Cartesian product of positive semidefinite cones and a polyhedral cone induced from the BBC constraints. BBCPOP aims to compute a tight lower bound for the optimal value of a large-scale POP in the class that is beyond the comfort zone of existing software packages. The robustness, reliability and efficiency of BBCPOP are demonstrated in comparison to the state-of-the-art software SDP package SDPNAL+ on randomly generated sparse POPs of degree 2 and 3 with up to a few thousands variables, and ones of degree 4, 5, 6. and 8 with up to a few hundred variables. Comparison with other BBC POPs that arise from combinatorial optimization problems such as quadratic assignment problems are also reported. The software package BBCPOP is available at https://sites.google.com/site/bbcpop1/.

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arXiv2000

Some Fundamental Properties of Successive Convex Relaxation Methods on LCP and Related Problems

General Successive Convex Relaxation Methods (SRCMs) can be used to compute the convex hull of any compact set, in an Euclidean space, described by a system of quadratic inequalities and a compact convex set which is not very complicated. Linear Complementarity Problems (LCPs) make an interesting and rich class of structured nonconvex optimization problems. In this paper, we study a few of the specialized lift-and-project methods and some of the possible ways of applying the general SCRMs to LCPs and related problems.

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arXiv1999

Some fundamental properties of successive convex relaxation methods on LCP and related problems

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