arXiv subjects

Naohiko Arima

Publications and source records attributed to Naohiko Arima.

6 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

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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