arXiv subjects

Sunyoung Kim

Publications and source records attributed to Sunyoung Kim.

At least 19 recordsLinked to original sources

arXiv2026

Order-2 Tightness of Block-Sparse SOS Relaxations for One-Layer ReLU Network Verification with a Matching Input-Sharing Graph

Azuma, Kim, and Yamashita formulated the verification problem for one-layer ReLU networks as a quadratically constrained quadratic program and established tight semidefinite relaxations for the edgeless case and for one-unit settings. In this work, we represent the sharing pattern of undecided ReLUs over a box input set through an input-sharing graph and focus on the case where this graph is a matching. We then derive an explicit, checkable sufficient condition for the tightness of the order-$2$ block-sparse SOS relaxation associated with the connected-component decomposition of this graph. Under the matching assumption, the global problem decomposes into isolated-vertex blocks and single-edge blocks. The key difficulty, which is absent from the edgeless case, is establishing tightness for a two-unit edge block. For regular rank-one edges, we show that the convex hull of each two-unit local set can be described exactly by two reduced one-unit hulls coupled through a common shared scalar. Combining the one-unit tightness result of Azuma et al. with Farkas' lemma and affine elimination, we obtain a local order-$2$ certificate for each edge block. Isolated-vertex blocks reduce to one-unit problems over box input sets and are therefore handled at the same order. We prove that, when the input-sharing graph is a matching and every edge satisfies the regular rank-one condition, the order-$2$ block-sparse SOS relaxation is tight. This extends the tight sparse relaxation result for the edgeless case to the first sparse setting with a nontrivial two-unit interaction.

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arXiv2026

Tight Conic Relaxations for Rank-one Doubly Nonnegative Matrix Completion

We study tight conic relaxations for a quadratically constrained quadratic programming (QCQP) formulation of rank-one doubly nonnegative (DNN) matrix completion. Motivated by sparse QCQPs whose lifted matrix variables include elements not directly specified by the objective or constraints, we interpret tightness as a rank-one completion property for the unspecified elements. For sparsity patterns whose blocks consist of cycles and edges, we prove that the dual formulations associated with the DNN and completely positive (CP) relaxations are equivalent. For cycle-type sparsity patterns, we derive explicit sufficient conditions under which the semidefinite programming (SDP) and DNN relaxations are tight. These sufficient conditions are stated explicitly in terms of local ratio bounds and cumulative-difference conditions on a rank-one optimal solution. We also show that adding suitable edges to the sparsity pattern relaxes the ratio conditions required for tightness. The results provide tractable certificates for when conic relaxations recover a rank-one optimal solution of the underlying QCQP.

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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

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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arXiv2026

Benchmarking Visual Feature Representations for LiDAR-Inertial-Visual Odometry Under Challenging Conditions

Accurate localization in autonomous driving is critical for successful missions including environmental mapping and survivor searches. In visually challenging environments, including low-light conditions, overexposure, illumination changes, and high parallax, the performance of conventional visual odometry methods significantly degrade undermining robust robotic navigation. Researchers have recently proposed LiDAR-inertial-visual odometry (LIVO) frameworks, that integrate LiDAR, IMU, and camera sensors, to address these challenges. This paper extends the FAST-LIVO2-based framework by introducing a hybrid approach that integrates direct photometric methods with descriptor-based feature matching. For the descriptor-based feature matching, this work proposes pairs of ORB with the Hamming distance, SuperPoint with SuperGlue, SuperPoint with LightGlue, and XFeat with the mutual nearest neighbor. The proposed configurations are benchmarked by accuracy, computational cost, and feature tracking stability, enabling a quantitative comparison of the adaptability and applicability of visual descriptors. The experimental results reveal that the proposed hybrid approach outperforms the conventional sparse-direct method. Although the sparse-direct method often fails to converge in regions where photometric inconsistency arises due to illumination changes, the proposed approach still maintains robust performance under the same conditions. Furthermore, the hybrid approach with learning-based descriptors enables robust and reliable visual state estimation across challenging environments.

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arXiv2026

Access Controlled Website Interaction for Agentic AI with Delegated Critical Tasks

Recent studies reveal gaps in delegating critical tasks to agentic AI that accesses websites on the user's behalf, primarily due to limited access control mechanisms on websites designed for agentic AI. In response, we propose a design of website-based interaction for AI agents with fine-grained access control for delegated critical tasks. Our approach encompasses a website design and implementation, as well as modifications to the access grant protocols in an open-source authorization service to tailor it to agentic AI, with delegated critical tasks on the website. The evaluation of our approach demonstrates the capabilities of our access-controlled website used by AI agents.

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arXiv2025

Tight Semidefinite Relaxations for Verifying Robustness of Neural Networks

For verifying the safety of neural networks (NNs), Fazlyab et al. (2019) introduced a semidefinite programming (SDP) approach called DeepSDP. This formulation can be viewed as the dual of the SDP relaxation for a problem formulated as a quadratically constrained quadratic program (QCQP). While SDP relaxations of QCQPs generally provide approximate solutions with some gaps, this work focuses on tight SDP relaxations that provide exact solutions to the QCQP for single-layer NNs. Specifically, we analyze tightness conditions in three cases: (i) NNs with a single neuron, (ii) single-layer NNs with an ellipsoidal input set, and (iii) single-layer NNs with a rectangular input set. For NNs with a single neuron, we propose a condition that ensures the SDP admits a rank-1 solution to DeepSDP by transforming the QCQP into an equivalent two-stage problem leads to a solution collinear with a predetermined vector. For single-layer NNs with an ellipsoidal input set, the collinearity of solutions is proved via the Karush-Kuhn-Tucker condition in the two-stage problem. In case of single-layer NNs with a rectangular input set, we demonstrate that the tightness of DeepSDP can be reduced to the single-neuron NNs, case (i), if the weight matrix is a diagonal matrix.

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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 $\eta$ of the SDP relaxation problem bounds the optimal value $\zeta$ of the QCQP from below, i.e., $\eta \leq \zeta$. The two problems are considered equivalent if $\eta = \zeta$. 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

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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arXiv2024

T-semidefinite programming relaxation with third-order tensors for constrained polynomial optimization

We study T-semidefinite programming (SDP) relaxation for constrained polynomial optimization problems (POPs). T-SDP relaxation for unconstrained POPs was introduced by Zheng, Huang and Hu in 2022. In this work, we propose a T-SDP relaxation for POPs with polynomial inequality constraints and show that the resulting T-SDP relaxation formulated with third-order tensors can be transformed into the standard SDP relaxation with block-diagonal structures. The convergence of the T-SDP relaxation to the optimal value of a given constrained POP is established under moderate assumptions as the relaxation level increases. Additionally, the feasibility and optimality of the T-SDP relaxation are discussed. Numerical results illustrate that the proposed T-SDP relaxation enhances numerical efficiency.

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arXiv2023

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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arXiv2023

Exact Matrix Completion via High-Rank Matrices in Sum-of-Squares Relaxations

We study exact matrix completion from partially available data with hidden connectivity patterns. Exact matrix completion was shown to be possible recently by Cosse and Demanet in 2021 with Lasserre's relaxation using the trace of the variable matrix as the objective function with given data structured in a chain format. In this study, we introduce a structure for the objective function so that the resulting sum-of-squares (SOS) relaxation, the dual of Lasserre's SDP relaxation, produces a rank-($N$-1) solution, where $N$ denotes the size of variable matrix in the SOS relaxation. Specifically, the arrowhead structure is employed for the coefficient matrix of the objective function. We show that a matrix can be exactly completed through the SOS relaxation when the connectivity of given data is not explicitly displayed or follows a chain format. The theoretical exactness is proved using the rank of the Gram matrix for the SOS relaxation. We also present numerical algorithms designed to find the coefficient matrix in the SOS relaxation. Numerical experiments illustrate the validity of the proposed algorithm.

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arXiv2023

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 $\eta_p = \zeta$, $\eta_d = \zeta$, and $\varphi = \zeta$, where $\zeta$, $\eta_p$, $\eta_d$, and $\varphi$ 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

Exact SDP relaxations for quadratic programs with bipartite graph structures

For nonconvex quadratically constrained quadratic programs (QCQPs), we first show that, under certain feasibility conditions, the standard semidefinite (SDP) relaxation is exact for QCQPs with bipartite graph structures. The exact optimal solutions are obtained by examining the dual SDP relaxation and the rank of the optimal solution of this dual SDP relaxation under strong duality. Our results on the QCQPs generalize the results on QCQP with sign-definite bipartite graph structures, QCQPs with forest structures, and QCQPs with nonpositive off-diagonal data elements. Second, we propose a conversion method from QCQPs with no particular structure to the ones with bipartite graph structures. As a result, we demonstrate that a wider class of QCQPs can be exactly solved by the SDP relaxation. Numerical instances are presented for illustration.

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arXiv2021

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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