arXiv subjects

Sunyoung Kim

Publications and source records attributed to Sunyoung Kim.

28 records · Page 2Linked to original sources

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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arXiv2020

Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity matrix from the data matrices of a QCQP with $n$ variables, the rank and positive semidefiniteness of the matrix are examined. We prove that if the rank of the aggregate sparsity matrix is not less than $n-1$ and the matrix remains positive semidefinite after replacing some off-diagonal nonzero elements with zeros, then the standard SDP relaxation provides an exact optimal solution for the QCQP under feasibility assumptions. In particular, we demonstrate that QCQPs with forest-structured aggregate sparsity matrix, such as the tridiagonal or arrow-type matrix, satisfy the exactness condition on the rank. The exactness is attained by considering the feasibility of the dual SDP relaxation, the strong duality of SDPs, and a sequence of QCQPs with perturbed objective functions, under the assumption that the feasible region is compact. We generalize our result for a wider class of QCQPs by applying simultaneous tridiagonalization on the data matrices. Moreover, simultaneous tridiagonalization is applied to a matrix pencil so that QCQPs with two constraints can be solved exactly by the SDP relaxation.

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arXiv2020

CareCall: a Call-Based Active Monitoring Dialog Agent for Managing COVID-19 Pandemic

Tracking suspected cases of COVID-19 is crucial to suppressing the spread of COVID-19 pandemic. Active monitoring and proactive inspection are indispensable to mitigate COVID-19 spread, though these require considerable social and economic expense. To address this issue, we introduce CareCall, a call-based dialog agent which is deployed for active monitoring in Korea and Japan. We describe our system with a case study with statistics to show how the system works. Finally, we discuss a simple idea which uses CareCall to support proactive inspection.

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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

A dual spectral projected gradient method for log-determinant semidefinite problems

We extend the result on the spectral projected gradient method by Birgin et al. in 2000 to a log-determinant semidefinite problem (SDP) with linear constraints and propose a spectral projected gradient method for the dual problem. Our method is based on alternate projections on the intersection of two convex sets, which first projects onto the box constraints and then onto a set defined by a linear matrix inequality. By exploiting structures of the two projections, we show the same convergence properties can be obtained for the proposed method as Birgin's method where the exact orthogonal projection onto the intersection of two convex sets is performed. Using the convergence properties, we prove that the proposed algorithm attains the optimal value or terminates in a finite number of iterations. The efficiency of the proposed method is illustrated with the numerical results on randomly generated synthetic/deterministic data and gene expression data, in comparison with other methods including the inexact primal-dual path-following interior-point method, the adaptive spectral projected gradient method, and the adaptive Nesterov's smooth method. For the gene expression data, our results are compared with the quadratic approximation for sparse inverse covariance estimation method. We show that our method outperforms the other methods in obtaining a better optimal value fast.

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arXiv2018

Solving Pooling Problems by LP and SOCP Relaxations and Rescheduling Methods

The pooling problem is an important industrial problem in the class of network flow problems for allocating gas flow in pipeline transportation networks. For P-formulation of the pooling problem with time discretization, we propose second order cone programming (SOCP) and linear programming (LP) relaxations and prove that they obtain the same optimal value as the semidefinite programming relaxation. The equivalence among the optimal values of the three relaxations is also computationally shown. Moreover, a rescheduling method is proposed to efficiently refine the solution obtained by the SOCP or LP relaxation. The efficiency of the SOCP and the LP relaxation and the proposed rescheduling method is illustrated with numerical results on the test instances from the work of Nishi in 2010, some large instances, and Foulds 3, 4, 5 test 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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