Sunyoung Kim

Sums of Squares and Semidefinite Program Relaxations for Polynomial Optimization Problems with Structured Sparsity

Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of the supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite program (SDP) relaxations are obtained. Numerical results from various test problems are included to show the improved performance of the SOS and SDP relaxations.

Algorithm 883: SparsePOP---A Sparse Semidefinite Programming Relaxation of Polynomial Optimization Problems

SparsePOP is a Matlab implementation of the sparse semidefinite programming (SDP) relaxation method for approximating a global optimal solution of a polynomial optimization problem (POP) proposed by Waki et al. [2006]. The sparse SDP relaxation exploits a sparse structure of polynomials in POPs when applying “a hierarchy of LMI relaxations of increasing dimensions” Lasserre [2006]. The efficiency of SparsePOP to approximate optimal solutions of POPs is thus increased, and larger-scale POPs can be handled.

Sparsity in sums of squares of polynomials

Abstract.Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming (SDP) relaxation of polynomial optimization problems. We discuss effective methods to obtain a simpler representation of a “sparse” polynomial as a sum of squares of sparse polynomials by eliminating redundancy.

PHoM – a Polyhedral Homotopy Continuation Method for Polynomial Systems

PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedral-linear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations f(x)=0. The second module CMPSc traces the solution curves of the homotopy equations to compute all isolated solutions of f(x)=0. The third module Verify checks whether all isolated solutions of f(x)=0 have been approximated correctly. We describe numerical methods used in each module and the usage of the package. Numerical results to demonstrate the performance of PHoM include some large polynomial systems that have not been solved previously.

Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion

A basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities. The sparsity, characterized with a chordal graph structure, can be detected in the variable matrix or in a linear or nonlinear matrix-inequality constraint of the problem. We classify the sparsity in two types, the domain-space sparsity (d-space sparsity) for the symmetric matrix variable in the objective and/or constraint functions of the problem, which is required to be positive semidefinite, and the range-space sparsity (r-space sparsity) for a linear or nonlinear matrix-inequality constraint of the problem. Four conversion methods are proposed in this framework: two for exploiting the d-space sparsity and the other two for exploiting the r-space sparsity. When applied to a polynomial semidefinite program (SDP), these conversion methods enhance the structured sparsity of the problem called the correlative sparsity. As a result, the resulting polynomial SDP can be solved more effectively by applying the sparse SDP relaxation. Preliminary numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems.

Generalized Lagrangian Duals and Sums of Squares Relaxations of Sparse Polynomial Optimization Problems

Sequences of generalized Lagrangian duals and their sums of squares (SOS) of polynomials relaxations for a polynomial optimization problem (POP) are introduced. The sparsity of polynomials in the POP is used to reduce the sizes of the Lagrangian duals and their SOS relaxations. It is proved that the optimal values of the Lagrangian duals in the sequence converge to the optimal value of the POP using a method from the penalty function approach. The sequence of SOS relaxations is transformed into a sequence of semidefinite programing (SDP) relaxations of the POP, which correspond to duals of modification and generalization of SDP relaxations given by Lasserre for the POP.

Computing all nonsingular solutions of cyclic- n polynomial using polyhedral homotopy continuation methods

All isolated solutions of the cyclic-n polynomial equations are not known for larger dimensions than 11. We exploit two types of symmetric structures in the cyclic-n polynomial to compute all isolated nonsingular solutions of the equations efficiently by the polyhedral homotopy continuation method and to verify the correctness of the generated approximate solutions. Numerical results on the cyclic-8 to the cyclic-12 polynomial equations, including their solution information, are given.

A Lagrangian---DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems

We propose an efficient computational method for linearly constrained quadratic optimization problems (QOPs) with complementarity constraints based on their Lagrangian and doubly nonnegative (DNN) relaxation and first-order algorithms. The simplified Lagrangian–completely positive programming (CPP) relaxation of such QOPs proposed by Arima, Kim, and Kojima in 2012 takes one of the simplest forms, an unconstrained conic linear optimization problem with a single Lagrangian parameter in a CPP matrix variable with its upper-left element fixed to 1. Replacing the CPP matrix variable by a DNN matrix variable, we derive the Lagrangian–DNN relaxation, and establish the equivalence between the optimal value of the DNN relaxation of the original QOP and that of the Lagrangian–DNN relaxation. We then propose an efficient numerical method for the Lagrangian–DNN relaxation using a bisection method combined with the proximal alternating direction multiplier and the accelerated proximal gradient methods. Numerical results on binary QOPs, quadratic multiple knapsack problems, maximum stable set problems, and quadratic assignment problems illustrate the superior performance of the proposed method for attaining tight lower bounds in shorter computational time.

Proteomic analysis of liver tissue from HBx-transgenic mice at early stages of hepatocarcinogenesis

The hepatitis B virus X‐protein (HBx), a multifunctional viral regulator, participates in the viral life cycle and in the development of hepatocellular carcinoma (HCC). We previously reported a high incidence of HCC in transgenic mice expressing HBx. In this study, proteomic analysis was performed to identify proteins that may be involved in hepatocarcinogenesis and/or that could be utilized as early detection biomarkers for HCC. Proteins from the liver tissue of HBx‐transgenic mice at early stages of carcinogenesis (dysplasia and hepatocellular adenoma) were separated by 2‐DE, and quantitative changes were analyzed. A total of 22 spots displaying significant quantitative changes were identified using LC‐MS/MS. In particular, several proteins involved in glucose and fatty acid metabolism, such as mitochondrial 3‐ketoacyl‐CoA thiolase, intestinal fatty acid‐binding protein 2 and cytoplasmic malate dehydrogenase, were differentially expressed, implying that significant metabolic alterations occurred during the early stages of hepatocarcinogenesis. The results of this proteomic analysis provide insights into the mechanism of HBx‐mediated hepatocarcinogenesis. Additionally, this study identifies possible therapeutic targets for HCC diagnosis and novel drug development for treatment of the disease.

Algorithm 920: SFSDP: A Sparse Version of Full Semidefinite Programming Relaxation for Sensor Network Localization Problems

SFSDP is a Matlab package for solving sensor network localization (SNL) problems. These types of problems arise in monitoring and controlling applications using wireless sensor networks. SFSDP implements the semidefinite programming (SDP) relaxation proposed in Kim et al. [2009] for sensor network localization problems, as a sparse version of the full semidefinite programming relaxation (FSDP) by Biswas and Ye [2004]. To improve the efficiency of FSDP, SFSDP exploits the aggregated and correlative sparsity of a sensor network localization problem. As a result, SFSDP can handle much larger problems than other software as well as three-dimensional anchor-free problems. SFSDP analyzes the input data of a sensor network localization problem, solves the problem, and displays the computed locations of sensors. SFSDP also includes the features of generating test problems for numerical experiments.