We propose a semidefinite relaxation for graph clustering known as Max-cut clustering. The clustering problem is formulated in terms of a discrete optimization problem and then relaxed to a SDP. The SDP is solved using a low-rank factorization trick that reduces the number of variables, and then using a simple projected gradient method. This is joint work with Nathan Srebro at the Toyota Techno...

Given an arbitrary matrix A in which all of the diagonal elements are zero, we would like to find x1, x2, . . . , xn ∈ {−1, 1} such that ∑n i=1 ∑n j=1 aijxixj is maximized. This problem has an important application in correlation clustering, and is also related to the well-known inequality of Grothendieck in functional analysis. While solving quadratic programs is NP-hard, we can approximate th...

The Komlós conjecture in discrepancy theory states that for some constant K and for any m× n matrix A whose columns lie in the unit ball there exists a vector x ∈ {−1,+1} such that ‖Ax‖∞ ≤ K. This conjecture also implies the Beck-Fiala conjecture on the discrepancy of bounded degree hypergraphs. Here we prove a natural relaxation of the Komlós conjecture: if the columns of A are assigned unit v...

SFSDP is a Matlab package for solving sensor network localization problems. The package contains four functions, SFSDP.m, SFSDPplus.m, generateProblem.m, test SFSDP.m, and some numerical examples. The function SFSDP.m is an Matlab implementation of the semidefinite programming (SDP) relaxation proposed in the recent paper by Kim, Kojima and Waki for sensor network localization problems, as a sp...

Recently, researchers have been interested in studying the semidefinite programming (SDP) relaxation model, where the matrix is both positive semidefinite and entry-wise nonnegative, for quadratically constrained quadratic programming (QCQP). Comparing to the basic SDP relaxation, this doubly-positive SDP model possesses additional O(n2) constraints, which makes the SDP solution complexity subs...

A reformulation of quadratically constrained binary programs as duals of set-copositive linear optimization problems is derived using either {0, 1}-formulations or {1, 1}-formulations. The latter representation allows an extension of the randomization technique by Goemans and Williamson. An application to the max-clique problem shows that the max-clique problem is equivalent to a linear program...

We consider the problem of scheduling unrelated parallel machines so as to minimize the total weighted completion time of jobs. Whereas the best previously known approximation algorithms for this problem are based on LP relaxations, we give a 2 –approximation algorithm that relies on a convex quadratic programming relaxation. For the special case of two machines we present a further improvement...

Building a linear fitting model for a given interval-valued data set is challenging since the minimization of the residue function leads to a huge combinatorial problem. To overcome such a difficulty, this article proposes a new semidefinite programming-based method for implementing linear fitting to interval-valued data. First, the fitting model is cast to a problem of quadratically constraine...

Given a 0-1 integer programming problem, several authors have introduced sequential relaxation techniques — based on linear and/or semidefinite programming — that generate the convex hull of integer points in at most n steps. In this paper, we introduce a sequential relaxation technique, which is based on p-order cone programming (1 ≤ p ≤ ∞). We prove that our technique generates the convex hul...

This short note extends the sparse SOS (sum of squares) and SDP (semidefinite programming) relaxation proposed by Waki, Kim, Kojima and Muramatsu for normal POPs (polynomial optimization problems) to POPs over symmetric cones, and establishes its theoretical convergence based on the recent convergence result by Lasserre on the sparse SOS and SDP relaxation for normal POPs. A numerical example i...