We consider the problem of maximizing a convex quadratic function over a bounded polyhedral set. We design a new framework based on SDP relaxations and cutting plane methods for solving the associated reference value problem. The major novelty is a new way to generate valid cuts through the doubly nonnegative (DNN) relaxation. We establish various theoretical properties of the DNN relaxation, including its equivalence with the Shor relaxation of an equivalent quadratically constrained problem, the strong duality, and the generation of valid cuts from an approximate solution of the DNN relaxation returned by an arbitrary SDP solver. Computational results on both real and synthetic data demonstrate the efficiency of the proposed method and its ability to solve high-dimensional problems with dense data. In particular, our new algorithm successfully solves in 3 days the reference value problem arising from computational biology for a dataset containing more than 300,000 instances of dimension 78. In contrast, CPLEX or Gurobi is estimated to require years of computational time for the same dataset on the same computing platform.
The equivalence between doubly nonnegative relaxation and semidefinite relaxation for binary quadratic programming problems
最近已经表明（Burer，Math. Program Ser.A120：479-495，2009），可以将一大类NP难的非凸二次规划问题建模为所谓的完全正规划问题，其是凸的但通常仍然是NP难的。通过双重非负松弛得到一个基本的易处理松弛，从而得到一个双重非负规划。本文证明了二元二次规划（BQP）问题的双重非负松弛等价于它的一个紧半正定松弛，当问题（BQP）化为最大割（MC）问题时，它的双重非负松弛等价于标准半正定松弛.此外，一些比较的数值结果报告。
The problem of minimizing a (nonconvex) quadratic form over the unit simplex, referred to as a standard quadratic program, admits an exact convex conic formulation over the computationally intractable cone of completely positive matrices. Replacing the intractable cone in this formulation by the larger but tractable cone of doubly nonnegative matrices, i.e., the cone of positive semidefinite and componentwise nonnegative matrices, one obtains the so-called doubly nonnegative relaxation, whose optimal value yields a lower bound on that of the original problem. We present a full algebraic charac…
This paper introduces an efficient algorithm for computing the best approximation of a given matrix onto the intersection of linear equalities, inequalities and the doubly nonnegative cone (the cone of all positive semidefinite matrices whose elements are nonnegative). In contrast to directly applying the block coordinate descent type methods, we propose an inexact accelerated (two-)block coordinate descent algorithm to tackle the four-block unconstrained nonsmooth dual program. The proposed algorithm hinges on the efficient semismooth Newton method to solve the subproblems, which have no clos…
We consider both facial reduction, \FRp, and symmetry reduction, \SRp, techniques for semidefinite programming, \SDPp. We show that the two together fit surprisingly well in an alternating direction method of multipliers, \ADMMp, approach. In fact, this approach allows for simply adding on nonnegativity constraints, and solving the doubly nonnegative, \DNN, relaxation of many classes of hard combinatorial problems. We also show that the singularity degree remains the same after \SRp, and that the \DNN relaxations considered here have singularity degree one, that is reduced to zero after \FRp. …
The doubly nonnegative (DNN) cone, being the set of all positive semidefinite matrices whose elements are nonnegative, is a popular approximation of the computationally intractable completely positive cone. The major difficulty for implementing a Newton-type method to compute the projection of a given large scale matrix onto the DNN cone lies in the possible failure of the constraint nondegeneracy, a generalization of the linear independence constraint qualification for nonlinear programming. Such a failure results in the singularity of the Jacobian of the nonsmooth equation representing the K…
Alternating direction methods of multipliers (ADMMs) are popular approaches to handle large scale semidefinite programs that gained attention during the past decade. In this paper, we focus on solving doubly nonnegative programs (DNN), which are semidefinite programs where the elements of the matrix variable are constrained to be nonnegative. Starting from two algorithms already proposed in the literature on conic programming, we introduce two new ADMMs by employing a factorization of the dual variable. It is well known that first order methods are not suitable to compute high precision optima…
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 a…
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-t…
In this paper we address the graph matching problem. Following the recent works of \cite{zaslavskiy2009path,Vestner2017} we analyze and generalize the idea of concave relaxations. We introduce the concepts of conditionally concave and probably conditionally concave energies on polytopes and show that they encapsulate many instances of the graph matching problem, including matching Euclidean graphs and graphs on surfaces. We further prove that local minima of probably conditionally concave energies on general matching polytopes (e.g., doubly stochastic) are with high probability extreme points …
In this paper, we study the polynomial optimization problem of multi-forms over the intersection of the multi-spheres and the nonnegative orthants. This class of problems is NP-hard in general, and includes the problem of finding the best nonnegative rank-one approximation of a given tensor. A Positivstellensatz is given for this class of polynomial optimization problems, based on which a globally convergent hierarchy of doubly nonnegative (DNN) relaxations is proposed. A (zero-th order) DNN relaxation method is applied to solve these problems, resulting in linear matrix optimization problems …
Two transforms of functions on a half-line are considered. It is proved that their composition gives a concave majorant for every nonnegative function. In particular, this composition is the identity transform on the class of nonnegative concave functions. Applications of this result to some problems of mathematical physics are indicated. Several open questions are formulated.
We revisit and strengthen splitting methods for solving doubly nonnegative, DNN, relaxations of the quadratic assignment problem, QAP. We use a modified restricted contractive splitting method, PRSM, approach. Our strengthened bounds and new dual multiplier estimates improve on the bounds and convergence results in the literature.
We study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful proper…
We formulate a doubly nonnegative (DNN) relaxation of the protein side-chain positioning (SCP) problem. We inherit the natural splitting of variables that stems from the facial reduction technique in the semidefinite relaxation. We solve the relaxation using a variant of the Peaceman-Rachford splitting method. Our numerical experiments show that we solve almost all instances of the NP-hard SCP problem to optimality.
Graph matching or quadratic assignment, is the problem of labeling the vertices of two graphs so that they are as similar as possible. A common method for approximately solving the NP-hard graph matching problem is relaxing it to a convex optimization problem over the set of doubly stochastic (DS) matrices. Recent analysis has shown that for almost all pairs of isomorphic and asymmetric graphs, the DS relaxation succeeds in correctly retrieving the isomorphism between the graphs. Our goal in this paper is to analyze the case of symmetric isomorphic graphs. This goal is motivated by shape match…
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 $η_p = ζ$, $η_d = ζ$, and $φ= ζ$, where $ζ$, $η_p$, $η_d$, and $φ$ denote the optimal…
In this paper, we show that the quadratic assignment problem (QAP) can be reformulated to an equivalent rank constrained doubly nonnegative (DNN) problem. Under the framework of the difference of convex functions (DC) approach, a semi-proximal DC algorithm (DCA) is proposed for solving the relaxation of the rank constrained DNN problem whose subproblems can be solved by the semi-proximal augmented Lagrangian method (sPALM). We show that the generated sequence converges to a stationary point of the corresponding DC problem, which is feasible to the rank constrained DNN problem. Moreover, numeri…
