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31
Solving Standard Quadratic Optimization Problems Via Linear, Semidefinite and Copositive Programming
 J. GLOBAL OPTIM
, 2001
"... The problem of minimizing a (nonconvex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive matr ..."
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Cited by 51 (6 self)
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The problem of minimizing a (nonconvex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive matrices via systems of linear inequalities, and, more refined, linear matrix inequalities (LMI's). In particular, we show that our approach leads to a polynomialtime approximation scheme for the standard quadratic optimization problem. This is an improvement on the previous complexity result by Nesterov [10] (that a 2/3approximation is always possible). Numerical examples from various applications are provided to illustrate our approach, which extends ideas of De Klerk and Pasechnik [5] for the maximal stable set problem in a graph.
Interiorpoint methods for optimization
, 2008
"... This article describes the current state of the art of interiorpoint methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twen ..."
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Cited by 17 (0 self)
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This article describes the current state of the art of interiorpoint methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twenty years.
INTERIOR POINTS OF THE COMPLETELY POSITIVE CONE
, 2008
"... A Matrix A is called completely positive if it can be decomposed as A = BBT with an entrywise nonnegative matrix B. The set of all such matrices is a convex cone which plays a role in certain optimization problems. A characterization of the interior of this cone is provided. ..."
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Cited by 16 (2 self)
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A Matrix A is called completely positive if it can be decomposed as A = BBT with an entrywise nonnegative matrix B. The set of all such matrices is a convex cone which plays a role in certain optimization problems. A characterization of the interior of this cone is provided.
PayoffMonotonic Game Dynamics and the Maximum Clique Problem
, 2006
"... Evolutionary gametheoretic models and, in particular, the socalled replicator equations have recently proven to be remarkably effective at approximately solving the maximum clique and related problems. The approach is centered around a classic result from graph theory that formulates the maximum c ..."
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Cited by 12 (8 self)
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Evolutionary gametheoretic models and, in particular, the socalled replicator equations have recently proven to be remarkably effective at approximately solving the maximum clique and related problems. The approach is centered around a classic result from graph theory that formulates the maximum clique problem as a standard (continuous) quadratic program and exploits the dynamical properties of these models, which, under a certain symmetry assumption, possess a Lyapunov function. In this letter, we generalize previous work along these lines in several respects. We introduce a wide family of gamedynamic equations known as payoffmonotonic dynamics, of which replicator dynamics are a special instance, and show that they enjoy precisely the same dynamical properties as standard replicator equations. These properties make any member of this family a potential heuristic for solving standard quadratic programs and, in particular, the maximum clique problem. Extensive simulations, performed on random as well as DIMACS benchmark graphs, show that
Reformulation and Convex Relaxation Techniques for Global Optimization
 4OR
, 2004
"... Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested i ..."
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Cited by 11 (9 self)
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Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested in determining the globally optimal point. This thesis is concerned with techniques for establishing such global optima using spatial BranchandBound (sBB) algorithms.
Attributed Tree Matching and Maximum Weight Cliques
 In ICIAP’9910th Int. Conf. on Image Analysis and Processing
, 1999
"... A classical way of matching relational structures consists of finding a maximum clique in a derived "association graph." However, it is not clear how to apply this approach to problems where the graphs are hierarchically organized, i.e. are trees, since maximum cliques are not constrained ..."
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Cited by 10 (2 self)
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A classical way of matching relational structures consists of finding a maximum clique in a derived "association graph." However, it is not clear how to apply this approach to problems where the graphs are hierarchically organized, i.e. are trees, since maximum cliques are not constrained to preserve the partial order. We have recently provided a solution to this problem by constructing the association graph using the graphtheoretic concept of connectivity. In this paper, we extend the approach to the problem of matching attributed trees. Specifically, we show how to derive a "weighted" association graph, and prove that the attributed tree matching problem is equivalent to finding a maximum weight clique in it. We then formulate the maximum weight clique problem in terms of a continuous optimization problem, which we solve using "replicator" dynamical systems developed in theoretical biology. This formulation is attractive because it can motivate analog and biological implementations....
A Complementary Pivoting Approach to the Maximum Weight Clique Problem
 SIAM J. OPTIM
, 2002
"... Given an undirected graph with positive weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (i.e., a clique) having largest total weight. The problem is known to be NPhard, even to approximate. Motivated by a recent quadratic program ..."
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Cited by 10 (3 self)
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Given an undirected graph with positive weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (i.e., a clique) having largest total weight. The problem is known to be NPhard, even to approximate. Motivated by a recent quadratic programming formulation, which generalizes an earlier remarkable result of Motzkin and Straus, in this paper we propose a new framework for the MWCP based on the corresponding linear complementarity problem (LCP). We show that, generically, all stationary points of the MWCP quadratic program exhibit strict complementarity. Despite this regularity result, however, the LCP turns out to be inherently degenerate, and we find that Lemke’s wellknown pivoting method, equipped with standard degeneracy resolution strategies, yields unsatisfactory experimental results. We exploit the degeneracy inherent in the problem to develop a variant of Lemke’s algorithm which incorporates a new and effective “lookahead ” pivot rule. The resulting algorithm is tested extensively on various instances of random as well as DIMACS benchmark graphs, and the results obtained show the effectiveness of our method.
Copositive optimization – recent developments and applications
 European Journal of Operational Research
, 2012
"... Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear const ..."
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Cited by 9 (1 self)
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Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of optimization is impressive, since problem classes both in the continuous and discrete world, as well as both deterministic and stochastic models are covered. Copositivity appears in local and global optimality conditions for quadratic optimization, but can also yield tighter bounds for NPhard combinatorial optimization problems. Here some of the recent success stories are told, along with principles, algorithms and applications. 1.
Copositivity detection by differenceofconvex decomposition and ωsubdivision
 MATHEMATICAL PROGRAMMING
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