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53
On the copositive representation of binary and continuous nonconvex quadratic programs
, 2007
"... In this paper, we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices. This result can be viewed as an extension of earlier separate results, which have established the copositive representation of ..."
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Cited by 89 (6 self)
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In this paper, we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices. This result can be viewed as an extension of earlier separate results, which have established the copositive representation of a small collection of NPhard problems. A simplification, which reduces the dimension of the linear conic program, and an extension to complementarity constraints are established, and computational issues are discussed.
BiQuadratic Optimization over Unit Spheres and Semidefinite Programming Relaxations
, 2008
"... Abstract. This paper studies the socalled biquadratic optimization over unit spheres min x∈R n,y∈R m bijklxiyjxkyl ..."
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Cited by 32 (15 self)
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Abstract. This paper studies the socalled biquadratic optimization over unit spheres min x∈R n,y∈R m bijklxiyjxkyl
Computable representations for convex hulls of lowdimensional quadratic forms
, 2007
"... Let C be the convex hull of points { ( 1) ( 1 x x)T  x ∈ F ⊂ ℜ n}. Representing or approximating C is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. If n ≤ 4 and F is a simplex then C has a computable representation in terms of matric ..."
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Cited by 30 (10 self)
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Let C be the convex hull of points { ( 1) ( 1 x x)T  x ∈ F ⊂ ℜ n}. Representing or approximating C is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. If n ≤ 4 and F is a simplex then C has a computable representation in terms of matrices X that are doubly nonnegative (positive semidefinite and componentwise nonnegative). If n = 2 and F is a box, then C has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulationlinearization technique (RLT). The simplex result generalizes known representations for the convex hull of {(x1, x2, x1x2)  x ∈ F} when F ⊂ ℜ 2 is a triangle, while the result for box constraints generalizes the wellknown fact that in this case the RLT constraints generate the convex hull of {(x1, x2, x1x2)  x ∈ F}. When n = 3 and F is a box, a representation for C can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3cube.
Copositive Programming  a Survey
, 2009
"... Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial ..."
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Cited by 29 (0 self)
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Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial and quadratic optimization. The purpose of this survey is to introduce the field to interested readers in the optimization community who wish to get an understanding of the basic concepts and recent developments in copositive programming, including modeling issues and applications, the connection to semidefinite programming and sumofsquares approaches, as well as algorithmic solution approaches for copositive programs.
Computing the stability number of a graph via linear and semidefinite programming
 SIAM Journal on Optimization
, 2007
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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 18 (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.
The complexity of optimizing over a simplex, hypercube or sphere: a short survey
"... We consider the computational complexity of optimizing various classes of continuous functions over a simplex, hypercube or sphere. These relatively simple optimization problems have many applications. We review known approximation results as well as negative (inapproximability) results from the rec ..."
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Cited by 15 (2 self)
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We consider the computational complexity of optimizing various classes of continuous functions over a simplex, hypercube or sphere. These relatively simple optimization problems have many applications. We review known approximation results as well as negative (inapproximability) results from the recent literature.
On the Equivalence of Algebraic Approaches to the Minimization of Forms on the Simplex
, 2003
"... We consider the problem of minimizing a form on the standard simplex [equivalently, the problem of minimizing an even form on the unit sphere]. Converging hierarchies of approximations for this problem can be constructed, that are based, respectively, on results by SchmudgenPutinar and by Polya ..."
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Cited by 13 (6 self)
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We consider the problem of minimizing a form on the standard simplex [equivalently, the problem of minimizing an even form on the unit sphere]. Converging hierarchies of approximations for this problem can be constructed, that are based, respectively, on results by SchmudgenPutinar and by Polya about representations of positive polynomials in terms of sums of squares. We show that the two approaches yield, in fact, the same approximations. The same type of argument also permits to establish some representation results a la Polya for positive polynomials on semialgebraic cones.