Results 1  10
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14
On Maximization of Quadratic Form over Intersection of Ellipsoids with Common Center
, 1998
"... . We demonstrate that if A 1 ; :::; Am are symmetric positive semidefinite n \Theta n matrices with positive definite sum and A is an arbitrary symmetric n \Theta n matrix, then the relative accuracy, in terms of the optimal value, of the semidefinite relaxation max X fTr(AX) j Tr(A i X) 1; i = 1 ..."
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Cited by 50 (3 self)
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. We demonstrate that if A 1 ; :::; Am are symmetric positive semidefinite n \Theta n matrices with positive definite sum and A is an arbitrary symmetric n \Theta n matrix, then the relative accuracy, in terms of the optimal value, of the semidefinite relaxation max X fTr(AX) j Tr(A i X) 1; i = 1; :::; m; X 0g (SDP) of the optimization program x T Ax ! max j x T A i x 1; i = 1; :::; m (P) is not worse than 1 \Gamma 1 2 ln(2m 2 ) . It is shown that this bound is sharp in order, as far as the dependence on m is concerned, and that a feasible solution x to (P) with x T Ax Opt(SDP) 2 ln(2m 2 ) () can be found efficiently. This somehow improves one of the results of Nesterov [4] where bound similar to (*) is established for the case when all A i are of rank 1. Keywords: Semidefinite relaxations, quadratic programming 1. Introduction Let A i , i = 1; :::; m, be positive semidefinite n \Theta n matrices with positive definite sum, and A be a n \Theta n symmetric matrix. Con...
Cones Of Matrices And Successive Convex Relaxations Of Nonconvex Sets
, 2000
"... . Let F be a compact subset of the ndimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each ..."
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Cited by 46 (20 self)
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. Let F be a compact subset of the ndimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets C k (k = 1, 2, . . . ) of R n such that (a) the convex hull of F # C k+1 # C k (monotonicity), (b) # # k=1 C k = the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding LovaszSchrijver liftandproject procedures with the use of SDP or LP relaxation applied to general quadratic optimization problems (QOPs) with infinitely many quadratic inequality constraints. Utilizing descriptions of sets based on cones of matrices and their duals, we establish the exact equivalence of the SDP relaxation and the semiinfinite convex QOP relaxation proposed originally by Fujie and Kojima. Using th...
Discretization and Localization in Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization
, 2000
"... . Based on the authors' previous work which established theoretical foundations of two, conceptual, successive convex relaxation methods, i.e., the SSDP (Successive Semidefinite Programming) Relaxation Method and the SSILP (Successive SemiInfinite Linear Programming) Relaxation Method, this pa ..."
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Cited by 25 (14 self)
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. Based on the authors' previous work which established theoretical foundations of two, conceptual, successive convex relaxation methods, i.e., the SSDP (Successive Semidefinite Programming) Relaxation Method and the SSILP (Successive SemiInfinite Linear Programming) Relaxation Method, this paper proposes their implementable variants for general quadratic optimization problems. These problems have a linear objective function c T x to be maximized over a nonconvex compact feasible region F described by a finite number of quadratic inequalities. We introduce two new techniques, "discretization" and "localization," into the SSDP and SSILP Relaxation Methods. The discretization technique makes it possible to approximate an infinite number of semiinfinite SDPs (or semiinfinite LPs) which appeared at each iteration of the original methods by a finite number of standard SDPs (or standard LPs) with a finite number of linear inequality constraints. We establish: ffl Given any open convex ...
A PROJECTED GRADIENT ALGORITHM FOR SOLVING THE MAXCUT SDP RELAXATION
"... In this paper, we present a projected gradient algorithm for solving the semidefinite programming (SDP) relaxation of the maximum cut (maxcut) problem. Coupled with a randomized method, this gives a very efficient approximation algorithm for the maxcut problem. We report computational results compar ..."
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Cited by 22 (8 self)
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In this paper, we present a projected gradient algorithm for solving the semidefinite programming (SDP) relaxation of the maximum cut (maxcut) problem. Coupled with a randomized method, this gives a very efficient approximation algorithm for the maxcut problem. We report computational results comparing our method with two earlier successful methods on problems with dimension up to 7,000.
Copositive Relaxation for General Quadratic Programming
 OPTIM. METHODS SOFTW
, 1998
"... We consider general, typically nonconvex, Quadratic Programming Problems. The Semidefinite relaxation proposed by Shor provides bounds on the optimal solution, but it does not always provide sufficiently strong bounds if linear constraints are also involved. To get rid of the linear sideconstraint ..."
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Cited by 15 (2 self)
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We consider general, typically nonconvex, Quadratic Programming Problems. The Semidefinite relaxation proposed by Shor provides bounds on the optimal solution, but it does not always provide sufficiently strong bounds if linear constraints are also involved. To get rid of the linear sideconstraints, another, stronger convex relaxation is derived. This relaxation uses copositive matrices. Special cases are discussed for which both relaxations are equal. At the end of the paper, the complexity and solvability of the relaxations are discussed.
Towards Implementations of Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization Problems
, 1999
"... Recently Kojima and Tuncel proposed new successive convex relaxation methods and their localizeddiscretized variants for general nonconvex quadratic optimization problems. Although an upper bound of the optimal objective function value within a previously given precision can be found theoretically ..."
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Cited by 12 (6 self)
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Recently Kojima and Tuncel proposed new successive convex relaxation methods and their localizeddiscretized variants for general nonconvex quadratic optimization problems. Although an upper bound of the optimal objective function value within a previously given precision can be found theoretically by solving a finite number of linear programs, several important implementation issues remain unsolved. In this paper, we discuss those issues, present practically implementable algorithms and report numerical results.
An Efficient Algorithm for Solving the MAXCUT SDP Relaxation
 School of ISyE, Georgie Tech, Atlanta, GA 30332
, 1998
"... In this paper we present a projected gradient algorithm for solving the semidefinite programming (SDP) relaxation of the maximum cut (MAXCUT) problem. Coupled with a randomized method, this gives a very efficient approximation algorithm for the MAXCUT problem. We report computational results compari ..."
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Cited by 10 (1 self)
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In this paper we present a projected gradient algorithm for solving the semidefinite programming (SDP) relaxation of the maximum cut (MAXCUT) problem. Coupled with a randomized method, this gives a very efficient approximation algorithm for the MAXCUT problem. We report computational results comparing our method with two earlier successful methods on problems with dimension up to 3000.
APPROXIMATING MAXIMUM STABLE SET AND MINIMUM GRAPH COLORING PROBLEMS WITH THE POSITIVE SEMIDEFINITE RELAXATION
"... We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in m ..."
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Cited by 9 (1 self)
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We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in most graphs and the structure inherent in the problem formulation. From the solution to the relaxation, we apply a randomized algorithm to find approximate maximum stable sets and a modification of a popular heuristic to find graph colorings. We obtained high quality answers for graphs with over 1000 vertices and almost 7000 edges.
Global Quadratic Optimization on the Sets With Simplex Structure
, 1999
"... In the first part of this paper we prove that the global quadratic optimization problem over a simplex can be solved with a constant relative accuracy. In the second part we consider some natural extensions of the result. ..."
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Cited by 2 (0 self)
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In the first part of this paper we prove that the global quadratic optimization problem over a simplex can be solved with a constant relative accuracy. In the second part we consider some natural extensions of the result.