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78
Solving LargeScale Sparse Semidefinite Programs for Combinatorial Optimization
 SIAM JOURNAL ON OPTIMIZATION
, 1998
"... We present a dualscaling interiorpoint algorithm and show how it exploits the structure and sparsity of some large scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational re ..."
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Cited by 119 (11 self)
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We present a dualscaling interiorpoint algorithm and show how it exploits the structure and sparsity of some large scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational results of interiorpoint algorithms for approximating the maximum cut semidefinite programs with dimension upto 3000.
New Results on Quadratic Minimization
, 2001
"... In this paper we present several new results on minimizing an indefinite quadratic function under quadratic/linear constraints. The emphasis is placed on the case where the constraints are two quadratic inequalities. This formulation is known as the extended trust region subproblem and the computati ..."
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Cited by 65 (8 self)
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In this paper we present several new results on minimizing an indefinite quadratic function under quadratic/linear constraints. The emphasis is placed on the case where the constraints are two quadratic inequalities. This formulation is known as the extended trust region subproblem and the computational complexity of this problem is still unknown. We consider several interesting cases related to this problem and show that for those cases the corresponding SDP relaxation admits no gap with the true optimal value, and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization. For the extended trust region subproblem itself, we introduce a parameterized problem and prove the existence of a trajectory which will lead to an optimal solution. Combining with a result obtained in the first part of the paper, we propose a polynomialtime solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint.
On Lagrangian relaxation of quadratic matrix constraints
 SIAM J. MATRIX ANAL. APPL
, 2000
"... Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent ..."
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Cited by 53 (18 self)
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Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g., convex programs and trust region subproblems, the Lagrangian relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective. In this paper we consider a certain QQP where the quadratic constraints correspond to the matrix orthogonality condition XXT = I. For this problem we show that the Lagrangian dual based on relaxing the constraints XXT = I and the seemingly redundant constraints XT X = I has a zero duality gap. This result has natural applications to quadratic assignment and graph partitioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix. We also show that the technique of relaxing quadratic matrix constraints can be used to obtain a strengthened semidefinite relaxation for the maxcut problem.
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 49 (19 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...
Approximation bounds for quadratic optimization with homogeneous quadratic constraints
 SIAM J. Optim
, 2007
"... Abstract. We consider the NPhard problem of finding a minimum norm vector in ndimensional real or complex Euclidean space, subject to m concave homogeneous quadratic constraints. We show that a semidefinite programming (SDP) relaxation for this nonconvex quadratically constrained quadratic program ..."
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Cited by 49 (24 self)
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Abstract. We consider the NPhard problem of finding a minimum norm vector in ndimensional real or complex Euclidean space, subject to m concave homogeneous quadratic constraints. We show that a semidefinite programming (SDP) relaxation for this nonconvex quadratically constrained quadratic program (QP) provides an O(m 2) approximation in the real case and an O(m) approximation in the complex case. Moreover, we show that these bounds are tight up to a constant factor. When the Hessian of each constraint function is of rank 1 (namely, outer products of some given socalled steering vectors) and the phase spread of the entries of these steering vectors are bounded away from π/2, we establish a certain “constant factor ” approximation (depending on the phase spread but independent of m and n) for both the SDP relaxation and a convex QP restriction of the original NPhard problem. Finally, we consider a related problem of finding a maximum norm vector subject to m convex homogeneous quadratic constraints. We show that an SDP relaxation for this nonconvex QP provides an O(1 / ln(m)) approximation, which is analogous to a result of Nemirovski et al. [Math. Program., 86 (1999), pp. 463–473] for the real case. Key words. semidefinite programming relaxation, nonconvex quadratic optimization, approximation bound
Global Quadratic Optimization via Conic Relaxation
, 1998
"... We present a convex conic relaxation for a problem of maximizing an indefinite quadratic form over a set of convex constraints on the squared variables. We show that for all these problems we get at least 12 37 relative accuracy of the approximation. In the second part of the paper we derive the ..."
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Cited by 39 (0 self)
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We present a convex conic relaxation for a problem of maximizing an indefinite quadratic form over a set of convex constraints on the squared variables. We show that for all these problems we get at least 12 37 relative accuracy of the approximation. In the second part of the paper we derive the conic relaxation by another approach based on the second order optimality conditions. We show that for l p balls, p # 2, intersected by a linear subspace, it is possible to guarantee (1  2 p )relative accuracy of the solution. As a consequence, we prove (1 1 e ln n )relative accuracy of the conic relaxation for an indefinite quadratic maximization problem over an ndimensional unit box with homogeneous linear equality constraints. We discuss the implications of the results for the discussion around the question P = NP. # CORE, Catholic University of Louvain, 34 voie du Roman Pays, 1348 LouvainlaNeuve, Belgium; email: nesterov@core.ucl.ac.be 1 1
Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations
 Computational Optimization and Applications
, 2003
"... Abstract We show that SDP (semidefinite programming) and SOCP (second order cone programming) relaxations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a generalization of the results by S. Zhang for a subclass of quadratic maximization problems tha ..."
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Cited by 34 (6 self)
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Abstract We show that SDP (semidefinite programming) and SOCP (second order cone programming) relaxations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a generalization of the results by S. Zhang for a subclass of quadratic maximization problems that have nonnegative offdiagonal coefficient matrices of quadratic objective functions and diagonal coefficient matrices of quadratic constraint functions. A new SOCP relaxation is proposed for the class of nonconvex quadratic optimization problems by extracting valid quadratic inequalities for positive semidefinite cones. Its effectiveness to obtain optimal values is shown to be the same as the SDP relaxation theoretically. Numerical results are presented to demonstrate that the SOCP relaxation is much more efficient than the SDP relaxation.
Complex Quadratic Optimization and Semidefinite Programming
 SIAM Journal on Optimization
, 2006
"... In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max3cut model used in a recent paper of Goemans and Williamson. We first develop a closedform f ..."
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Cited by 31 (12 self)
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In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max3cut model used in a recent paper of Goemans and Williamson. We first develop a closedform formula to compute the probability of a complexvalued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized (with a specific rounding rule) solution based on the optimal solution of the complex SDP relaxation problem. In particular, we study the limit of that model, in which the problem remains NPhard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomizationrounding procedure guarantees a worstcase performance ratio of π/4 ≈ 0.7854, which is better than the ratio of 2/π ≈ 0.6366 for its counterpart in the real case due to Nesterov. Furthermore, if the objective matrix is realvalued positive semidefinite with nonpositive offdiagonal elements, then the performance ratio improves to 0.9349.