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21
Approximating Quadratic Programming With Bound Constraints
 Mathematical Programming
, 1997
"... We consider the problem of approximating the global maximum of a quadratic program (QP) with n variables subject to bound constraints. Based on the results of Goemans and Williamson [4] and Nesterov [6], we show that a 4=7 approximate solution can be obtained in polynomial time. Key words. Quadratic ..."
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Cited by 68 (13 self)
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We consider the problem of approximating the global maximum of a quadratic program (QP) with n variables subject to bound constraints. Based on the results of Goemans and Williamson [4] and Nesterov [6], we show that a 4=7 approximate solution can be obtained in polynomial time. Key words. Quadratic programming, global maximizer, approximation algorithm This author is supported in part by NSF grant DMI9522507. 1 Introduction Consider the quadratic programming (QP) problem ¯ q(Q) := Maximize q(x) := x T Qx (QP) Subject to \Gammae x e; where Q 2 ! n\Thetan is given and e 2 ! n is the vector of all ones. Let ¯ x = ¯ x(Q) be a maximizer of the problem. In this paper, without loss of generality, we assume that ¯ x 6= 0. Normally, there is a linear term in the objective function: q(x) = x T Qx + c T x. However, the problem can be homogenized as Maximize q(x) := x T Qx + tc T x Subject to \Gammae x e; \Gamma1 t 1 by adding a scalar variable t. There always is an opti...
On Lagrangian relaxation of quadratic matrix constraints
 SIAM J. Matrix Anal. Appl
, 2000
"... Abstract. 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 equ ..."
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Cited by 50 (19 self)
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Abstract. 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. Key words. Lagrangian relaxations, quadratically constrained quadratic programs, semidefinite programming, quadratic assignment, graph partitioning, maxcut problems
On Cones of Nonnegative Quadratic Functions
, 2001
"... We derive LMIcharacterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized copositivity. These matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain. As a domain, we consider for ..."
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Cited by 38 (10 self)
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We derive LMIcharacterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized copositivity. These matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain. As a domain, we consider for instance the intersection of a (upper) levelset of a quadratic function and a halfplane. We arrive at a generalization of Yakubovich's Sprocedure result. As an application we show that optimizing a general quadratic function over the intersection of an ellipsoid and a halfplane can be formulated as SDP, thus proving the polynomiality of this class of optimization problems, which arise, e.g., from the application of the trust region method for nonlinear programming. Other applications are in control theory and robust optimization. Keywords: LMI, SDP, CoPositive Cones, Quadratic Functions, SProcedure, Matrix Decomposition.
OptimizationBased Animation
, 2002
"... A new paradigm for rigid body simulation is presented and analyzed. Current techniques for rigid body simulation run slowly on scenes with many bodies in close proximity. Each time two bodies collide or make or break a static contact, the simulator must interrupt the numerical integration of velocit ..."
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Cited by 36 (1 self)
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A new paradigm for rigid body simulation is presented and analyzed. Current techniques for rigid body simulation run slowly on scenes with many bodies in close proximity. Each time two bodies collide or make or break a static contact, the simulator must interrupt the numerical integration of velocities and accelerations. Even for simple scenes, the number of discontinuities per frame time can rise to the millions. An efficient optimizationbased animation (OBA) algorithm is presented which can simulate scenes with many convex threedimensional bodies settling into stacks and other “crowded” arrangements. This algorithm simulates Newtonian (second order) physics and Coulomb friction, and it uses quadratic programming (QP) to calculate new positions, momenta, and accelerations strictly at frame times. The extremely small integration steps inherent to traditional simulation techniques are avoided. Contact points are synchronized at the end of each frame. Resolving contacts with friction is known to be a difficult problem. Analytic force calculation can have ambiguous or nonexisting solutions. Purely impulsive techniques avoid these ambiguous cases, but still require an excessive and computationally expensive number of updates in the case of
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 29 (5 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.
Strong Duality for a TrustRegion Type Relaxation of the Quadratic Assignment Problem
, 1998
"... Lagrangian duality underlies many efficient algorithms for convex minimization problems. A key ingredient is strong duality. Lagrangian relaxation also provides lower bounds for nonconvex problems, where the quality of the lower bound depends on the duality gap. Quadratically constrained quadratic p ..."
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Cited by 15 (10 self)
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Lagrangian duality underlies many efficient algorithms for convex minimization problems. A key ingredient is strong duality. Lagrangian relaxation also provides lower bounds for nonconvex problems, where the quality of the lower bound depends on the duality gap. Quadratically constrained quadratic programs (QQPs) provide important examples of nonconvex programs. For the simple case of one quadratic constraint (the trust region subproblem) strong duality holds. In addition, necessary and sufficient (strengthened) second order optimality conditions exist. However, these duality results already fail for the two trust region subproblem. Surprisingly, there are classes of more complex, nonconvex QQPs where strong duality holds. One example is the special case of orthogonality constraints, which arise naturally in relaxations for the quadratic assignment problem (QAP). In this paper we show that strong duality also holds for a relaxation of QAP where the orthogonality constraint is replaced ...
A fast impulsive contact suite for rigid body simulation
 IEEE Transactions on Visualization and Computer Graphics
, 2004
"... Abstract—A suite of algorithms is presented for contact resolution in rigid body simulation under the Coulomb friction model: Given a set of rigid bodies with many contacts among them, resolve dynamic contacts (collisions) and static (persistent) contacts. The suite consists of four algorithms: 1) p ..."
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Cited by 8 (1 self)
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Abstract—A suite of algorithms is presented for contact resolution in rigid body simulation under the Coulomb friction model: Given a set of rigid bodies with many contacts among them, resolve dynamic contacts (collisions) and static (persistent) contacts. The suite consists of four algorithms: 1) partial sequential collision resolution, 2) final resolution of collisions through the solution of a single convex QP (positive semidefinite quadratic program), 3) resolution of static contacts through the solution of a single convex QP, 4) freezing of “stationary ” bodies. This suite can generate realisticlooking results for simple examples yet, for the first time, can also tractably resolve contacts for a simulation as large as 1,000 cubes in an “hourglass. ” Freezing speeds up this simulation by more than 25 times. Thanks to excellent commercial QP technology, the contact resolution suite is simple to implement and can be “plugged into” any simulation algorithm to provide fast and realisticlooking animations of rigid bodies. Index Terms—Quadratic programming, computer graphics, physicallybased modeling, simulation, animation. æ
Approximating Global Quadratic Optimization With Convex Quadratic Constraints
 Journal of Global Optimization
, 1998
"... We consider the problem of approximating the global maximum of a quadratic program (QP) subject to convex nonhomogeneous quadratic constraints. We prove an approximation quality bound that is related to a condition number of the convex feasible set; and it is the currently best for approximating ce ..."
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Cited by 8 (0 self)
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We consider the problem of approximating the global maximum of a quadratic program (QP) subject to convex nonhomogeneous quadratic constraints. We prove an approximation quality bound that is related to a condition number of the convex feasible set; and it is the currently best for approximating certain problems, such as quadratic optimization over the assignment polytope, according to the best of our knowledge.
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 8 (2 self)
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Abstract. This paper studies the socalled biquadratic optimization over unit spheres min x∈R n,y∈R m bijklxiyjxkyl
An efficient rescaled perceptron algorithm for conic systems
, 2006
"... The classical perceptron algorithm is an elementary rowaction/relaxation algorithm for solving a homogeneous linear inequality system Ax> 0. A natural condition measure associated with this algorithm is the Euclidean width τ of the cone of feasible solutions, and the iteration complexity of the per ..."
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Cited by 7 (5 self)
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The classical perceptron algorithm is an elementary rowaction/relaxation algorithm for solving a homogeneous linear inequality system Ax> 0. A natural condition measure associated with this algorithm is the Euclidean width τ of the cone of feasible solutions, and the iteration complexity of the perceptron algorithm is bounded by 1/τ 2, see Rosenblatt 1962 [20]. Dunagan and Vempala [5] have developed a rescaled version of the perceptron algorithm with an improved complexity of O(n ln(1/τ)) iterations (with high probability), which is theoretically efficient in τ, and in particular is polynomialtime in the bitlength model. We explore extensions of the concepts of these perceptron methods to the general homogeneous conic system Ax ∈ int K where K is a regular convex cone. We provide a conic extension of the rescaled perceptron algorithm based on the notion of a deepseparation oracle of a cone, which essentially computes a certificate of strong separation. We show that the rescaled perceptron algorithm is theoretically efficient if an efficient deepseparation oracle is available for the feasible region. Furthermore, when K is the crossproduct of basic cones that are either halfspaces or secondorder cones, then a deepseparation oracle is available and hence the rescaled perceptron algorithm is theoretically efficient. When the basic cones of K include semidefinite cones, then a probabilistic deepseparation oracle for K can be constructed that also yields a theoretically efficient version of the rescaled perceptron algorithm. Key words: