Quadratic Programming with Box Constraints

Angelis, P. L. De; Pardalos, Pãnos M.; Toraldo, Gerardo

doi:10.1007/978-1-4757-2600-8_5

Cited by 39 publications

(15 citation statements)

References 39 publications

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“…Proof Since P (x) = 1 2 x Qx is a polynomial in x with no negative coefficients, the Baum-Eagon Theorem [3] yields P ( (14). Hence (a) is established.…”

Section: Theorem 24 If Q Is Non-negative With Positive Diagonal Thenmentioning

confidence: 89%

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Multi-Standard Quadratic Optimization: interior point methods and cone programming reformulation

Bomze

Schachinger

2009

Comput Optim Appl

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A Standard Quadratic Optimization Problem (StQP) consists of maximizing a (possibly indefinite) quadratic form over the standard simplex. Likewise, in a multi-StQP we have to maximize a (possibly indefinite) quadratic form over the Cartesian product of several standard simplices (of possibly different dimensions). Among many other applications, multi-StQPs occur in Machine Learning Problems. Several converging monotone interior point methods are established, which differ from the usual ones used in cone programming. Further, we prove an exact cone programming reformulation for establishing rigorous yet affordable bounds and finding improving directions.

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“…Proof Since P (x) = 1 2 x Qx is a polynomial in x with no negative coefficients, the Baum-Eagon Theorem [3] yields P ( (14). Hence (a) is established.…”

Section: Theorem 24 If Q Is Non-negative With Positive Diagonal Thenmentioning

confidence: 89%

“…. , m, we arrive at another key class in quadratic programming, namely box-constrained QPs with applications, e.g., in the maximum-cut problem and other combinatorial optimization fields, be it as an exact reformulation or as a relaxation problem leading to rigorous bounds; for a survey we refer to [14].…”

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confidence: 99%

Multi-Standard Quadratic Optimization: interior point methods and cone programming reformulation

Bomze

Schachinger

2009

Comput Optim Appl

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“…There are numerous methods for solving (1) and more general nonconvex quadratic programs, including local methods (Gould and Toint, 2002) and global methods (Pardalos, 1991). For a survey of methods to globally solve (1), see De Angelis et al (1997) as well as Vandenbussche and Nemhauser (2005a,b) and Burer and Vandenbussche (2006).…”

Section: Introductionmentioning

confidence: 99%

Relaxing the optimality conditions of box QP

Burer

Chen

2009

Comput Optim Appl

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We present semidefinite relaxations of nonconvex, box-constrained quadratic programming, which incorporate the first-and second-order necessary optimality conditions. We compare these relaxations with a basic semidefinite relaxation due to Shor, particularly in the context of branch-and-bound to determine a global optimal solution, where it is shown empirically that the new relaxations are significantly stronger. We also establish theoretical relationships between the new relaxations and Shor's relaxation.

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“…There are numerous methods for solving (QPB) and more general nonconvex quadratic programs, including local methods [8] and global methods [14]. For a survey of methods to globally solve (QPB), see [6]. Existing global optimization techniques for (QPB) can be classified into two groups.…”

Section: Introductionmentioning

confidence: 99%