Checking local optimality in constrained quadratic programming is NP-hard

Pardalos, Pãnos M.; Schnitger, Georg

doi:10.1016/0167-6377(88)90049-1

Cited by 167 publications

(66 citation statements)

References 3 publications

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“…Confirming that the MFCQ holds when the LICQ does not requires solution of a linear program (2.10); verifying assumption (iii) for all p such that J * A p ≥ 0 requires finding the global minimizer of a possibly indefinite quadratic form over a cone, an NP-hard problem [75,82], not to mention the issue of how to check that (iii) holds for all λ ∈ M λ . If, however, the gradients of the active constraints at x * are linearly independent and strict complementarity holds, Theorem 2.23 leads immediately to the following result, which we state separately for future reference.…”

Section: Sufficient Optimality Conditionsmentioning

confidence: 99%

Interior Methods for Nonlinear Optimization

Forsgren

Gill

Wright³

2002

SIAM Rev.

577

363

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Abstract.Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interior-point techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization. This picture changed dramatically with Karmarkar's widely publicized announcement in 1984 of a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interior methods for nonlinearly constrained optimization.

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Section: Sufficient Optimality Conditionsmentioning

confidence: 99%

Interior Methods for Nonlinear Optimization

Forsgren

Gill

Wright³

2002

SIAM Rev.

577

363

View full text Add to dashboard Cite

show abstract

“…Obviously, (1)- (4) is feasible if and only if the constrained quadratic minimization problem (5)- (8) has a solution at most k C. It is well known that constrained quadratic programming is NP-hard in general [18], even without integrality constraints. More specifically, we have a constrained concave minimization problem, which is generally known to be NP-hard as well [9].…”

Section: Quadratic Programming Relaxationmentioning

confidence: 99%

“…Now we are ready to prove the performance bound of Theorem 1. First, use (17) together with (18) and (19) to obtain…”

Section: Qp Based Greedy Algorithmmentioning

confidence: 99%

“…More precisely, the mathematical program is an integer, concave minimization problem with linear constraints. Although such problems are NP-hard to solve in general [18,9], even without integrality constraints, we show how to solve this quadratic programming relaxation with arbitrary precision in polynomial time; a result of interest in its own. Based on the solution of this mathematical program, we assign resources to the jobs.…”

Section: Introduction and Related Workmentioning

confidence: 99%

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Scheduling Parallel Jobs with Linear Speedup

Grigoriev

Uetz

2006

Approximation and Online Algorithms

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We consider a scheduling problem where a set of jobs is apriori distributed over parallel machines. The processing time of any job is dependent on the usage of a scarce renewable resource, e.g. personnel. An amount of k units of that resource can be allocated to the jobs at any time, and the more of that resource is allocated to a job, the smaller its processing time. The dependence of processing times on the amount of resources is linear for any job. The objective is to find a resource allocation and a schedule that minimizes the makespan. Utilizing an integer quadratic programming relaxation, we show how to obtain a (3 + ε)-approximation algorithm for that problem, for any ε > 0. This generalizes and improves previous results, respectively. Our approach relies on a fully polynomial time approximation scheme to solve the quadratic programming relaxation. This result is interesting in itself, because the underlying quadratic program is NP-hard to solve. We also derive lower bounds, and discuss further generalizations of the results.

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“…6, for a precise definition of a dead point). The difficulty arises because the verification of such a point as a local minimizer of (1.1) is an NP-hard problem-see Murty and Kabadi [12] and Pardalos and Schnitger [13]. Unfortunately, even if lower values of ϕ do exist in the neighborhood of a dead point, any number of constraints may need to be deleted simultaneously in order to compute a direction of improvement.…”

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confidence: 99%

On the Identification of Local Minimizers in Inertia-Controlling Methods for Quadratic Programming

Forsgren

Gill

Murray

1991

SIAM J. Matrix Anal. & Appl.

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Abstract. The verification of a local minimizer of a general (i.e., nonconvex) quadratic program is in general an NP-hard problem. The difficulty concerns the optimality of certain points (which we call dead points) at which the first-order necessary conditions for optimality are satisfied, but strict complementarity does not hold. One important class of methods for solving general quadratic programming problems are called inertia-controlling quadratic programming (ICQP) methods. We derive a computational scheme for proceeding at a dead point that is appropriate for a general ICQP method.

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Checking local optimality in constrained quadratic programming is NP-hard

Cited by 167 publications

References 3 publications

Interior Methods for Nonlinear Optimization

Interior Methods for Nonlinear Optimization

Scheduling Parallel Jobs with Linear Speedup

On the Identification of Local Minimizers in Inertia-Controlling Methods for Quadratic Programming

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