New bounds on the unconstrained quadratic integer programming problem

Halikias, George; Jaimoukha, Imad M.; Malik, Usman; Gungah, S. K.

doi:10.1007/s10898-007-9155-z

Cited by 11 publications

(4 citation statements)

References 16 publications

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“…In [19] (for other approaches see e.g., [20]), a result for tightening this duality gap is provided. While for zonotopes where n m, the combined approach from [19, Corollary 2 & Lemma 3] is not applicable, the following example demonstrates the usefulness for cases where n ≈ m.…”

Section: Theorem 1 (Enclosing Ellipsoid)mentioning

confidence: 99%

Scalable Zonotope-Ellipsoid Conversions using the Euclidean Zonotope Norm

Gaßmann

Althoff

2020

2020 American Control Conference (ACC)

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Set-based computations become increasingly popular for safety-critical systems to ensure properties of controllers and observers. To efficiently compute various set operations, one often uses different set representations and conversions between them. Two popular set representations, for which scalable conversion algorithms do not yet exist, are zonotopes and ellipsoids. We provide computational approaches for all four conversion cases, i.e., overapproximations and underapproximations from zonotopes to ellipsoids and vice versa. By using upper bounds on the maximum and lower bounds on the minimum Euclidean norm of a given zonotope, our approaches have polynomial complexity and thus can be used for high-dimensional spaces. We show that the tightness of our approaches directly depends on the tightness of the Euclidean norm. Numerical experiments demonstrate the usefulness of our proposed methods.

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Section: Theorem 1 (Enclosing Ellipsoid)mentioning

confidence: 99%

Scalable Zonotope-Ellipsoid Conversions using the Euclidean Zonotope Norm

Gaßmann

Althoff

2020

2020 American Control Conference (ACC)

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“…To avoid the conservatism associated with this approach, and at the same time handle the combinatorial nature of the discrete problem, we develop a procedure for representing a general class of uncertainties involving all combinations of binary variables and then use an elimination lemma and an extension of a semidefinite relaxation procedure for binary (0, 1) variables [26], [27], [28], to derive conditions for their solution. That is, we extend the robustness results in [14], [15], which deal with continuous norm-bounded structured uncertainties, to discrete structured uncertainties.…”

Section: Uncertaintymentioning

confidence: 99%

“…This is in common with the corresponding results in [14], [15] for continuous uncertainties. While the results in [27], [28] can be used to investigate the circumstances under which our conditions are also necessary, this falls outside the scope of this work. Our numerical experience, reported in Section V, indicates that they are sufficiently tight for practical systems.…”

Section: A Tractable Solution To the Fault-tolerantmentioning

confidence: 99%

A Semidefinite Relaxation Procedure for Fault-Tolerant Observer Design

Sevilla

Jaimoukha

Chaudhuri

et al. 2015

IEEE Trans. Automat. Contr.

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Abstract-A fault-tolerant observer design methodology is proposed. The aim is to guarantee a minimum level of closed-loop performance under all possible sensor fault combinations while optimizing performance under the nominal, fault-free condition. A novel approach is proposed to tackle the combinatorial nature of the problem, which is computationally intractable even for a moderate number of sensors, by recasting the problem as a robust performance problem, where the uncertainty set is composed of all combinations of a set of binary variables. A procedure based on an elimination lemma and an extension of a semidefinite relaxation procedure for binary variables is then used to derive sufficient conditions (necessary and sufficient in the case of one binary variable) for the solution of the problem which significantly reduces the number of matrix inequalities needed to solve the problem. The procedure is illustrated by considering a fault-tolerant observer switching scheme in which the observer outputs track the actual sensor fault condition. A numerical example from an electric power application is presented to illustrate the effectiveness of the design.

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“…Malik et al propose a spectral decomposition based method to estimate the relaxation gap for improving the lower bound [10]. Halikias et al [7] adopt the spectral decomposition based method for improving the SDR bound, and this type of methods is further studied by Lu et al [9]. Another type of convex relaxations is to reformulate the BQP problem into a copositive programming problem [4] for obtaining tighter relaxations.…”

Section: Introductionmentioning

confidence: 99%

Convex reformulation for binary quadratic programming problems via average objective value maximization

Lü

Guo

2014

Optim Lett

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Quadratic convex reformulation is an important method for improving the performance of a branch-and-bound based binary quadratic programming solver. In this paper, we study a new convex reformulation method. By this reformulation, the efficiency of a branch-and-bound algorithm can be improved significantly. We also compare this new reformulation method with other proposed methods, whose effectiveness has been proven. Numerical experimental results show that our reformulation method performs better than the compared methods for certain types of binary quadratic programming problems.

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New bounds on the unconstrained quadratic integer programming problem

Cited by 11 publications

References 16 publications

Scalable Zonotope-Ellipsoid Conversions using the Euclidean Zonotope Norm

Scalable Zonotope-Ellipsoid Conversions using the Euclidean Zonotope Norm

A Semidefinite Relaxation Procedure for Fault-Tolerant Observer Design

Convex reformulation for binary quadratic programming problems via average objective value maximization

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