Optimization of a Quadratic Function with a Circulant Matrix

Phuong, Nguyen Thi Hoai; Tụy, Hoàng; Al-Khayyal, Faiz A.

doi:10.1007/s10589-006-6448-y

Cited by 5 publications

(7 citation statements)

References 11 publications

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“…Here, the matrix C is circulant. In Table 3, we compare our algorithm with the algorithm presented in [13]. The latter is based on a problem-specific enumeration scheme.…”

Section: Non-convex Ternary Instancesmentioning

confidence: 99%

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Semidefinite relaxations for non-convex quadratic mixed-integer programming

Buchheim

Wiegele

2012

Math. Program.

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Abstract. We present semidefinite relaxations for unconstrained nonconvex quadratic mixed-integer optimization problems. These relaxations yield tight bounds and are computationally easy to solve for mediumsized instances, even if some of the variables are integer and unbounded. In this case, the problem contains an infinite number of linear constraints; these constraints are separated dynamically. We use this approach as a bounding routine in an SDP-based branch-and-bound framework. In case of a convex objective function, the new SDP bound improves the bound given by the continuous relaxation of the problem. Numerical experiments show that our algorithm performs well on various types of non-convex instances.

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“…Here, the matrix C is circulant. In Table 3, we compare our algorithm with the algorithm presented in [13]. The latter is based on a problem-specific enumeration scheme.…”

Section: Non-convex Ternary Instancesmentioning

confidence: 99%

“…The latter is based on a problem-specific enumeration scheme. The timings for the algorithm of Phuong et al are taken from [13], as the code is not available. They used a Pentium IV 2.53 GHz, a slightly slower machine than we used for our experiments.…”

Section: Non-convex Ternary Instancesmentioning

confidence: 99%

Semidefinite relaxations for non-convex quadratic mixed-integer programming

Buchheim

Wiegele

2012

Math. Program.

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“…In [5] the blade optimisation problem is conveniently restricted to locating the optimal arrangement of a fixed number of blades belonging to two different typologies. Furthermore, it is shown that such restriction enables the casting of the problem in a C-UBQP form.…”

Section: An Examplementioning

confidence: 99%

“…Furthermore, it is shown that such restriction enables the casting of the problem in a C-UBQP form. Eventually, [5] provides truly optimal (non-heuristic) solutions for some sample cases involving 40 discrete optimisation variables. Notably, it would be hard to deliver truly optimal solutions for larger problems since the combinatorial nature of the problem results in computation times exponentially increasing with the number of variables.…”

Section: An Examplementioning

confidence: 99%

“…It is shown that a certain type of Unconstrained Binary Quadratic Programming (UBQP) problems can be solved in heuristic terms by suitably designed ∆Σ modulators. The approach is practically demonstrated by considering an optimisation problem coming from the domain of compression systems [4], [5] and the solutions obtained by the ∆Σ approach are found to be as good as the truly optimal ones for those cases where the latter are known. Notably, the intuition that the operation of ∆Σ modulators is tied to the ability of solving a discrete optimisation problem is ready present in the Literature [6].…”

Section: Introductionmentioning

confidence: 99%

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A heuristic solution to the optimisation of flutter control in compression systems (and to some more binary quadratic programming problems) via ΔΣ modulation circuits

Callegari

Bizzarri

2010

Proceedings of 2010 IEEE International Symposium on Circuits and Systems

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An example of how circuit related techniques can help solving optimisatin problems originating from completely different domains is provided. It is shown that a specific class of Unconstrained Binary Quadratic Programming (UBQP) problems, including those arising in the optimisation of flutter control via blade mistuning, can be solved by means of ∆Σ modulators. This is done in steps, first restating the UBQP problem as a specific signal processing problem, and then attacking the latter via the design of a ∆Σ modulator with a suitably derived Noise Transfer Function. A (heuristically) optimal solution for the original problem is finally obtained from the modulator output stream. The method is validated by two numerical examples arising in the design of turbo-machines.

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On the synthesis of periodic signals by discrete pulse-trains and optimisation techniques

Bizzarri

Callegari

Rovatti

et al. 2009

2009 European Conference on Circuit Theory and Design

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In this paper, the synthesis of discrete sequences capable of representing analog periodic waveforms and of being transformed into them by mere low-pass filtering is considered. The problem is known to be solvable by modulation techniques such as Pulse Width Modulation or ΣΔ. Nonetheless, how close to an optimum such techniques can go is an interesting issue to explore, since the design of such modulators relies on assumptions that can only be approximately respected. To this aim, we develop an exact optimisation framework capable of providing the best and most parsimonious coding possible, which is then used as a benchmark. It is shown that the optimisation problem has very strong features, which could potentially be exploited to speed up convergence or to develop efficient heuristics. At the same time it is verified that the ΣΔ performance lag with regards to the optimum is generally very little, so that almost no margin exists for alternative heuristics unless the latter are exploited to optimise on merit factors beyond mere SNR

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Optimization of a Quadratic Function with a Circulant Matrix

Cited by 5 publications

References 11 publications

Semidefinite relaxations for non-convex quadratic mixed-integer programming

Semidefinite relaxations for non-convex quadratic mixed-integer programming

A heuristic solution to the optimisation of flutter control in compression systems (and to some more binary quadratic programming problems) via ΔΣ modulation circuits

On the synthesis of periodic signals by discrete pulse-trains and optimisation techniques

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Optimization of a Quadratic Function with a Circulant Matrix

Cited by 5 publications

References 11 publications

Semidefinite relaxations for non-convex quadratic mixed-integer programming

Semidefinite relaxations for non-convex quadratic mixed-integer programming

A heuristic solution to the optimisation of flutter control in compression systems (and to some more binary quadratic programming problems) via &#x0394;&#x03A3; modulation circuits

On the synthesis of periodic signals by discrete pulse-trains and optimisation techniques

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A heuristic solution to the optimisation of flutter control in compression systems (and to some more binary quadratic programming problems) via ΔΣ modulation circuits