Global optimization algorithms for bound constrained problems

Pajor, László

doi:10.14232/phd.625

Cited by 5 publications

(8 citation statements)

References 78 publications

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“…Proof. This follows directly using Theorem 3.1, Corollary 2.3, the fact that the largest root of P α,β d+1 is equal to −ξ β,α d+1 , and the closed form expression (13) for the roots of the Chebyshev polynomials of the first kind.…”

Section: Tight Lower Bound For the Lasserre Hierarchymentioning

confidence: 91%

Worst-Case Examples for Lasserre’s Measure–Based Hierarchy for Polynomial Optimization on the Hypercube

Klerk

Laurent

2020

Mathematics of OR

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We study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864 − 885], and a related hierarchy by De Klerk, Hess and Laurent [SIAM J. Optim. 27(1), (2017) pp. 347 − 367]. For polynomial optimization over the hypercube we show a refined convergence analysis for the first hierarchy. We also show lower bounds on the convergence rate for both hierarchies on a class of examples. These lower bounds match the upper bounds and thus establish the true rate of convergence on these examples. Interestingly, these convergence rates are determined by the distribution of extremal zeroes of certain families of orthogonal polynomials.

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Section: Tight Lower Bound For the Lasserre Hierarchymentioning

confidence: 91%

Worst-Case Examples for Lasserre’s Measure–Based Hierarchy for Polynomial Optimization on the Hypercube

Klerk

Laurent

2020

Mathematics of OR

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“…The description and even various implementations of the examined problems can be found in several resources and online collections. For example, the compact mathematical formulation and known optima of all of the mentioned problems can be found in Appendix A at Pal [13]. Our computational results are summarized in Table 3.…”

Section: Computational Test Results On Global Optimization Problemsmentioning

confidence: 99%

“…The solver needs an initial search interval for every variable, so we set the [−100, 100] interval as initial box of every variable in our self-made test cases (Table 2), and the usual boxes for the standard problems (see for example Appendix A in [13]). In the transformed cases, the bounds were transformed appropriately.…”

Section: On the Advantages Of The Transformationsmentioning

confidence: 99%

Nonlinear Symbolic Transformations for Simplifying Optimization Problems

Antal¹,

Csendes²

2016

Acta Cybern

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The theory of nonlinear optimization traditionally studies numeric computations. However, increasing attention is being paid to involve computer algebra into mathematical programming. One can identify two possibilities of applying symbolic techniques in this field. Computer algebra can help the modeling phase by producing alternate mathematical models via symbolic transformations. The present paper concentrates on this direction. On the other hand, modern nonlinear solvers use more and more information about the structure of the problem through the optimization process leading to hybrid symbolic-numeric nonlinear solvers. This paper presents a new implementation of a symbolic simplification algorithm for unconstrained nonlinear optimization problems. The program can automatically recognize helpful transformations of the mathematical model and detect implicit redundancy in the objective function.We report computational results obtained for standard global optimization test problems and for other artificially constructed instances. Our results show that a heuristic (multistart) numerical solver takes advantage of the automatically produced transformations.New theoretical results will also be presented, which help the underlying method to achieve more complicated transformations.

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“…Minden hivatkozott feladat tömör matematikai leírása az ismert optimumokkal együtt megtalálható pl. Pál László értekezésének A. függelékében is [68]. Egy-két kivétellel a feladatok részletes bemutatásától eltekintek, annál is inkább, mivel a dolgozat eredményeinek megértéséhez nem feltétlenül szükséges ezek ismerete.…”

Section: Teszthalmazunclassified

“…például [68] A. függelékét). Az átalakított feladatoknál a befoglaló intervallumot is transzformáltam az átírásnak megfelelően.…”

Section: áLlítás Ha Azunclassified

A matematikai modellezés hatása nemlineáris optimalizálási feladatok megoldásának hatékonyságára

Antal

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Global optimization algorithms for bound constrained problems

Cited by 5 publications

References 78 publications

Worst-Case Examples for Lasserre’s Measure–Based Hierarchy for Polynomial Optimization on the Hypercube

Worst-Case Examples for Lasserre’s Measure–Based Hierarchy for Polynomial Optimization on the Hypercube

Nonlinear Symbolic Transformations for Simplifying Optimization Problems

A matematikai modellezés hatása nemlineáris optimalizálási feladatok megoldásának hatékonyságára

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