Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm

Nataraj, P. S. V.; Arounassalame, M.

doi:10.1007/s10898-009-9485-0

Cited by 24 publications

(20 citation statements)

References 16 publications

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“…Remark 2: The above theorem says that the minimum and maximum coefficients of b i (x) provide lower and upper bounds for the range of p. This forms the Bernstein range enclosure, defined by B(x) in equation (11). Figure 1 shows for a univariate polynomial p, its Bernstein coefficients (b 0 , b 1 , .…”

Section: Bernstein Global Optimization Algorithmmentioning

confidence: 99%

“…This procedure is based on the well-known Bernstein form of polynomials [9], and uses several nice properties associated with this Bernstein form. Optimization procedures based on this Bernstein form, also called Bernstein global optimization algorithms, have shown good promise to solve hard nonconvex NLP and MINLP problems (see, for instance, [10], [11], [12]). They are therefore very promising for NMPC applications.…”

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confidence: 99%

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Nonlinear model predictive control based on Bernstein global optimization with application to a nonlinear CSTR

Patil

Maciejowski

Ling

2016

2016 European Control Conference (ECC)

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We present a model predictive control based tracking problem for nonlinear systems based on global optimization. Specifically, we introduce a 'Bernstein global optimization' procedure and demonstrate its applicability to the aforementioned control problem. This Bernstein global optimization procedure is applied to predictive control of a nonlinear CSTR system. Its strength and benefits are compared with those of a sub-optimal procedure, as implemented in MATLAB using fmincon function, and two well established global optimization procedures, BARON and BMIBNB.

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Section: Bernstein Global Optimization Algorithmmentioning

confidence: 99%

mentioning

confidence: 99%

Nonlinear model predictive control based on Bernstein global optimization with application to a nonlinear CSTR

Patil

Maciejowski

Ling

2016

2016 European Control Conference (ECC)

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“…The latter ones require the ability to compute tight bounds for the range of the objective function and the functions describing the constraints over the considered search region. In the case of polynomial optimization problems, one can make use of the expansion of a polynomial into Bernstein polynomials, see [7], [16], [18], [19], [21], [22]. Then the minimum and maximum of the coefficients of this expansion, the so-called Bernstein coefficients, provide bounds for the range of the polynomial over the search region.…”

Section: Introductionmentioning

confidence: 99%

Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials

Titi

Garloff

2017

Applied Mathematics and Computation

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In this paper, multivariate polynomials in the Bernstein basis over a simplex (simplicial Bernstein representation) are considered. Two matrix methods for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented. Also matrix methods for the calculation of the Bernstein coefficients over subsimplices generated by subdivision of the standard simplex are proposed and compared with the use of the de Casteljau algorithm. The evaluation of a multivariate polynomial in the power and in the Bernstein basis is considered as well. All the methods solely use matrix operations such as multiplication, transposition, and reshaping; some of them rely also on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. The latter one enables the use of the Fast Fourier Transform hereby reducing the amount of arithmetic operations.

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“…However, this method needs the repeated evaluation of Bernstein coefficients, which is a computationally costlier process. The Bernstein coefficient contraction algorithm was proposed in [9] for solving global optimization problems. We combine the advantage of the Bernstein Krawczyk algorithm, and the Bernstein coefficient contraction algorithm and propose a new algorithm to solve the electrical circuit analysis problems, which are modeled as a system of polynomials.…”

Section: Introductionmentioning

confidence: 99%

“…A Bernstein algorithm that uses the Krawczyk operator [11] for pruning step was proposed in [9]. A major drawback of this method is that it requires the repeated computation of Bernstein coefficients for the new domains.…”

Section: Introductionmentioning

confidence: 99%

Analysis of nonlinear electrical circuits using bernstein polynomials

Arounassalame

2012

Int. J. Autom. Comput.

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In electrical circuit analysis, it is often necessary to find the set of all direct current (d.c.) operating points (either voltages or currents) of nonlinear circuits. In general, these nonlinear equations are often represented as polynomial systems. In this paper, we address the problem of finding the solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n-dimensional box. Branch and Bound algorithms based on interval methods can give guaranteed enclosures for the solution. However, because of repeated evaluations of the function values, these methods tend to become slower. Branch and Bound algorithm based on Bernstein coefficients can be used to solve the systems of polynomial equations. This avoids the repeated evaluation of function values, but maintains more or less the same number of iterations as that of interval branch and bound methods. We propose an algorithm for obtaining the solution of polynomial systems, which includes a pruning step using Bernstein Krawczyk operator and a Bernstein Coefficient Contraction algorithm to obtain Bernstein coefficients of the new domain. We solved three circuit analysis problems using our proposed algorithm. We compared the performance of our proposed algorithm with INTLAB based solver and found that our proposed algorithm is more efficient and fast.

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Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm

Abstract: Bernstein polynomials, Constrained global optimization, Subdivision, Pruning,

Cited by 24 publications

References 16 publications

Nonlinear model predictive control based on Bernstein global optimization with application to a nonlinear CSTR

Nonlinear model predictive control based on Bernstein global optimization with application to a nonlinear CSTR

Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials

Analysis of nonlinear electrical circuits using bernstein polynomials

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