A new subdivision algorithm for the Bernstein polynomial approach to global optimization

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

doi:10.1007/s11633-007-0342-7

Cited by 37 publications

(17 citation statements)

References 13 publications

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“…Following the notations given in [15,16], let l ∈ N be the number of variables and x = (x1, x2, · · · , x l ) ∈ R l . A multi-index I is defined as I = (i1, i2, · · · , i l ) ∈ N l and multi-power x I is defined as…”

Section: The Bernstein Formmentioning

confidence: 99%

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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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Section: The Bernstein Formmentioning

confidence: 99%

“…The subdivision strategy is generally more efficient than the degree elevation strategy [15,17] and is therefore preferred. A subdivision in the r-th direction (1 r l) is a bisection perpendicular to this direction.…”

Section: Subdivision Proceduresmentioning

confidence: 99%

Analysis of nonlinear electrical circuits using bernstein polynomials

Arounassalame

2012

Int. J. Autom. Comput.

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“…In many practical situations in engineering, data are only known to lie within intervals and only ranges of values are computed experimentally [13] . In such cases interval computation yield desired results [14,15] . Interval analysis method provides a direct means of solving the two highly nonlinear equations (11) and (12) to find the value of capacitance and predicting the behaviour of a self-excited induction generator under balanced operating conditions.…”

Section: Evolution Of Equations and Interval Analysismentioning

confidence: 99%

A reliable and accurate calculation of excitation capacitance value for an induction generator based on interval computation technique

Thakur¹,

Agarwal

Nataraj³

2011

Int. J. Autom. Comput.

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A squirrel cage induction generator (SCIG) offers many advantages for wind energy conversion systems but suffers from poor voltage regulation under varying operating conditions. The value of excitation capacitance (Cexct) is very crucial for the selfexcitation and voltage build-up as well as voltage regulation in SCIG. Precise calculation of the value of Cexct is, therefore, of considerable practical importance. Most of the existing calculation methods make use of the steady-state model of the SCIG in conjunction with some numerical iterative method to determine the minimum value of Cexct. But this results in over estimation, leading to poor transient dynamics. This paper presents a novel method, which can precisely calculate the value of Cexct by taking into account the behavior of the magnetizing inductance during saturation. Interval analysis has been used to solve the equations. In the proposed method, a range of magnetizing inductance values in the saturation region are included in the calculation of Cexct, required for the self-excitation of a 3-φ induction generator. Mathematical analysis to derive the basic equation and application of interval method is presented. The method also yields the magnetizing inductance value in the saturation region which corresponds to an optimum Cexct(min) value. The proposed method is experimentally tested for a 1.1 kW induction generator and has shown improved results.

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“…Following the notations given in [6,7] where N as the degree of p. We can expand a given multivariate polynomial into Bernstein polynomial to obtain bounds for its range over a l -dimensional box Without loss of generality, we consider the unit box since any non-empty box x of R l can be mapped affi nely onto this box. The I th Bernstein polynomial of degree N is defi ned as (4) where, for (5) The Bernstein coeffi cients b 1 (u) of p over the unit box u are given by (6) Thus the Bernstein form of a multivariate polynomial p is defi ned by (7) The Bernstein coefficients are collected in an array where .…”

Section: Bernstein Formsmentioning

confidence: 99%

“…By repeatedly applying domain subdivision, one can make the lower bound equal to the global minimum of the polynomial over the box. Motivated by these features, algorithms for unconstrained global optimization based on the Bernstein form are proposed in [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

An algorithm for constrained global optimization of multivariate polynomials using the Bernstein form and John optimality conditions

Nataraj

Arounassalame

2009

OPSEARCH

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We propose an algorithm for constrained global optimization of multivariate polynomials using the Bernstein form of polynomials. The proposed algorithm is of the branch and prune type, where branching is done using subdivision and pruning is done using the John optimality conditions for constrained minima. A main feature of this algorithm is that the branching and pruning operations are done with the Bernstein polynomial coeffi cients. The performance of the proposed algorithm is compared with those of existing global optimization techniques, on a few examples. The obtained results show the superiority of the proposed method over existing methods, in terms of number of iterations and computational time.

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A new subdivision algorithm for the Bernstein polynomial approach to global optimization

Cited by 37 publications

References 13 publications

Analysis of nonlinear electrical circuits using bernstein polynomials

Analysis of nonlinear electrical circuits using bernstein polynomials

A reliable and accurate calculation of excitation capacitance value for an induction generator based on interval computation technique

An algorithm for constrained global optimization of multivariate polynomials using the Bernstein form and John optimality conditions

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