Algorithms for unconstrained global optimization of nonlinear (polynomial) programming problems: The single and multi-segment polynomial B-spline approach

In this paper, we propose basic and improved algorithms based on polynomial B-spline form for constrained global optimization of multivariate polynomial functions. The proposed algorithms are based on a branch-and-bound framework. In improved algorithm we introduce several new ingredients, such as B-spline box consistency and B-spline hull consistency algorithm to prune the search regions and make the search more efficient. The performance of the basic and improved algorithm is tested and compared on set of test problems. The results of the tests show the superiority of the improved algorithm over the basic algorithm in terms of the chosen performance metrics. We compare optimal value of global minimum obtained using the proposed algorithms with CENSO, GloptiPoly and several state-of-the-art NLP solvers, on set of $11$ test problems. The results of the tests show the superiority of the proposed algorithm and CENSO solver (open source solver for global optimization of B-spline constrained problem) in that it always captures the global minimum to the user-specified accuracy.

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“…𝐷 𝑜 (x), 𝐷 𝑔𝑖 (x) and 𝐷 ℎ𝑗 (x) respectively. The algorithm in [8] is suggested for the computation. ;…”

Section: Begin Algorithmmentioning

confidence: 99%

Constrained global optimization of multivariate polynomials using polynomial B-spline form and B-spline consistency prune approach

Gawali¹,

Patil²,

Zidna

et al. 2021

RAIRO-Oper. Res.

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“…We will identify N as a type of dependence function (linear/non-linear) and the severity of this non-linearity [24]. For N = 1, we have a linear model.…”

Section: 2 Identification Of N and A Imentioning

confidence: 99%

Experimental Investigation of the Triple-Junction Effect on the Electric Discharge Characteristics and Modeling of the Maximum Discharge Current

Saim

Bitam-Megherbi

2021

ECTI-EEC

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The electrical discharge and breakdown characteristics of a triple junction (TJ) in high voltage (HV) electrical insulation are becoming an important issue in recent industrial applications. In this paper, a comparative study is conducted between three insulation materials used in HV: silicone, porcelain, and heat tempered glass, with electrode shape parameters such as inter-electrode distance (d), applied voltage (V), and geometry investigated for positive polarity, using an experimental approach. For optimum test organization to support our research and highlight the parameters and their interactions, we have established a mathematical regression model of the discharge current. It is polynomial, with good performance and a high level of precision. Besides, this yields one monovariable and another two-variable quadratic model. The latter can simulate the electric discharge simultaneously with two variables: V and d, and an adjusted determination coefficient () of nearly 0.99. Consequently, the two-variable quadratic model has been adapted to V and d ranges of 10 to 50 kV and 1.9 to 8 cm, respectively. Finally, this study provides us with new applicable knowledge about the TJ effect on electrical insulation with recommendations to optimize the design for higher performance.

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“…This algorithm takes the coefficients of polynomial in power form, degree of every polynomial variable, domain box and the limit of tolerance as input and provides the DC operating points, i.e., set of solution as an output. The algorithm proposed in [16] is used to obtain an array of B-Spline coefficients D(x) over a given search box x. We then initialize a list L with the item {x, D(x)} and solution list L sol to the empty list.…”

Section: Proposed Algorithm For Solving Systems Of Nonlinear Polynomial Equationsmentioning

confidence: 99%

“…The coefficients of the polynomial B-Spline form are known as B-Spline coefficients. In [16], a polynomial B-Spline coefficient computation algorithm is presented for solving global optimization problems. The use of this algorithm avoids evaluation of polynomial function derivative required to get interval enclosure of Krawczyk contractor as interval enclosure is obtained from B-Spline coefficients of polynomial B-Spline form of polynomial function in power form.…”

Section: Introductionmentioning

confidence: 99%

“…Begin Algorithm 1 {Determine the B-Spline coefficients} Determine the B-Spline coefficients array of the nonlinear polynomials D i (x) on the initial box x, where i = 1, 2, • • • , n, and n is the number of polynomial equations. (Use algorithms given in[16])2 {Initialize lists} Set L ← {x, D(x)}, L sol ← {}3 {Start iteration} if L = ∅ then go to 12 else pick the last item from L, denote it as {b, D(b)}, and delete this item entry from L end 4 {Box feasibility test} for i = 1 to n do if min (D i (b)) > 0 then Delete this box and go to 3 else if max (D i (b)) < 0 then Delete this box and go to 3 else go to 5 end end {Accepting the new DC operating points} if wid(b) < then store b in L sol and go to 3 B-Spline Krawczyk operator algorithm} Execute B-Spline Krawczyk operator algorithm, [K] = BKO (D i (b), b) 7 {Determine the new contracted bounds} The contracted bounds are computed as b new = b ∩ K 8 {Discard the new contracted bounds} if b new = ∅ then discard the item {b, D(b)} go to 3 else go to 9 end 9 {Subdivision decision} if any(wid(b new ) < 0.8 × wid(b)) then go to 10 else go to 11 end 10 {Compute B-Spline coefficients for contracted new bound b new and store the item {b new , D i (b new )} in the list L } Compute the B-Spline coefficients over the new contracted bound b new , i.e. D i (b new ) and store the item {b new , D i (b new )} in the list L and go to 3 11 {Generate two items} (a) Obtain subdivision direction r using maximum width rule for the box b .…”

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confidence: 99%

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An Integrative Approach for Analysis of Nonlinear Electrical Circuits Using-Polynomial B-Spline Expansion and B-Spline Krawczyk Operator

Gawali¹,

Zidna

Nataraj

2021

Int. J. Appl. Comput. Math

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This paper addresses the problem of finding a set of all direct current (DC) operating points of a nonlinear circuit, which is a crucial step in its development and requires the solution of a nonlinear system of polynomial equations. We propose a novel algorithm for finding the set of all solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n dimensional box. The proposed algorithm is based on the following techniques: (i) B-Spline expansion to obtain a polynomial B-Spline form of the original polynomial in power form; (ii) B-Spline Krawczyk contractor for domain pruning. To avoid the repeated evaluation of function value the algorithm suggested uses B-Spline coefficients to find the value of Krawczyk operator and the computation of derivative of polynomial function. We solved three circuit analysis problems using the proposed algorithm and compared the performance of proposed algorithm with INTLAB-based solver and found that the former is more efficient in terms of computation time and number of iterations.

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Algorithms for unconstrained global optimization of nonlinear (polynomial) programming problems: The single and multi-segment polynomial B-spline approach

Cited by 8 publications

References 10 publications

Constrained global optimization of multivariate polynomials using polynomial B-spline form and B-spline consistency prune approach

Constrained global optimization of multivariate polynomials using polynomial B-spline form and B-spline consistency prune approach

Experimental Investigation of the Triple-Junction Effect on the Electric Discharge Characteristics and Modeling of the Maximum Discharge Current

An Integrative Approach for Analysis of Nonlinear Electrical Circuits Using-Polynomial B-Spline Expansion and B-Spline Krawczyk Operator

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