The Effect of Hessian Evaluations in the Global Optimization αBB Method

Hladík, Milan

doi:10.1007/978-3-319-67168-0_6

Cited by 3 publications

(1 citation statement)

References 30 publications

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“…The efficiency of symbolic computation of the Hessian matrix was studied in Hladík [23]. Symbolic evaluation has a big potential in determining much tighter underestimators, however, it is very difficult to implement symbolic expression simplifications automatically by a computer program.…”

Section: Theorem 1 (Scaled Gerschgorin Inclusion)mentioning

confidence: 99%

An extension of the $$\alpha \hbox {BB}$$ α BB -type underestimation to linear parametric Hessian matrices

Hladík

2015

J Glob Optim

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The classical αBB method is a global optimization method the important step of which is to determine a convex underestimator of an objective function on an interval domain. Its particular point is to enclose the range of a Hessian matrix in an interval matrix. To have a tighter estimation of the Hessian matrices, we investigate a linear parametric form enclosure in this paper. One way to obtain this form is by using a slope extension of the Hessian entries. Numerical examples indicate that our approach can sometimes significantly reduce overestimation on the objective function. However, the slope extensions highly depend on a choice of the center of linearization. We compare some naive choices and also propose a heuristic one, which performs well in executed examples, but it seems there is no one global winner.

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Section: Theorem 1 (Scaled Gerschgorin Inclusion)mentioning

confidence: 99%

An extension of the $$\alpha \hbox {BB}$$ α BB -type underestimation to linear parametric Hessian matrices

Hladík

2015

J Glob Optim

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Linear interval parametric approach to testing pseudoconvexity

2020

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Hybrid PSO-α BB Global Optimisation for C² Box-Constrained Multimodal NLPs

2022

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αBB is an elegant deterministic branch and bound global optimisation that guarantees global optimum convergence with minimal parameter tuning. However, the method suffers from a slow convergence speed calling for computational improvements in several areas. The current paper proposes hybridising the branch and bound process with particle swarm optimisation to improve its global convergence speed when solving twice differentiable (C 2 ) box-constrained multimodal functions. This hybridisation complemented with interval analysis leads to an early discovery of the global optimum, quicker pruning of suboptimal regions in the problem space, thus improving global convergence. Also, when used as a heuristic search algorithm, the hybrid algorithm yields superior solution accuracy owing to the combined search capabilities of PSO and the branch and bound framework. Computational experiments have been conducted on CEC 2017/2019 test sets and on n-dimensional classical test sets yielding improved convergence speed in the complete search configuration and superior solution accuracy in the heuristic search configuration. INDEX TERMS αBB, particle swarm optimisation (PSO), box-constrained NLPs, hybridisation.

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The Effect of Hessian Evaluations in the Global Optimization αBB Method

Cited by 3 publications

References 30 publications

An extension of the $$\alpha \hbox {BB}$$ α BB -type underestimation to linear parametric Hessian matrices

An extension of the $$\alpha \hbox {BB}$$ α BB -type underestimation to linear parametric Hessian matrices

Linear interval parametric approach to testing pseudoconvexity

Hybrid PSO-α BB Global Optimisation for C² Box-Constrained Multimodal NLPs

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