On the functional form of convex underestimators for twice continuously differentiable functions

Floudas, Christodoulos A.; Kreinovich, Vladik

doi:10.1007/s11590-006-0003-8

Cited by 13 publications

(5 citation statements)

References 19 publications

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“…, γ n are computed via the classical αBB method. Theoretical justification for αBB and γBB relaxation terms is given in [13]. Convex relaxations of quadratic functions were investigated in [7], linear relaxations in [9], and a generalization of McCormick relaxations in [28].…”

Section: Convex Underestimatorsmentioning

confidence: 99%

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The Effect of Hessian Evaluations in the Global Optimization αBB Method

Hladík

2017

Modeling, Simulation and Optimization of Complex Processes HPSC 2015

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We consider convex underestimators that are used in the global optimization αBB method and its variants. The method is based by augmenting the original nonconvex function by a relaxation term that is derived from an interval enclosure of the Hessian matrix. In this paper, we discuss the advantages of symbolic computation of the Hessian matrix. Symbolic computation often allows simplifications of the resulting expressions, which in turn means less conservative underestimators. We show by examples that even a small manipulation with the symbolic expressions, which can be processed automatically by computers, can have a large effect on the quality of underestimators.

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Section: Convex Underestimatorsmentioning

confidence: 99%

“…, γ n are computed via the classical αBB method. Theoretical justification for αBB and γBB relaxation terms is given in [13].…”

Section: Convex Underestimatorsmentioning

confidence: 99%

The Effect of Hessian Evaluations in the Global Optimization αBB Method

Hladík

2017

Modeling, Simulation and Optimization of Complex Processes HPSC 2015

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“…Akrotirianakis and Floudas [11] discussed their computational experience for the γ BB underestimators on box-constrained problems. Floudas and Kreinovich [60,61] proved that the classical and the exponential αBB underestimators are the only two functional forms possessing both shift-invariance and sign-invariance, and are therefore the two natural choices for any αBB-like convexification methodology.…”

Section: Literature Reviewmentioning

confidence: 99%

A Generalization of the Classical αBB Convex Underestimation via Diagonal and Nondiagonal Quadratic Terms

Skjäl

Westerlund

Misener

et al. 2012

J Optim Theory Appl

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The classical αBB method determines univariate quadratic perturbations that convexify twice continuously differentiable functions. This paper generalizes αBB to additionally consider nondiagonal elements in the perturbation Hessian matrix. These correspond to bilinear terms in the underestimators, where previously all nonlinear terms were separable quadratic terms. An interval extension of Gerschgorin's circle theorem guarantees convexity of the underestimator. It is shown that underestimation parameters which are optimal, in the sense that the maximal underestimation error is minimized, can be obtained by solving a linear optimization model.Theoretical results are presented regarding the instantiation of the nondiagonal underestimator that minimizes the maximum error. Two special cases are analyzed to convey an intuitive understanding of that optimally-selected convexifier. Illustrative examples that convey the practical advantage of these new αBB underestimators are presented.

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“…, n, are determined such that g(x) is convex. A generalization of this method using non-diagonal shifting appeared in Akrotirianakis et al [5] and Skjäl et al [38,39], and a different form of an understimator using exponentials instead of a quadratic function as a perturbation function was discussed in [3,4,14,15]. Other forms of convex and linear relaxations were investigated in Anstreicher [7], Domes and Neumaier [10], and Scott et al [37], for instance.…”

Section: Introductionmentioning

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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On the functional form of convex underestimators for twice continuously differentiable functions

Cited by 13 publications

References 19 publications

The Effect of Hessian Evaluations in the Global Optimization αBB Method

The Effect of Hessian Evaluations in the Global Optimization αBB Method

A Generalization of the Classical αBB Convex Underestimation via Diagonal and Nondiagonal Quadratic Terms

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

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