Duality results for nonlinear single minimax location problems via multi-composed optimization

Wanka, Gert; Wilfer, Oleg

doi:10.1007/s00186-017-0603-3

Cited by 7 publications

(7 citation statements)

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“…In [27] the authors approached nonlinear minmax location problems by means of the conjugate duality in the case where the distances were measured by gauge functions. In this section we investigate such location problems in a more general setting, namely, where the distances are measured by perturbed minimal time functions.…”

Section: Location Problem With Perturbed Minimal Time Functionsmentioning

confidence: 99%

“…With these newly introduced functions we can write the optimization problem (P S h,T ) as a multi-composed optimization problem (cf. [25,27])…”

Section: Remark 32mentioning

confidence: 99%

“…, n, are proper, convex and lower semicontinuous, it is obvious that the function F is proper, R n +convex and R n + -epi-closed. In addition, as the function G is linear continuous, it follows that the function F does not need to be monotone as asked in the general theory in [10,25,27,29].…”

Section: Remark 32mentioning

confidence: 99%

“…Additionally to this dual problem, we consider the following one, that is equivalent to it in the sense that they share the same optimal objective value and this can be proven similarly to the proof of [27,Theorem 9],…”

Section: Remark 310mentioning

confidence: 99%

“…To be able to deal with the considered general nonlinear minmax location problems by means of conjugate duality we rewrite them as multi-composed optimization problems, for which we have recently proposed a duality approach in [10,25,27,29]. The corresponding necessary and sufficient optimality conditions are then derived.…”

mentioning

confidence: 99%

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A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality

Grad

Wilfer

2019

J Glob Optim

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We investigate via a conjugate duality approach general nonlinear minmax location problems formulated by means of an extended perturbed minimal time function, necessary and sufficient optimality conditions being delivered together with characterizations of the optimal solutions in some particular instances. A parallel splitting proximal point method is employed in order to numerically solve such problems and their duals. We present the computational results obtained in matlab on concrete examples, successfully comparing these, where possible, with earlier similar methods from the literature. Moreover, the dual employment of the proximal method turns out to deliver the optimal solution to the considered primal problem faster than the direct usage on the latter. Since our technique successfully solves location optimization problems with large data sets in high dimensions, we envision its future usage on big data problems arising in machine learning.Keywords Gauge (Minkowski) function · Minimal time function · Minmax multifacility location problem · Sylvester problem · Apollonius problem · Proximal point algorithm · Epigraphical projection · Projection operator · Machine learning 1 PreliminariesIn this paper we investigate nonlinear minmax location problems that are generalizations of the classical Sylvester problem in location theory-not to be confused with Sylvester's line

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Section: Location Problem With Perturbed Minimal Time Functionsmentioning

confidence: 99%

“…With these newly introduced functions we can write the optimization problem (P S h,T ) as a multi-composed optimization problem (cf. [25,27])…”

Section: Remark 32mentioning

confidence: 99%

Section: Remark 32mentioning

confidence: 99%

Section: Remark 310mentioning

confidence: 99%

mentioning

confidence: 99%