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

Abstract

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

A note on the subdifferential of convex multi-composite functionals

“…In this section, we apply Theorem 3.1 to the optimality conditions of a constrained convex minmax location problem with perturbed minimal time functions and set-up costs. Such problems were recently investigated via conjugate duality approach; see [10] for more details.…”

“…Remark 4.1. Note that the decomposition of the objective function of problem (M L P) is completely different from that of [10], that is, the special construction of g, h (1) Let (x 1 , ...,…”

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