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.
Highlights
In 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
The original location optimization problem cannot be directly numerically solved by means of the usual algorithms because the involved functions often lack differentiability, while a direct employment of some proximal point method is not possible because of the complicated structure of the objective function, that consists of the maximum of n functions, each containing a composition of functions
Given the fact that the algorithm we propose is able to successfully solve location optimization problems with large data sets in high dimensions faster than its counterparts from the literature makes us confident regarding a future usage of this technique on big data problems arising in machine learning, for instance those approached by means of support vector techniques
Summary
In 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. The computational results show that the primal optimal solutions are obtained faster when numerically solving the dual problems One of these concrete examples was numerically solved in [13,17] by means of a subgradient method and a comparison of the computational results is provided as well, stressing once again the superiority of the algorithm proposed in the present paper. Let us point out that by the operations we defined on a Hausdorff locally convex space attached with a maximal element and on the extended real space, there holds 0 f = δdom f and (0Z∗ F) = δdom F for any f : X → R and F : X → Z. Last but not least denote the optimal objective value of an optimization problem (P) by v(P) and note that when an infimum/supremum is attained we write min/max instead of inf/sup. For more on convex optimization in Hilbert spaces we warmly recommend [3]
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