An Efficient Random Algorithm for Box Constrained Weighted Maximin Dispersion Problem

Jinjin Huang

doi:10.4236/apm.2019.94015

Abstract

The box-constrained weighted maximin dispersion problem is to find a point in an n-dimensional box such that the minimum of the weighted Euclidean distance from given m points is maximized. In this paper, we first reformulate the maximin dispersion problem as a non-convex quadratically constrained quadratic programming (QCQP) problem. We adopt the successive convex approximation (SCA) algorithm to solve the problem. Numerical results show that the proposed algorithm is efficient.

Highlights

The weighted maximin problem model with box constraints is as follows:{ } ( ) m= ax x∈χ f x : mini= 1,m ωi x − xi 2 (1){ ( ) } where χ = y ∈ Rn y12, yn2,1 T ∈κ, κ is a convex cone; x1, xm ∈ Rn are m given points; these m points are equivalent to m locations; ωi > 0 for i = 1, m and ⋅ denotes the Euclidean norm
The box-constrained weighted maximin dispersion problem is to find a point in an n-dimensional box such that the minimum of the weighted Euclidean distance from given m points is maximized
Numerical results show that the proposed algorithm is efficient

Summary

Introduction

The weighted maximin problem model with box constraints is as follows:. { ( ) } where χ = y ∈ Rn y12 , , yn , T ∈κ , κ is a convex cone; x1, , xm ∈ Rn are m given points; these m points are equivalent to m locations; ωi > 0 for i = 1, , m and ⋅ denotes the Euclidean norm. In paper [5], they consider the 2 problem of finding a point in a unit n-dimensional p - ball (p ≥ 2) such that the minimum of the weighted Euclidean distance from given m points is maximized They show in paper [6] that the SDP-relaxation-based approximation algorithm provides the first theoretical approximation bound of ( ) 1− O ln (m) n . We model the maximin dispersion problem as a Quadratically constrained quadratic programming (QCQP), noting that (1) is a non-smooth, non-convex optimization problem, because the point-wise minimum of convex quadratics is non-differentiable and non-concave We solve this problem with a general approximation framework, which is successive convex approximation (SCA), which can be summarized as follows: each quadratic component of (1) is locally linearized at the current iteration to construct its convex approximation function, so we obtain a convex subproblem.

Technical Preliminaries

Algorithm of Generation

Numerical Results

Conclusion