The weighted maximin problem model with box constraints is as follows:
where , is a convex cone; are m given points; these m points are equivalent to m locations; for and denotes the Euclidean norm. In our numerical calculation, is equal to 1. The goal is to find a point in a closed set such that the minimum of the weighted Euclidean distance from given set of points in is maximized. The weighted maximin problem has been widely used in spatial management, facility location, and pattern recognition.
The weighted maximin dispersion problem with box constraints is known to be NP-hard in general  . For the low-dimensional cases ( and being a polyhedral set), it is solvable in polynomial time   . For , a heuristic algorithm   is used to solve this problem.
In paper  , they look for approximate solution through convex relaxation,
and prove that the approximation bound is , where depends on , when or , . In paper  , they
use the linear programming relaxation method to give the approximate bounds of the ball problem, which is . In paper  , they consider the problem of finding a point in a unit n-dimensional (p ≥ 2) such that the minimum of the weighted Euclidean distance from given m points is maximized. They show in paper  that the SDP-relaxation-based approximation algorithm provides the first theoretical approximation bound of .
In this paper, firstly, 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. The solution of each subproblem is then used as the point about which we construct a convex surrogate function in the next iteration, repeat the steps, and then adopt the random block coordinate descent method (RBCDM) to obtain the solution of subproblem.
The remainder of the paper is organized as follows. In Section 2, we give technical preliminaries. In Section 3, we first reformulate maximin dispersion problem as a QCQP problem. Then, we describe the overall SCA approach and use the proposed methods (RBCDM) for solving each subproblem. In Section 4, we present some numerical results. Conclusions are made in Section 5.
2. Technical Preliminaries
The following concepts or definitions are adopted in our paper.
We use to denote the space of n dimensional real valued vectors, and , we denote the ith component of x by . Thus, each is a column vector
Let and be a set, then the distance of the point y from the set is defined as
3. Algorithm of Generation
We now reformulation (1) into the following equivalent form,
and we finally obtain
and we will work with this formulation, note that the problem still remains non-convex.
We first introduce our algorithm ideas. First, we construct a convex optimization function of the non-convex objective function (3) by locally linearizing each quadratic component of (3) about the iterate point , we obtain a n-dimensional convex subproblem. Second, we adopt random block coordinate descent method (RBCDM) to transform the n-dimensional convex subproblem into one-dimensional convex subproblem to reduce the computational complexity, here, the optimization variables be decomposed into n independent blocks. At each iteration of this method, random one of the components of variable is optimized, while the remaining variables are held fixed, until all components of a variable are updated, remember as a round, repeat the above steps until we achieve the effect we want. Such block structure can lead to low-complexity algorithms. Finally, to solve the one-dimensional subproblem.
Defining , where . Since is concave for , on locally linearizing about the current iterate point , we can obtain a global upper-bound of original objective . At the point , we construct a convex approximation function of at as follows:
where , for .
Define , the piecewise linear function is an upper bound of the original function at , which is tight at  . We replace with its piecewise linear approximation about to obtain the non-smooth, convex subproblem.
This subproblem is computationally expensive, so we transform the high-dimensional problem into one-dimensional problem to reduce the complexity.
The concrete steps are as follows: We random update the jth component of x at the current iterate point and keep the other components unchanged，it must be noted that the is a component of random selection, let , so we have
obtain the one-dimensional convex subproblem
In order to solve the solution of one-dimensional piecewise linear function (5), we first arrange the of the m lines from small to large, i.e. . For the convenience of description, we remember these m lines as , where . The following is the algorithmic frameworks for solving one-dimensional subproblem.
4. Numerical Results
In order to benchmark the performance of our proposed algorithms, we do some simple numerical comparisons. We do numerical experiments on 4 random instances when dimension n takes different values, respectively, such as n = 100, 500, 1000, 2000. The corresponding m we chose smaller than n, the same as n, and bigger than n. where all weights are equal to 1, all the numerical tests are implemented in MATLAB R2016a and run on a laptop with 2.50 GHz processor and 4 GB RAM.
All the input points orderly form an matrix. We randomly generate this matrix using the following matlab scripts:
Table 1. Algorithmic frameworks of subproblem.
Table 2. Numerical results.
We report the numerical results in Table 1. The columns present the optimal objective function values of convex relaxation  of the 26 instances. In  , they first reformulate (1) as an equivalent smooth optimization problem as following
product the following convex relaxation (CR) when :
we solved it with CVX solver  .
The next column present the statistical results over the 1000 runs of the general algorithm proposed in  , the subcolumns “max”, “min”, “ave” and “time 1” give the best, the worst, the average objective function values and running time found among 1000 tests, respectively. The last column is the result of our algorithm, where we choose 0 vector as the initial point. Finally, add a rounding (i.e., if , then , otherwise , for for the solution obtained by the iteration. The subcolumns “f(x1)”, “f(x2)” and “time 2” represents the numerical result corresponding to no add rounding, add rounding and running time of our algorithm, respectively. Numerical results show that the effect of “f(x2)” is the best. Table 2 shows that the qualities of the solutions returned by our algorithm are generally higher than those obtained by the general algorithm in  .
In this paper, we reformulate the maximin dispersion problem as QCQP problem and the original non-convex problem is approximated by a sequence of convex problems. Then, we adopt the random block coordinate descent method (RBCDM) to obtain the solution of subproblem. Numerical results show that the proposed algorithm is efficient.