An efficient cutting plane algorithm for the smallest enclosing circle problem

Abstract

In this paper, we consider the problem of computing the smallest enclosing circle. An efficient cutting plane algorithm is derived. It is based on finding the valid cut and reducing the problem into solving a series of linear programs. The numerical performance of this algorithm outperforms other existing algorithms in our computational experiments.

A reformulation-linearization based algorithm for the smallest enclosing circle problem

“…Computational results in 2D. In this section, through extensive numerical studies, we demonstrate that the Algorithm I proposed in Section 2 outperforms second order cone optimization (1) solved by the CPLEX Solver and SDPT3, and the cutting plane algorithm given in [5].…”

“…Moreover, it is not hard to see there is no relation between the number of iterations and the number of circles. Now, we show the numerical performance comparison of the Algorithm I presented in this paper, SOCP, SDPT3 and CP given in [5]. From Table 3, one can see that all algorithms are able to get accurate results for all test problem scales and the average CPU time of SDPT3 is much slower than SOCP, CP, and Algorithm I.…”

“…Xu, Freund, and Sun [14] propose a quadratic programming approach whose numerical performance outperforms those of the subgradient approach, the randomized incremental algorithm, and second order cone reformulation (1) solved by SeDuMi ( [11]). More recently, Jiang, Luo, and Ling [5] develop an efficient cutting plane algorithm and their computational experiments demonstrate the cutting plane algorithm outperforms the quadratic programming approach given in [14] and second order cone reformulation (1) solved by the CPLEX Solver. Furthermore, efficient algorithms for computing the small enclosing ball of a set of m balls in R n are considered in [16,9].…”

“…The algorithm developed in Section 2 is inspired by the Reformulation-Linearization Technique (RLT) in [6,7,8] and the algorithm based on approximation of quadratic terms in [1]. Because the computational results show that the cutting plane algorithm in [5] performs better than the quadratic programming approach in [14], we compare the numerical performances of the algorithm proposed in this paper with those of the cutting plane algorithm in [5] and second order cone reformulation (1) solved by the CPLEX Solver and SDPT3. In Section 3, we demonstrate that the algorithm proposed in this paper outperforms the other two approaches through extensive numerical studies.…”

“…SDPT3 represents the method to solve second order cone optimization (1) by using SDPT3. CP denotes the best algorithm in [5]. Algorithm I denotes the method presented in this paper.…”