2019/01/22 by Xia, Yong, Yang, Meijia, Wang, Shu
#90C20 #90C26 #90C47 #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1901.07645
We study the n-dimensional problem of finding the smallest ball enclosing the intersection of p given balls, the so-called Chebyshev center problem (CCB). It is a minimax optimization problem and the inner maximization is a uniform quadratic optimization problem (UQ). When p<=n, (UQ) is known to enjoy a strong duality and consequently (CCB) is solved via a standard convex quadratic programming (SQP). In this paper, we first prove that (CCB) is NP-hard and the special case when n = 2 is strongly polynomially solved. With the help of a newly introduced linear programming relaxation (LP), the (SQP) relaxation is reobtained more directly and the first approximation bound for the solution obtained by (SQP) is established for the hard case p>n. Finally, also based on (LP), we show that (CCB) is polynomially solved when either n or p-n(> 0) is fixed.