Convergence of the method of Chebyshev centers and some applications

Nenakhov, É. I.; Primak, M. E.

doi:10.1007/bf01074784

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The Chebyshev Points Of Convex Sets In E N2

Introduction1

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1998

2020

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Cited by 5 publications

(3 citation statements)

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“…Later, the Chebyshev point methods were developed by Kiev mathematicians Zukhovitskii, Polyak, Primak and others [3][4][5][6]8].…”

Section: The Chebyshev Points Of Convex Sets In E Nmentioning

confidence: 99%

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Imbedding and cutting methods for global optimization and their applications

Bulatov¹

2009

Optimization

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“…Later, the Chebyshev point methods were developed by Kiev mathematicians Zukhovitskii, Polyak, Primak and others [3][4][5][6]8].…”

Section: The Chebyshev Points Of Convex Sets In E Nmentioning

confidence: 99%

“…There are various known versions of the base algorithm developed by different authors [2,[4][5][6]8].…”

Section: The Chebyshev Points Of Convex Sets In E Nmentioning

confidence: 99%

Imbedding and cutting methods for global optimization and their applications

Bulatov¹

2009

Optimization

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“…Chebyshev centering problem has some applications in optimization. It serves as a key subproblem in each iteration for solving constrained optimization and equilibrium problems [5,18]. The other example is that the problem of finding the optimal 1-network 1 of a compact set C ⊆ R 2 can be reduced to finding the Chebyshev center and Chebyshev radius for C [15].…”

Section: Introductionmentioning

confidence: 99%

Chebyshev center of the intersection of balls: complexity, relaxation and approximation

2020

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We study the n-dimensional problem of finding the smallest ball enclosing the intersection of p given balls, the so-called Chebyshev center problem (CC B ). 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 (CC B ) is solved via a standard convex quadratic programming (SQP). In this paper, we first prove that (CC B ) is NP-hard and the special case when n = 2 is strongly polynomially solvable. 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 (CC B ) is polynomially solvable when either n or p − n(> 0) is fixed.

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