Minimal Ellipsoid Circumscribing a Polytope Defined by a System of Linear Inequalities

Authors:
Jun-Ya Gotoh;Hiroshi Konno
Affiliations:
Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba-City, Japan 305-8573;Department of Industrial and Systems Engineering, Chuo University, Tokyo, Japan 112-8551
Venue:
Journal of Global Optimization
Year:
2006

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Abstract

In this paper, we will propose algorithms for calculating a minimal ellipsoid circumscribing a polytope defined by a system of linear inequalities. If we know all vertices of the polytope and its cardinality is not very large, we can solve the problem in an efficient manner by a number of existent algorithms. However, when the polytope is defined by linear inequalities, these algorithms may not work since the cardinality of vertices may be huge. Based on a fact that vertices determining an ellipsoid are only a fraction of these vertices, we propose algorithms which iteratively calculate an ellipsoid which covers a subset of vertices. Numerical experiment shows that these algorithms perform well for polytopes of dimension up to seven.