Understanding Badly and Well-Behaved Linear Matrix Inequalities Via Semi-infinite Optimization

Qinghong Zhang ()
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Qinghong Zhang: Northern Michigan University

Journal of Optimization Theory and Applications, 2024, vol. 203, issue 2, No 28, 1820-1846

Abstract: Abstract In this paper, we use a linear semi-infinite optimization approach to study badly and well-behaved linear matrix inequalities. We utilize a result on uniform LP duality of linear semi-infinite optimization problems to prove recent results obtained by Pataki. Such an approach not only provides alternative proofs of known results, but also gives new insights about badly and well-behaved linear matrix inequalities in terms of a cone and a linear subspace associated with the corresponding linear semi-infinite systems. Furthermore, when the linear matrix inequality constraint of the primal semidefinite optimization problem is badly behaved, we give a characterization of objective functions for the primal linear semidefinite optimization problem with which strong duality holds.

Keywords: Linear semidefinite optimization; Linear semi-infinite optimization; Uniform LP duality; Badly behaved linear matrix inequalities; Well-behaved linear matrix inequalities; 90C22; 90C34; 90C46; 49N15 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02405-6

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