Rounding on the standard simplex: regular grids for global optimization
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Given a point on the standard simplex, we calculate a proximal point on the regular grid which is closest with respect to any norm in a large class, including all \(\ell ^p\)-norms for \(p\ge 1\). We show that the minimal \(\ell ^p\)-distance to the regular grid on the standard simplex can exceed one, even for very fine mesh sizes in high dimensions. Furthermore, for \(p=1\), the maximum minimal distance approaches the \(\ell ^1\)-diameter of the standard simplex. We also put our results into perspective with respect to the literature on approximating global optimization problems over the standard simplex by means of the regular grid.
KeywordsRounding Regular grid Approximation Maximin distance Proximal point
The authors are grateful to two anonymous referees who pointed out additional related references, as well as to Oliver Stein for valuable remarks. E. Alper Yıldırım was supported in part by Turkish Scientific and Technological Research Council (TÜBİTAK) Grant 112M870 and by TÜBA-GEBİP (Turkish Academy of Sciences Young Scientists Award Program).
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