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On the resolution of certain discrete univariate max–min problems

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Abstract

We analyze a class of discrete, univariate, and strictly quasiconcave max–min problems. A distinctive feature of max–min-type optimization problems is the nonsmoothness of the objective being maximized. Here we exploit strict quasiconcavity of the given set of functions to prove existence and uniqueness of the optimizer, and to provide computable bounds for it. The analysis inspires an efficient algorithm for computing the optimizer without having to resort to any regularization or heuristics. We prove correctness of the proposed algorithm and briefly discuss the effect of tolerances and approximate computation. Our study finds direct application in the context of certain mesh deformation methods, wherein the optimal perturbation for a vertex is computed as the solution of a max–min problem of the type we consider here. We include examples demonstrating improvement of simplicial meshes while adopting the proposed algorithm for resolution of the optimization problems involved, and use these numerical experiments to examine the performance of the algorithm.

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Notes

  1. More precisely, we require the number of pairwise intersections of functions within the interval \([{\underline{\varLambda }}_{y_0},{\overline{\varLambda }}_{y_0}]\) to be finite.

  2. We say that \(f_\alpha \) is active over the interval \({\text {I}}\) if \(F=f_\alpha \) on \({\text {I}}\).

  3. Arguing as we did in Propositions 1 and 2, we can shows that \(f_{\alpha _s}\) has a unique maximizer over \([\lambda _{s-1},\lambda _s]\). For the purpose of arriving at Eq. (32) however, the inclusion noted in Eq. (31) is sufficient.

  4. We thank Prof. Adrian Lew (Stanford University) for helpful discussions regarding Proposition 10.

  5. We only request that Q be defined over triangles whose vertices do not all coincide.

  6. In general, n is less than or equal to the valence of a vertex since the quality curves of two or more elements may coincide. For the representative data in Fig. 5c, n equals the vertex valence.

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  1. Department of Mechanical Engineering, Indian Institute of Science, Bangalore, India

    Ramsharan Rangarajan

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  1. Ramsharan Rangarajan
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Rangarajan, R. On the resolution of certain discrete univariate max–min problems. Comput Optim Appl 68, 163–192 (2017). https://doi.org/10.1007/s10589-017-9903-z

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