2,396 research outputs found

    Some remarks on barycentric-sum problems over cyclic groups

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    We derive some new results on the k-th barycentric Olson constants of abelian groups (mainly cyclic). This quantity, for a finite abelian (additive) group (G,+), is defined as the smallest integer l such that each subset A of G with at least l elements contains a subset with k elements {g_1, ..., g_k} satisfying g_1 + ... + g_k = k g_j for some 1 <= j <= k.Comment: to appear in European Journal of Combinatoric

    On the quantitative isoperimetric inequality in the plane with the barycentric distance

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    In this paper we study the following quantitative isoperimetric inequality in the plane: λ02(Ω)Cδ(Ω)\lambda_0^2(\Omega) \leq C \delta(\Omega) where δ\delta is the isoperimetric deficit and λ0\lambda_0 is the barycentric asymmetry. Our aim is to generalize some results obtained by B. Fuglede in \cite{Fu93Geometriae}. For that purpose, we consider the shape optimization problem: minimize the ratio δ(Ω)/λ02(Ω)\delta(\Omega)/\lambda_0^2(\Omega) in the class of compact connected sets and in the class of convex sets

    From error bounds to the complexity of first-order descent methods for convex functions

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    This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near the set of minimizers (as those of functions with H\"olderian growth). A counterexample shows that the equivalence is no longer true for extremely flat functions. This fact reveals the relevance of an approach based on KL inequality. In a second stage, we show how KL inequalities can in turn be employed to compute new complexity bounds for a wealth of descent methods for convex problems. Our approach is completely original and makes use of a one-dimensional worst-case proximal sequence in the spirit of the famous majorant method of Kantorovich. Our result applies to a very simple abstract scheme that covers a wide class of descent methods. As a byproduct of our study, we also provide new results for the globalization of KL inequalities in the convex framework. Our main results inaugurate a simple methodology: derive an error bound, compute the desingularizing function whenever possible, identify essential constants in the descent method and finally compute the complexity using the one-dimensional worst case proximal sequence. Our method is illustrated through projection methods for feasibility problems, and through the famous iterative shrinkage thresholding algorithm (ISTA), for which we show that the complexity bound is of the form O(qk)O(q^{k}) where the constituents of the bound only depend on error bound constants obtained for an arbitrary least squares objective with 1\ell^1 regularization
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