0000000000934365

AUTHOR

Alexandre Cabot

showing 2 related works from this author

Convergence rate of a relaxed inertial proximal algorithm for convex minimization

2018

International audience; In a Hilbert space setting, the authors recently introduced a general class of relaxed inertial proximal algorithms that aim to solve monotone inclusions. In this paper, we specialize this study in the case of non-smooth convex minimization problems. We obtain convergence rates for values which have similarities with the results based on the Nesterov accelerated gradient method. The joint adjustment of inertia, relaxation and proximal terms plays a central role. In doing so, we highlight inertial proximal algorithms that converge for general monotone inclusions, and which, in the case of convex minimization, give fast convergence rates of values in the worst case.

[ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC]Class (set theory)Control and OptimizationInertial frame of referenceLyapunov analysis0211 other engineering and technologies02 engineering and technologyManagement Science and Operations Research01 natural sciencessymbols.namesakenonsmooth convex minimizationrelaxationweak-convergence0101 mathematics[MATH]Mathematics [math]point algorithmMathematics021103 operations researchWeak convergence[QFIN]Quantitative Finance [q-fin]Applied MathematicsHilbert space[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]dynamicsmaximally monotone operatorsInertial proximal method010101 applied mathematicsMonotone polygonRate of convergenceConvex optimizationmaximal monotone-operatorssymbolsRelaxation (approximation)[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]subdifferential of convex functionsAlgorithm
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Envelopes for sets and functions II: generalized polarity and conjugacy

2018

International audience; Let X,Y be two nonempty sets, Φ an extended real-valued bivariate coupling function on X × Y and Γ a subset of X × Y. The present paper provides extensions to the well-known generalized Φ-conjugacy and Γ-polarity of diverse results of our previous work [2] related to φ-conjucacy and Λ-polarity, where Λ is a subset of a vector space E and φ is a function on E defining the particular coupling function (x,y)→φ(x−y) on E × E. A particular attention is devoted to the conjugacy functions (resp. polarity sets) which are mutually generating. Finally, for a superadditive conjugacy function Φ, we obtain a full description of the class of Φ-envelopes.

regularizationconvexityLegendre-Fenchel conjugateMutually generating conjugacy functionsΦ-envelopeΓ- polarMutually generating polarity sets.MSC: 52A41 49J53 41A65[MATH] Mathematics [math][MATH]Mathematics [math]Generalized conjugacy
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