A dual method for minimizing a nonsmooth objective over one smooth inequality constraint

Ron Shefi; Marc Teboulle

doi:10.1007/s10107-015-0952-8

A dual method for minimizing a nonsmooth objective over one smooth inequality constraint

Ron Shefi, Marc Teboulle^*

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the class of nondifferentiable convex problems which minimizes a nonsmooth convex objective over a single smooth constraint. Exploiting the smoothness of the feasible set and using duality, we introduce a simple first order algorithm proven to globally converge to an optimal solution with a O(1 / ε) efficiency estimate. The performance of the algorithm is demonstrated by solving large instances of the convex sparse recovery problem.

Original language	English
Pages (from-to)	137-164
Number of pages	28
Journal	Mathematical Programming
Volume	159
Issue number	1-2
DOIs	https://doi.org/10.1007/s10107-015-0952-8
State	Published - 1 Sep 2016

Keywords

Complexity/rate of convergence analysis
Duality
First order methods
Nonsmooth convex minimization
Sparse recovery
l-norm minimization

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10.1007/s10107-015-0952-8

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title = "A dual method for minimizing a nonsmooth objective over one smooth inequality constraint",

abstract = "We consider the class of nondifferentiable convex problems which minimizes a nonsmooth convex objective over a single smooth constraint. Exploiting the smoothness of the feasible set and using duality, we introduce a simple first order algorithm proven to globally converge to an optimal solution with a O(1 / ε) efficiency estimate. The performance of the algorithm is demonstrated by solving large instances of the convex sparse recovery problem.",

keywords = "Complexity/rate of convergence analysis, Duality, First order methods, Nonsmooth convex minimization, Sparse recovery, l-norm minimization",

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AU - Teboulle, Marc

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N2 - We consider the class of nondifferentiable convex problems which minimizes a nonsmooth convex objective over a single smooth constraint. Exploiting the smoothness of the feasible set and using duality, we introduce a simple first order algorithm proven to globally converge to an optimal solution with a O(1 / ε) efficiency estimate. The performance of the algorithm is demonstrated by solving large instances of the convex sparse recovery problem.

AB - We consider the class of nondifferentiable convex problems which minimizes a nonsmooth convex objective over a single smooth constraint. Exploiting the smoothness of the feasible set and using duality, we introduce a simple first order algorithm proven to globally converge to an optimal solution with a O(1 / ε) efficiency estimate. The performance of the algorithm is demonstrated by solving large instances of the convex sparse recovery problem.

KW - Complexity/rate of convergence analysis

KW - Duality

KW - First order methods

KW - Nonsmooth convex minimization

KW - Sparse recovery

KW - l-norm minimization

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JO - Mathematical Programming

JF - Mathematical Programming

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A dual method for minimizing a nonsmooth objective over one smooth inequality constraint

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Keywords

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