Concave programming for minimizing the zero-norm over polyhedral sets

Rinaldi, Francesco; Schoen, Fabio; Sciandrone, Marco

doi:10.1007/s10589-008-9202-9

Cited by 71 publications

(85 citation statements)

References 10 publications

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“…In the same way as in [32] and [53] we will use a successive linearization algorithm (SLA) algorithm to solve the concave minimization problem at each iteration in r. This algorithm is a nitely timestep Franck & Wolf algorithm, [45]. …”

Section: Algorithmmentioning

confidence: 99%

“…Approximating the 0 -norm by smooth functions through an homotopy method starting from the 1 -norm has been studied in the PhD thesis [53] and in [55,54,41]. In these works, the authors consider a selection of minimization problems using smooth functions such that (t + r) p with r > 0 and 0 < p < 1,…”

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confidence: 99%

“…−p with r > 0 and 1 < p, log(t + r) with 0 < r << 1 or 1 − e −rt with r > 0 and p ∈ N. The subproblems of the homotopy algorithm are solved using a Frank and Wolfe approach [55], also called SLA in [32], and this method is further studied in [40].…”

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confidence: 99%

“…Focusing on concave functions is a logical choice considering that the p -norm is itself concave. The motivation here is to keep the good properties of the method from [55] and related work, a homotopy method between the 1 -norm and the 0 -norm problems, and smoothness at the origin. In particular, such a theoretical study has not been done in the literature.…”

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confidence: 99%

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A Smoothing Method for Sparse Optimization over Polyhedral Sets

Migot

Haddou

2015

Advances in Intelligent Systems and Computing

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In this paper, we investigate a class of heuristic schemes to solve the NP-hard problem of minimizing 0-norm over a polyhedral set. A wellknown approximation is to consider the convex problem of minimizing 1-norm. We are interested in nding improved results in cases where the problem in 1-norm does not provide an optimal solution to the 0-norm problem. We consider a relaxation technique using a family of smooth concave functions depending on a parameter. Some other relaxations have already been tried in the literature and the aim of this paper is to provide a more general context. This motivation allows deriving new theoretical results that are valid for general constraint set. We use a homotopy algorithm, starting from a solution to the problem in 1-norm and ending in a solution of the problem in 0-norm. We show the existence of the solutions of the subproblem, convergence results, a kind of monotonicity of the solutions as well as error estimates leading to an exact penalization theorem. We also provide keys for implementing the algorithm and numerical simulations.Mathematics Subject Classication. 90-08 and 65K05

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Section: Algorithmmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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A Smoothing Method for Sparse Optimization over Polyhedral Sets

Migot

Haddou

2015

Advances in Intelligent Systems and Computing

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“…To make the problem tractable and, since the zero norm is a stepwise function, continuous, differentiable and concave approximations have been proposed to deal with the problem, [33,199,200,235].…”

Section: Relevance Of Featuresmentioning

confidence: 99%

Supervised classification and mathematical optimization

Carrizosa

Morales

2013

Computers & Operations Research

124

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Data Mining techniques often ask for the resolution of optimization problems. Supervised Classification, and, in particular, Support Vector Machines, can be seen as a paradigmatic instance. In this paper, some links between Mathematical Optimization methods and Supervised Classification are emphasized. It is shown that many different areas of Mathematical Optimization play a central role in off-the-shelf Supervised Classification methods. Moreover, Mathematical Optimization turns out to be extremely useful to address important issues in Classification, such as identifying relevant variables, improving the interpretability of classifiers or dealing with vagueness/noise in the data.

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