An active-set algorithmic framework for non-convex optimization problems over the simplex

In this paper, we describe a new active-set algorithmic framework for minimizing a non-convex function over the unit simplex. At each iteration, the method makes use of a rule for identifying active variables (i.e., variables that are zero at a stationary point) and specific directions (that we name...

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Bibliographic Details
Published inComputational optimization and applications Vol. 77; no. 1; pp. 57 - 89
Main Authors Cristofari, Andrea, De Santis, Marianna, Lucidi, Stefano, Rinaldi, Francesco
Format Journal Article
LanguageEnglish
Published New York Springer US 01.09.2020
Springer Nature B.V
Subjects
Algorithms
Computational geometry
Convergence
Convex and Discrete Geometry
Convexity
Iterative methods
Management Science
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Statistics
Non-convex optimization
65K05
90C30
Active-set methods
90C06
Large-scale optimization
Unit simplex
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Summary:In this paper, we describe a new active-set algorithmic framework for minimizing a non-convex function over the unit simplex. At each iteration, the method makes use of a rule for identifying active variables (i.e., variables that are zero at a stationary point) and specific directions (that we name active-set gradient related directions) satisfying a new “nonorthogonality” type of condition. We prove global convergence to stationary points when using an Armijo line search in the given framework. We further describe three different examples of active-set gradient related directions that guarantee linear convergence rate (under suitable assumptions). Finally, we report numerical experiments showing the effectiveness of the approach.
ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-020-00195-x