Analysis of Linear Convergence of a (1 + 1)-ES with Augmented Lagrangian Constraint Handling

Asma Atamna 1 Anne Auger 1 Nikolaus Hansen 1
1 TAO - Machine Learning and Optimisation
LRI - Laboratoire de Recherche en Informatique, UP11 - Université Paris-Sud - Paris 11, Inria Saclay - Ile de France, CNRS - Centre National de la Recherche Scientifique : UMR8623
Abstract : We address the question of linear convergence of evolution strategies on constrained optimization problems. In particular, we analyze a (1 + 1)-ES with an augmented Lagrangian constraint handling approach on functions defined on a continuous domain, subject to a single linear inequality constraint. We identify a class of functions for which it is possible to construct a homogeneous Markov chain whose stability implies linear convergence. This class includes all functions such that the augmented Lagrangian of the problem, centered with respect to its value at the optimum and the corresponding Lagrange multiplier, is positive homogeneous of degree 2 (thus including convex quadratic functions as a particular case). The stability of the constructed Markov chain is empirically investigated on the sphere function and on a moderately ill-conditioned ellipsoid function.
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Asma Atamna, Anne Auger, Nikolaus Hansen. Analysis of Linear Convergence of a (1 + 1)-ES with Augmented Lagrangian Constraint Handling. GECCO 2016 - Genetic and Evolutionary Computation Conference, Jul 2016, Denver, United States. pp.213-220. ⟨hal-01318807⟩

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