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

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.

Linearly Convergent Evolution Strategies via Augmented Lagrangian Constraint Handling

“…An update rule was presented for the penalty parameter and the algorithm was observed to converge on the sphere function and on a moderately ill-condition ellipsoid function, with one linear constraint. This algorithm was analyzed in [3] using tools from the Markov chain theory. The authors constructed a homogeneous Markov chain and deduced linear convergence under the stability of this Markov chain.…”

“…Recently, an augmented Lagrangian approach to handle constraints within ES algorithms was proposed with the motivation to design an algorithm converging linearly [2]. The algorithm was analyzed theoretically and sufficient conditions for linear convergence, posed in terms of stability conditions of an underlying Markov chain, were formulated [3]. In those works, however, only the case of a single linear constraint was considered.…”

“…This general approach was exploited in [5] to prove the linear convergence of the (1, λ)-ES with selfadaptation on the sphere function and in [6] to prove the linear convergence of the (1 + 1)-ES with 1/5th success rule. This general methodology to prove linear convergence in the case of unconstrained optimization was generalized to constrained optimization, in the case where a single constraint is handled via an adaptive augmented Lagrangian approach [3]. The underlying algorithm being a (1 + 1)-ES.…”

“…In this work, we generalize the study in [3] to the case of multiple linear inequality constraints. We analyze a (µ/µw, λ)-ES with an augmented Lagrangian constraint handling approach in the case of active constraints.…”

“…We analyze a (µ/µw, λ)-ES with an augmented Lagrangian constraint handling approach in the case of active constraints. The analyzed algorithm is an extension of the one analyzed in [3], where we generalize the original update rule for the penalty factor in [2] to the case of multiple constraints. We construct a homogeneous Markov chain for problems such that the corresponding augmented Lagrangian, centered at the optimum of the problem and the corresponding Lagrange multipliers, is positive homogeneous of degree 2, given some invariance properties are satisfied by the algorithm.…”