Towards an Augmented Lagrangian Constraint Handling Approach for the (1+1)-ES

Abstract

We consider the problem of devising an approach for handling inequality constraints in evolution strategies that allows converging linearly to optimal solutions on sphere functions with a single linear constraint. An analysis of the single-step behaviour of the (1 + 1)-ES shows that the task of balancing improvements in the objective with those in the constraint function is quite delicate, and that adaptive approaches need to be carefully designed in order to avoid failure. Based on the understanding gained, we propose a simple augmented Lagrangian approach and experimentally demonstrate good performance on a broad range of sphere functions as well as on moderately ill-conditioned ellipsoids with a single linear constraint.

Linearly Convergent Evolution Strategies via Augmented Lagrangian Constraint Handling

“…More recently, an augmented Lagrangian approach was combined with a (1 + 1)-ES for the case of a single linear constraint [2]. 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.…”

“…The authors constructed a homogeneous Markov chain and deduced linear convergence under the stability of this Markov chain. In [4], the augmented Lagrangian constraint handling mechanism in [2] is implemented for CMA-ES and a general framework for building a general augmented Lagrangian based randomized algorithm for constrained optimization in the case of one constraint is presented.…”

“…It exhibits linear convergence on wide classes of functions when solving unconstrained optimization problems. However, many practical problems come with constraints and the question of how to handle them properly to particularly preserve the linear convergence is an important one [2]. Recently, an augmented Lagrangian approach to handle constraints within ES algorithms was proposed with the motivation to design an algorithm converging linearly [2].…”

“…However, many practical problems come with constraints and the question of how to handle them properly to particularly preserve the linear convergence is an important one [2]. 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].…”

“…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.…”