Linearly Convergent Evolution Strategies via Augmented Lagrangian Constraint Handling

We analyze linear convergence of an evolution strategy for constrained optimization with an augmented Lagrangian constraint handling approach. We study the case of multiple active linear constraints and use a Markov chain approach---used to analyze randomized optimization algorithms in the unconstrained case---to establish linear convergence under sufficient conditions. More specifically, we exhibit a class of functions on which a homogeneous Markov chain (defined from the state variables of the algorithm) exists and whose stability implies linear convergence. This class of functions is defined such that the augmented Lagrangian, centered in its value at the optimum and the associated Lagrange multipliers, is positive homogeneous of degree $2$, and includes convex quadratic functions. Simulations of the Markov chain are conducted on linearly constrained sphere and ellipsoid functions to validate numerically the stability of the constructed Markov chain.