Analysis of linear convergence of a (1 + 1)-ES with augmented Lagrangian constraint handling

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.