We consider the problem of minimizing a function f subject to a single inequality constraint g(x)  0, in a black-box scenario. We present a covariance matrix adaptation evolution strategy using an adaptive augmented Lagrangian method to handle the constraint. We show that our algorithm is an instance of a general framework that allows to build an adaptive constraint handling algorithm from a general randomized adaptive algorithm for unconstrained optimization. We assess the performance of our algorithm on a set of linearly constrained functions, including convex quadratic and ill-conditioned functions, and observe linear convergence to the optimum.

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.

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.