Adaptive Ranking Based Constraint Handling for Explicitly Constrained Black-Box Optimization

A novel explicit constraint handling technique for the covariance matrix adaptation evolution strategy (CMA-ES) is proposed. The proposed constraint handling exhibits two invariance properties. One is the invariance to arbitrary element-wise increasing transformation of the objective and constraint functions. The other is the invariance to arbitrary affine transformation of the search space. The proposed technique virtually transforms a constrained optimization problem into an unconstrained optimization problem by considering an adaptive weighted sum of the ranking of the objective function values and the ranking of the constraint violations that are measured by the Mahalanobis distance between each candidate solution to its projection onto the boundary of the constraints. Simulation results are presented and show that the CMA-ES with the proposed constraint handling exhibits the affine invariance and performs similarly to the CMA-ES on unconstrained counterparts.Comment: 9 page

Analysis of the (μ/μI,λ)(\mu/\mu_I,\lambda)-CSA-ES with Repair by Projection Applied to a Conically Constrained Problem

Theoretical analyses of evolution strategies are indispensable for gaining a deep understanding of their inner workings. For constrained problems, rather simple problems are of interest in the current research. This work presents a theoretical analysis of a multi-recombinative evolution strategy with cumulative step size adaptation applied to a conically constrained linear optimization problem. The state of the strategy is modeled by random variables and a stochastic iterative mapping is introduced. For the analytical treatment, fluctuations are neglected and the mean value iterative system is considered. Non-linear difference equations are derived based on one-generation progress rates. Based on that, expressions for the steady state of the mean value iterative system are derived. By comparison with real algorithm runs, it is shown that for the considered assumptions, the theoretical derivations are able to predict the dynamics and the steady state values of the real runs.Comment: This is a PREPRINT of an article that has been accepted for publication in the journal MIT Press Evolutionary Computation (ECJ). 25 pages + supplementary material. The work was supported by the Austrian Science Fund FWF under grant P29651-N3

Linearly Convergent Evolution Strategies via Augmented Lagrangian Constraint Handling

International audienceWe 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 ran-domized 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