Augmented Lagrangian Constraint Handling for CMA-ES — Case of a Single Linear Constraint

Abstract

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.

Linearly Convergent Evolution Strategies via Augmented Lagrangian Constraint Handling

“…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.…”

“…This idea was already put forward in [4] for the case of a single constraint, and we generalize it here to the case of m constraints.…”

Analysis of the (1,λ)-σ-Self-Adaptation Evolution Strategy with repair by projection applied to a conically constrained problem

“…for l ← 1 to λ do 7:σ l ← σ (g) e τ N (0,1) 8:x l ← x (g) +σ l N (0, I) 9: if not isFeasible(x l ) then see Algorithm 2 10:x l ← projectOntoCone(x l ) see (4), (5) 11:…”

“…For this, the one-generation behavior was analyzed. A multi-recombinative variant of this algorithm has been presented in [5] for a single linear constraint and multiple linear constraints in [6]. Markov chains have been used in both cases for a theoretical investigation.…”

Adaptive ranking based constraint handling for explicitly constrained black-box optimization

“…It is applicable to problems with arbitrary quantifiable constraints, but it assumes that the objective function is defined in the infeasible domain. This method has been first proposed for (1 + 1)-ES [7] and extended to the CMA-ES in a single constraint case [8], where the median success rule is applied for the step-size adaptation.…”