Section: New Results

Novel Constraint Handling

Participants : Asma Atamna, Anne Auger, Nikolaus Hansen.

In the context of constrained optimization, we have investigated to use augmented Lagrangian approaches to handle constraints. The originality of the approach is that the parameters of the augmented Lagrangian are adapted online. We have shown sufficient conditions for linear convergence of the ensuing methods with linear constraints [5]. Those sufficient conditions rely on finding a Markov chain candidate to be stable. This Markov chain derives from invariance properties of the algorithm. At the same time we have proposed an algorithm variant for the $(\mu /\mu ,\lambda )$-CMA-ES and an arbitrary number of constraints.

In [10], we have investigated the linear convergence question on the point of view of invariance. We have analyzed the invariances of adaptive algorithms handling constraints with augmented Lagrangian: we have shown that invariance to monotonic transformation of the objective functions is lost but that a subclass of invariance can and should be preserved, namely affine transformation of the objective function and scaling of the constraint by a positive constant.