Discrete Evolutionary Algorithms for Optimizing Sphere

Approaching mixed-integer black box optimization (MI-BBO) problems requires algorithms that can handle both the continuous as well as the discrete variables. Recent work in this area has made progress by looking at algorithms for continuous-variable problems being used for discrete search spaces.With this work we approach the opposite direction: we analyze algorithms for discrete-variable problems being used for a problem in a continuous search space. Concretely, we define the (1+1) EA and Random Local Search (RLS) with a step size of η ∈ ℝ>0, for optimizing continuous variables. The parameter η needs to be adjusted over the run of the algorithm to allow for arbitrary approximation of (local) optima. We show that the (1+1) EA with a fixed schedule for adjusting η optimizing Sphere can achieve an approximation of ε within O(nlog(n)log(n/ε)).In order to improve over the fixed schedule for adjusting η, we switch to RLS and consider a self-adjusting rule for η. We compare the performance of this algorithm experimentally to the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) on the functions Sphere, Ellipsoidal and Rosenbrock from the 2009 BBOB functions suite. We then extend the RLS with self-adjusting operator to handle mixed-binary problems and compare it with CMA-ES with Margin (CMA-ESwM) on SphereOneMax.