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Quality gain is the expected relative improvement of the function value in asingle step of a search algorithm. Quality gain analysis reveals thedependencies of the quality gain on the parameters of a search algorithm, basedon which one can derive the optimal values for the parameters. In this paper,we investigate evolution strategies with weighted recombination on generalconvex quadratic functions. We derive a bound for the quality gain and twolimit expressions of the quality gain. From the limit expressions, we derivethe optimal recombination weights and the optimal step-size, and find that theoptimal recombination weights are independent of the Hessian of the objectivefunction. Moreover, the dependencies of the optimal parameters on the dimensionand the population size are revealed. Differently from previous works where thepopulation size is implicitly assumed to be smaller than the dimension, ourresults cover the population size proportional to or greater than thedimension. Numerical simulation shows that the asymptotically optimal step-sizewell approximates the empirically optimal step-size for a finite dimensionalconvex quadratic function.

Quality Gain Analysis of the Weighted Recombination Evolution Strategy on General Convex Quadratic Functions