In this work, we introduce multiplicative drift analysis as a suitable way to analyze the runtime of randomized search heuristics such as evolutionary algorithms.We give a multiplicative version of the classical drift theorem. This allows easier analyses in those settings where the optimization progress is roughly proportional to the current distance to the optimum.To display the strength of this tool, we regard the classical problem how the (1+1) Evolutionary Algorithm optimizes an arbitrary linear pseudo-Boolean function. Here, we first give a relatively simple proof for the fact that any linear function is optimized in expected time O(n log n), where n is the length of the bit string. Afterwards, we show that in fact any such function is optimized in expected time at most (1 + o(1))1.39en ln(n), again using multiplicative drift analysis. We also prove a corresponding lower bound of (1 − o(1))en ln(n) which actually holds for all functions with a unique global optimum.We further demonstrate how our drift theorem immediately gives natural proofs (with better constants) for the best known runtime bounds for the (1+1) Evolutionary Algorithm on combinatorial problems like finding minimum spanning trees, shortest paths, or Euler tours. * Carola Winzen is a recipient of the Google Europe Fellowship in Randomized Algorithms, and this work is supported in part by this Google Fellowship.

Drift analysis is one of the strongest tools in the analysis of evolutionary algorithms. Its main weakness is that it is often very hard to find a good drift function.In this paper, we make progress in this direction. We prove a multiplicative version of the classical drift theorem. This allows easier analyses in those settings, where the optimization progress is roughly proportional to the current objective value.Our drift theorem immediately gives natural proofs for the best known run-time bounds for the (1+1) Evolutionary Algorithm computing minimum spanning trees and shortest paths, since here we may simply take the objective function as drift function.As a more challenging example, we give a relatively simple proof for the fact that any linear function is optimized in time O(n log n). In the multiplicative setting, a simple linear function can be used as drift function (without taking any logarithms).However, we also show that, both in the classical and the multiplicative setting, drift functions yielding good results for all linear functions exist only if the mutation probability is at most c/n for a small constant c.

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We extend the work of Lehre and Witt (GECCO 2010) on the unbiased black-box model by considering higher arity variation operators.In particular, we show that already for binary operators the black-box complexity of LeadingOnes drops from Θ(n 2 ) for unary operators to O(n log n). For OneMax, the Ω(n log n) unary black-box complexity drops to O(n) in the binary case. For k-ary operators, k ≤ n, the OneMax-complexity further decreases to O(n/ log k).