Pseudo-Boolean monotone functions are unimodal functions which are trivial to optimize for some hillclimbers, but are challenging for a surprising number of evolutionary algorithms (EAs). A general trend is that EAs are efficient if parameters like the mutation rate are set conservatively, but may need exponential time otherwise. In particular, it was known that the @math -EA and the @math -EA can optimize every monotone function in pseudolinear time if the mutation rate is @math for some @math . The second part of the statement was also known for the @math -EA. In this paper we show that the first statement does not apply to the @math -EA. More precisely, we prove that for every constant @math there is a constant integer @math such that the @math -EA with mutation rate @math and population size @math needs superpolynomial time to optimize some monotone functions. Thus, increasing the population size by just a constant has devastating effects on the performance. This is in stark contrast to many other benchmark functions on which increasing the population size either increases the performance significantly, or affects performance mildly. The reason why larger populations are harmful lies in the fact that larger populations may temporarily decrease selective pressure on parts of the population. This allows unfavorable mutations to accumulate in single individuals and their descendants. If the population moves sufficiently fast through the search space, such unfavorable descendants can become ancestors of future generations, and the bad mutations are preserved. Remarkably, this effect only occurs if the population renews itself sufficiently fast, which can only happen far away from the optimum. This is counter-intuitive since usually optimization gets harder as we approach the optimum.
The analysis of EAs on monotone functions started in 2010 by the work of Doerr, Jansen, Sudholt, Winzen and Zarges @cite_15 @cite_10 . Their contribution was twofold: firstly, they showed that the , which flips each bit independently with static mutation rate @math , needs time @math on all monotone functions if the mutation parameter @math is a constant strictly smaller than one. This result was already implicit in @cite_6 .
@cite_15 @cite_10 @cite_6
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We study the well-known black-box optimisation algorithm (1+1)-EA on a novel type of noise model. In our noise model, the fitness function is linear with positive weights, but the absolute values of the weights may fluctuate in each round. Thus in every state, the fitness function indicates that one-bits are preferred over zero-bits. In particular, hillclimbing heuristics should be able to find the optimum fast. We show that the (1+1)-EA indeed finds the optimum in time @math if the mutation parameter is @math for a constant @math However, we also show that for @math the (1+1)-EA needs superpolynomial time to find the optimum. Thus the choice of mutation parameter is critical even for optimisation tasks in which there is a clear path to the the optimum. A similar threshold phenomenon has recently been shown for noise-free monotone fitness functions. Extending previous analyses on function classes like linear functions, we analyze how the simple (1+1) evolutionary algorithm optimizes pseudo-Boolean functions that are strictly monotone. Contrary to what one would expect, not all of these functions are easy to optimize. The choice of the constant c in the mutation probability p(n) = c n can make a decisive difference. We show that if c > 1, then the (1+1) EA finds the optimum of every such function in Θ(n log n) iterations. For c = 1, we can still prove an upper bound of O(n3 2). However, for c > 33, we present a strictly monotone function such that the (1+1) EA with overwhelming probability does not find the optimum within 2Ω(n) iterations. This is the first time that we observe that a constant factor change of the mutation probability changes the run-time by more than constant factors. Extending previous analyses on function classes like linear functions, we analyze how the simple 1+1 evolutionary algorithm optimizes pseudo-Boolean functions that are strictly monotonic. These functions have the property that whenever only 0-bits are changed to 1, then the objective value strictly increases. Contrary to what one would expect, not all of these functions are easy to optimize. The choice of the constant c in the mutation probability pn=c n can make a decisive difference. We show that if c iterations. For c=1, we can still prove an upper bound of On3 2. However, for , we present a strictly monotonic function such that the 1+1 EA with overwhelming probability needs iterations to find the optimum. This is the first time that we observe that a constant factor change of the mutation probability changes the runtime by more than a constant factor. This paper analyzes monotone comparative statics predictions in several classes of stochastic optimization problems. The main results characterize necessary and sufficient conditions for comparative statics predictions to hold based on properties of primitive functions, that is, utility functions and probability distributions. The results apply when the primitives satisfy one of the following two properties: (i) a single-crossing property, which arises in applications such as portfolio investment problems and auctions, or (ii) log-supermodularity, which arises in the analysis of demand functions, affiliated random variables, stochastic orders, and orders over risk aversion.