Choosing a suitable algorithm from the myriads of different search heuristics is difficult when faced with a novel optimization problem. In this work, we argue that the purely academic question of what could be the best possible algorithm in a certain broad class of black-box optimizers can give fruitful indications in which direction to search for good established optimization heuristics. We demonstrate this approach on the recently proposed DLB benchmark, for which the only known results are O(n^3) runtimes for several classic evolutionary algorithms and an O(n^2 log n) runtime for an estimation-of-distribution algorithm. Our finding that the unary unbiased black-box complexity is only O(n^2) suggests the Metropolis algorithm as an interesting candidate and we prove that it solves the DLB problem in quadratic time. Since we also prove that better runtimes cannot be obtained in the class of unary unbiased algorithms, we shift our attention to algorithms that use the information of more parents to generate new solutions. An artificial algorithm of this type having an O(n log n) runtime leads to the result that the significance-based compact genetic algorithm (sig-cGA) can solve the DLB problem also in time O(n log n). Our experiments show a remarkably good performance of the Metropolis algorithm, clearly the best of all algorithms regarded for reasonable problem sizes.

Choosing a suitable algorithm from the myriads of different search heuristics is difficult when faced with a novel optimization problem. In this work, we argue that the purely academic question of what could be the best possible algorithm in a certain broad class of black-box optimizers can give fruitful indications in which direction to search for good established optimization heuristics. We demonstrate this approach on the recently proposed DLB benchmark, for which the only known results are O(n 3 ) runtimes for several classic evolutionary algorithms and an O(n 2 log n) runtime for an estimation-of-distribution

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We prove an identity in distribution between two kinds of partition functions for the log-gamma directed polymer model: (1) the point-to-point partition function in a quadrant and (2) the point-to-line partition function in an octant. As an application, we prove that the point-to-line free energy of the log-gamma polymer in an octant obeys a phase transition depending on the strength of the noise along the boundary. This transition of (de)pinning by randomness was first predicted in physics by Kardar in 1985 and proved rigorously for zero temperature models by Baik and Rains in 2001. While it is expected to arise universally for models in the Kardar–Parisi–Zhang universality class, this is the first positive temperature model for which this transition can be rigorously established.