One-fifth Success Rule Research Articles

Global Linear Convergence of Evolution Strategies on More than Smooth Strongly Convex Functions

Evolution strategies (ESs) are zeroth-order stochastic black-box optimization heuristics invariant to monotonic transformations of the objective function. They evolve a multivariate normal distribution, from which candidate solutions are generated. Among different variants, CMA-ES is nowadays recognized as one of the state-of-the-art zeroth-order optimizers for difficult problems. Despite ample empirical evidence that ESs with a step-size control mechanism converge linearly, theoretical guarantees of linear convergence of ESs have been established only on limited classes of functions. In particular, theoretical results on convex functions are missing, where zeroth-order and also first-order optimization methods are often analyzed. In this paper, we establish almost sure linear convergence and a bound on the expected hitting time of an ES family, namely, the $(1+1)_\kappa$-ES, which includes the (1+1)-ES with (generalized) one-fifth success rule and an abstract covariance matrix adaptation with bounded condition number, on a broad class of functions. The analysis holds for monotonic transformations of positively homogeneous functions and of quadratically bounded functions, the latter of which particularly includes monotonic transformation of strongly convex functions with Lipschitz continuous gradient. As far as the authors know, this is the first work that proves linear convergence of ES on such a broad class of functions.

Self-Adjusting Mutation Rates with Provably Optimal Success Rules

The one-fifth success rule is one of the best-known and most widely accepted techniques to control the parameters of evolutionary algorithms. While it is often applied in the literal sense, a common interpretation sees the one-fifth success rule as a family of success-based updated rules that are determined by an update strength F and a success rate. We analyze in this work how the performance of the (1+1) Evolutionary Algorithm on Leading Ones depends on these two hyper-parameters. Our main result shows that the best performance is obtained for small update strengths \(F=1+o(1)\) and success rate 1/e. We also prove that the running time obtained by this parameter setting is, apart from lower order terms, the same that is achieved with the best fitness-dependent mutation rate. We show similar results for the resampling variant of the (1+1) Evolutionary Algorithm, which enforces to flip at least one bit per iteration.

Enhancing Cartesian genetic programming through preferential selection of larger solutions

We demonstrate how the efficiency of Cartesian genetic programming methods can be enhanced through the preferential selection of phenotypically larger solutions among equally good solutions. The advantage is demonstrated in two qualitatively different problems: the eight-bit parity problems and the “Paige” regression problem. In both cases, the preferential selection of larger solutions provides an advantage in term of the performance and of speed, i.e. number of evaluations required to evolve optimal or high-quality solutions. Performance can be further enhanced by self-adapting the mutation rate through the one-fifth success rule. Finally, we demonstrate that, for problems like the Paige regression in which neutrality plays a smaller role, performance can be further improved by preferentially selecting larger solutions also among candidates with similar fitness.

Fuzzy adaptive genetic algorithm for multi-objective assembly line balancing problems

This paper aims at multi-objective straight and U-shaped assembly line balancing problems with the fuzzy task processing times. The problems are referred to herein as f-SALBP and f-SULBP and the objectives that are considered to be satisfied are: (a) minimizing the numbers of stations, (b) maximizing the fuzzy line efficiency, (c) minimizing the fuzzy idleness percentage, and (d) minimizing the fuzzy smoothness index. In fact, the f-SALBP and f-SULBP are SALBP and SULBP generalization in fuzzy circumstance, respectively. Initially, the two problems are formulated and due to the uncertainty, variability and imprecision that often occurred in real-world production systems, the processing time of tasks are supposed as triangular fuzzy numbers. Then, to solve the problem, a hybrid multi-objective genetic algorithm is proposed. A One-Fifth Success Rule (OFSR) is deployed for the selection and mutation operators to improve the genetic algorithm's performance. The results in the genetic algorithm are being controlled in convergence and diversity simultaneously by means of controlling the selective pressure (SP) and mutation rate. Likewise, a fuzzy controller to SP is employed for the OFSR toward a better implementation of the genetic algorithm. In addition, the Taguchi design of experiments is used for parameter control and calibration. Finally, the numerical examples are presented to compare the performance of the proposed method with the existing ones. The results show significantly better performance for the proposed algorithm.

Modified genetic algorithm for simple straight and U-shaped assembly line balancing with fuzzy processing times

This paper aims at the straight and U-shaped assembly line balancing. Due to the uncertainty, variability and imprecision in actual production systems, the processing time of tasks are presented in triangular fuzzy numbers. In this case, it is intended to optimize the efficiency and idleness percentage of the assembly line as well as and concurrently with minimizing the number of workstations. To solve the problem, a modified genetic algorithm is proposed. One-fifth success rule in selection operator to improve the genetic algorithm performance. This leads genetic algorithm being controlled in convergence and diversity simultaneously by the means of controlling the selective pressure. Also a fuzzy controller in selective pressure employed for one-fifth success rule better implementation in genetic algorithm. In addition, Taguchi design of experiments used for parameter control and calibration. Finally, numerical examples are presented to compare the performance of proposed method with existing ones. Results show the high performance of the proposed algorithm.

A variable scaling hybrid differential evolution for solving large-scale power dispatch problems

A variable scaling hybrid differential evolution (VSHDE) is used to solve the large-scale power dispatch problems. The hybrid differential evolution (HDE) method has been presented as a method using the parallel processors of the two-membered evolution strategy ((1+1)-ES). In this way, the global search ability for the HDE can be inspected. To accelerate the search for the global solution, the concept of the variable scaling factor based on the one-fifth success rule of evolution strategies is embedded in the original HDE. The use of the variable scaling factor in the VSHDE can overcome the drawback of the need for fixed and random scaling factors in an HDE. To realise the dynamic economic dispatch (DED) system, the valve-point loading effect, system load demand, power losses, spinning reserve capacity, ramp rate limits and prohibited operation zones are considered here. Two test problems and two DED systems including those of 10 units and 20 units are used to compare the performance of the proposed method with an HDE. Numerical results show that the performance of the proposed method is better than that of the HDE method.