In this paper we focus on the linearly constrained continuous optimization. A one of the state-of-the-art stochastic algorithms for ill-conditioned and nonseparable unconstrained problems, namely the covariance matrix adaptation evolution strategy (CMA-ES) is applied to solve linearly constrained continuous optimization problems. We extend the box constraint handling technique that turns a box constrained optimization problem into an unconstrained optimization problem by introducing an artificial fitness landscape, where a penalty function is added to the function value at the nearest feasible solution. The penalty function is adapted during search so as to create an artificial landscape outside the feasible domain that makes the function as easily solvable by the CMA-ES as possible. Treating a box constraint as a special case of linear constraints, we generalize the box constraint handling to apply the same technique to an arbitrary linear constrained problem. Moreover, the adaptation of the penalty coefficient is accelerated. The resulting linear constraint handling technique exhibits an invariant performance on problems with linear constraints under a linear transformation of the coordinate system, showing that a linearly constrained problem can be essentially as efficiently solvable by the CMA-ES as a box constrained problem.
In recent years, accidents and product recalls caused by product failures have become major problems in many industries worldwide. To predict how changes of a product recall system affects safety in the society and to get valuable suggestions to improve product recall systems, we simulated the recall process in society using social simulation model. This research is important because the current product recall systems are not designed by mathematical and predictive approaches such as a computer simulation, but designed by empirical approaches. As a simulation model, we propose Layered Co-evolution Model with Logic Value Typed Genetic Programming (GP). We evaluated the proposed method by using the multi-agent simulation in an artificial society where producer agents and consumer agents both compete and cooperate with each other. This experiment discovered that the producer agents and the consumer agents are able to co-evolve toward a convergence point in Layered Co-evolution Model through the interactions between both types of agents. From the experiment, it is also understood that Logic Value Typed GP, which uses logic values and logic operators, has the advantages over the existing GP method that uses real number values. The Logic Value Typed GP is more stable in the evolutionary process and more efficient in terms of agents' learning process. In addition, we predicted that making the accident-compensation-level stricter decreases the frequency of product accidents in the whole artificial society. This is the result of the producer agents increasing the frequency of product recalls or raising production costs under such a stricter level. This prediction is useful for realizing a safer society.
The population size, i.e., the number of candidate solutions per iteration, is the only parameter for the covariance matrix adaptation evolution strategy (CMA-ES) that needs to be tuned depending on the ruggedness and the uncertainty of the objective function. The population size has a great impact on the performance of the CMA-ES, however, it is prohibitively expensive in black-box scenario to tune the population size in advance. Moreover, a reasonable population size is not constant during the optimization. In this paper, we propose a novel strategy to adapt the population size. We introduce the evolution path in the parameter space of the Gaussian distribution, which accumulates successive parameter updates. Based on the length of the evolution path with respect to the Fisher metric, we quantify the accuracy of the parameter update. The population size is then updated so that the quantified accuracy is kept in the constant range during search. The proposed strategy is evaluated on test functions including rugged functions and noisy functions where a larger population size is known to help to find a better solution. The experimental results show that the population size is kept as small as the default population size on unimodal functions, and it is increased at the early stage of the optimization of multimodal functions and decreased after the sampling distribution is concentrated in a single valley of a local optimum. On noisy test functions, the proposed strategy start increasing the population size when the noise-to-signal ratio becomes relatively high. The proposed strategy is compared with the CMA-ES and the state-of-the-art uncertainty handling in the CMA-ES, namely UH-CMA-ES, with a hand-tuned population sizes.