Evolutionary Multiobjective Optimization Based on Gaussian Process Modeling

Multiobjective optimization is the process of simultaneously optimizing two or more conflicting objectives and is used for solving real-world optimization problems in various fields, from product design to process optimization. One of the most effective ways of solving problems with multiple objectives is to use multiobjective evolutionary algorithms (MOEAs). MOEAs draw inspiration from adaptation processes occurring in nature. In order to find the best solutions, they perform numerous solution evaluations. If these solution evaluations are time-consuming, the optimization process can take a lot of time. To obtain the results of such an optimization problem faster (or to obtain them in a reasonable amount of time), surrogate models can be used to approximate the objective functions of the problem. Instead of performing time-consuming exact evaluation to evaluate a solution, the solution can be approximated with a surrogate model. Using solution approximations can significantly accelerate the optimization process, but can also spoil the results if the solution approximations are inaccurate. When comparing approximated solutions, a solution can incorrectly appear to be dominated by inaccurate and over-optimistic approximations. To reduce the possibility of incorrect comparisons, we propose new relations under uncertainty that, in addition to the approximated values, consider also the confidence intervals for the approximations. The relations under uncertainty were compared with the Pareto dominance relations in the experiments that confirmed that the use of the proposed relations reduces the possibility of incorrect comparisons. We included the relations under uncertainty in a new MOEA called Differential Evolution for Multiobjective Optimization based on Gaussian Process modeling (GP-DEMO). GP-DEMO is based on Differential Evolution for Multiobjective Optimization (DEMO), a steady-state algorithm known to be very effective in solving numerical multiobjective optimization problems. GP-DEMO was compared with DEMO and also with another surrogate-model-based MOEA called Generational Evolution Control (GEC). These algorithms were tested on 12 benchmark problems of different complexities and on two real-world problems – optimization of continuous steel casting and finding the correlation between a simulated and a measured electrocardiogram (ECG). The empirical analysis of their results showed that GP-DEMO and DEMO produce similar results, but GP-DEMO needs considerably less exact evaluations. In comparison to GEC, GP-DEMO achieves better results and the number of exact evaluations depends on the type of the optimization problem. In order to determine when to use GP-DEMO instead of DEMO, we calculated for every test problem how long a single exact solution evaluation should last for the optimization times of GP-DEMO and DEMO to be equal. So for an arbitrary optimization problem we can, depending on the assessed complexity and the duration of a single exact solution evaluation, estimate which of the two algorithms is more suitable.