Set Oriented Numerics
Nowadays, the importance of solving multi-objective optimization problems (MOPs) becomes crucial for different application areas such as chemistry, design, manufacture, medicine, among others. However, the task of solving this kind of problems is not straightforward, since several objectives have to be optimized at the same time. The solution set of a MOP is called the Pareto set, which typically forms a \((k-1)\)-dimensional surface, where \(k\) is the number of objectives that have to be optimized. The most common way to obtain a finite approximation of the solution set is by using deterministic techniques. Nevertheless,other techniques such as evolutionary algorithms have rep orted very promising results, and even being more robust than the mathematical techniques. In order to assess the finite approximation obtained by any algorithm researchers have proposed some performance indicators, which help us to select the best approximation between a set of them according to our needs. The Hypervolume is one of the most widely used indicators, since it has some desirable properties. Recently, the Hypervolume Newton method was proposed as standalone technique able to converge the whole population toward the best hypervolume distribution. One of the highlights of this method is its ability to converge quadratically which makes it a natural candidate to use within a global approach such as evolutionary algorithms. It is known, the success of hybridizing indicator based evolutionary algorithms with local searcher techniques for improving the performance or even refining the final solution. Here, we present the first integration of the proposed Hypervolume Newton Method into an evolutionary algorithm. To do this we first present the formulation of the Hypervolume Hessian matrix, since in previous works an approximation was used. Then, we formulate the Hypervolume Newton Method algorithm and we will show applications on some tests. In order to extend the applicability of the Hypervolume Newton method, we introduce a constrained handling technique for inequality constraints as well. Finally, we will present the numerical results of the memetic strategy against the evolutionary algorithms without our lo cal search.
- Adrian Sosa