Section: New Results

A multiobjective algorithm framework and the COMO-CMA-ES algorithm

One classical way to solve multiobjective optimization problems is to transform the multiple objective functions into a single one, also known as scalarization, and to solve multiple versions of such scalarizations to achieve multiple trade-off points. Another approach, coming from the field of evolutionary multiobjective optimization is to formulate a single-objective set problem via indicators: the goal here is to find the set of solutions (of a given size) that maximizes a certain quality. The hypervolume indicator has been regularly used to define the quality of the solution set because it has favorable theoretical properties.

The “classical” definition of the hypervolume indicator and how it is used in multiobjective solvers, however, has some disadvantages: (i) the resulting single-objective problem is of high dimension and the gradient of the hypervolume indicator is zero in dominated areas of the search space—not giving a solver enough information about where to search for good solutions. In [7], we discussed and visualized these disadvantages and proposed a new quality criterion which is based on the hypervolume indicator but solves the mentioned disadvantages. The implementation of this idea and its combination with the single-objective solver CMA-ES resulted in the COMO-CMA-ES which shows improved performance over several existing multiobjective solvers on a wide range of convex quadratic functions.