Evolution strategies under constraints

Optimization under constraints

Research in constrained optimization is concerned with the search for the optimal parameters of a complex system or objective function within a restricted parameter range.

The general mathematical description of a constrained optimization problem is:

The objective function F(y) usually represents a quantity to be minimized (cost or energy functions) or maximized (profit or utility functions) over an admissible range of search space parameter vectors M . The admissible range M is determined by problem-specific constraints g(y) and h(y). These formulate necessary constraints on the parameter vector y and thus limit the possible parameter range, e.g. in terms of resource availability, physical constraints or structural dependencies.

Optimization problems under constraints can be found in all scientific disciplines dealing with unknown parameters . Within the Process and Product Engineering (PPE) research centre, applications include financial mathematics (portfolio optimization), operations research (logistics), statisticsand engineering (design automation). Often, the determination of analytical solutions to these optimization problems is limited or even impossible. In such situations efficient and effective numerical methods are required to find reasonable solutions.

Evolution Strategies (ES)

In particular, versions of the so-called covariance matrix adaptation ES (CMA-ES) are currently the most powerful, universally applicable direct search methods for non-restricted optimization problems. These strategies mimic the principles of Darwinian evolution to achieve further improvement in a population of candidate solutions. However, to date, the success of these direct search strategies tends to be limited to the unconstrained case. That is, the incorporation of equality constraints h(y) and inequality constraints g(y) into the design of ES is still at an early stage compared to other classes of evolutionary algorithms such as differential evolution or genetic algorithms.

Project goals 

The goal of the project is to advance the development of ES for constrained optimization on a theoretically sound basis.

This is done by combining the analysis of direct search methods with theoretically motivated algorithm design and by evaluating the developed strategies. Based on the findings from this research, a deeper understanding of the operating principles of direct search methods in restricted search spaces is expected. This will lead to both more powerful algorithms, and general design principles for evolutionary algorithms.