Constrained Optimization – The research field of Constrained Optimization is concerned with searching for the optimal parameters of a complex system or a target function with respect to a restricted parameter range.
The objective function F(y) usually represents a quantity that either is to be minimized (cost or energy functions) or maximized (profit or benefit functions) over a feasible set M of admissible search space parameter vectors. The feasible set M is determined by the problem-specific constraint functions g(y), and h(y). The constraints formulate necessary conditions on the parameter vector y and thereby limit the feasible parameter range, e.g. with respect to resource availability, physical limitations, or structural dependencies.
Optimization problems under constraints can be found in all scientific disciplines that deal with unknown parameters. Within the Research Centre for Process and Product Engineering (PPE) applications, among others, involve Financial Mathematics (Portfolio Optimization), Operations Research (Logistics), Statistics, and Engineering (Design Automation). Frequently, the derivation of analytical solutions to these optimization problems is limited or, in fact, rarely possible. In such situations, efficient and effective numerical methods are required to provide reasonable solutions.
Evolution Strategies – Especially versions of the so-called covariance matrix adaptation ES (CMA-ES) are arguably the currently best-performing, general purpose, direct search methods for unconstrained optimization. These strategies mimic the principles of Darwinian Evolution to generate onward improvement in a population of candidate solutions. However, up until now, the success of these direct search strategies is rather restricted to the unconstrained case. That is, the incorporation of equality constraints h(y) and inequality constraints g(y) in the design of ESs is still in an infant state when compared to other classes of Evolutionary Algorithms such as Differential Evolution or Genetic Algorithms.
Project Goals – It is the goal of this project to foster the development of ESs for constrained optimization on a theoretically-grounded basis. This is accomplished by connecting the analysis of direct search strategies with theoretically motivated algorithm design, and the evaluation of the strategies developed. Based on the knowledge gained through this research, a deeper understanding of the working principles of direct search strategies in constrained search spaces is expected. This will not only lead to better performing ESs, but also to general design principles for Evolutionary Algorithms.