**Variable-Size Design Space Optimization**

In many optimal design problems, the number of design variables, N, is itself a design variable. This class of optimization problems arises in many applications including (1) microgrid design, (2) space vehicle trajectory design, (2) optimal grouping, and (3) automated planning. To date, there is no computationally efficient and rational framework for solving this general Variable-Size Design Space (VSDS) optimization problem. The current practice is to solve the design optimization problem many times to cover all possible values of N, which is computationally expensive and in some cases prohibitive.

The research conducted in this area investigates three different novel approaches to handle this type of problems in a rigorous framework. In some applications the proposed concepts enable new problem formulations, while in other applications they enable exploiting more of the design space at a significantly lower computational cost. These approaches are:

- Hidden Genes Evolutionary Algorithm
- Dynamic-Size Multiple Population Evolutionary Algorithms
- Structured-Chromosome Evolutionary Algorithms

**Inverse problem**

An inverse problem is a general framework that is used to extract information about a system using observed measurements. Global optimization methods may be used to solve the inverse problem.

Several applications are characterized by a large number of variables, usually because of the discretizations of partial differential equations in the mathematical model, and a large date set (measurements). One of the state-of-the-art algorithms for data assimilation is the ensemble Kalman filter (EnKF). Evolutionary optimization algorithms perform global search and can be used to search for the optimal variables (model parameters) that minimize the penalty (error) function. The computational cost of the evolutionary algorithms, given the large size of the problem, is high and might be prohibitive in practical applications.

The research conducted in this area investigates the development of recursive evolutionary algorithms that are significantly lower in computational cost compared to standard evolutionary algorithms, yet they maintain the advantages of the standard evolutionary algorithms. The computational cost of the proposed algorithms is comparable to that of the EnKF. The proposed approach is:

- Dynamic Penalty Function Evolutionary algorithms