### 11.01.15 - 16.01.15, Seminar 15031

# Understanding Complexity in Multiobjective Optimization

### Diese Seminarbeschreibung wurde vor dem Seminar auf unseren Webseiten veröffentlicht und bei der Einladung zum Seminar verwendet.

## Motivation

This seminar carries on a series of four previous Dagstuhl Seminars (04461, 06501, 09041 and 12041) that were focused on Multiobjective Optimization. Our major goal is to further strengthen the links between the Evolutionary Multiobjective Optimization (EMO) and Multiple Criteria Decision Making (MCDM) communities, and to advance our understanding of different aspects concerning complexity in multiobjective optimization.

The need for a better understanding of complexity is pressing and timely, as recent work has sometimes shown opposing views regarding how problems scale and grow in difficulty, and their inherent challenges. On the one hand, we know that multiobjective optimization problems are complex problems by their very nature; optimization problems that are easy to solve in the single objective case are often intractable and highly complex already in the biobjective case. Moreover, recent work has pointed to further fundamental limitations in multiobjective optimization as we scale up to many objectives.

On the other hand, a multiobjective perspective can in a sense also help reduce complexity. For example, it often leads to a better understanding of a problem and hence supports the decision making process. Moreover, adding objectives to a problem does not always make it harder, because decomposing it can reduce the presence of local optima. And multiobjective approaches can also be used to support constraint handling, to model robustness criteria, or to approach bilevel optimization problems, simplifying these aspects. Further afield, too, in the machine learning community, we are seeing that the multiobjective optimization perspective is being used to get at the root of ill-posed problems in dimensionality reduction, pattern recognition and classification.

From the MCDM point of view, we observe that there is an intrinsic complexity in the process of understanding the optimization problem and building preferences on the solutions proposed by the multiobjective optimization. At the beginning of the decision process the Decision Maker (DM) has a rather vague idea of the decision problem at hand and, consequently, also the preferences are incomplete, approximate, uncertain or fuzzy.

Thus, better understanding complexity in multiobjective optimization is of central importance for the two communities, MCDM and EMO, and several related disciplines. It would enable us to wield existing methodologies with greater knowledge, control and effect, and should, more importantly, provide the foundations and impetus for the development of new, principled methods, in this area. Taking into account the above remarks, complexity in multiobjective optimization, as the main theme of the seminar, will be focused around three topics:

Focus 1: Complexity in preference: This topic is mainly concerned with elicitation, representation and exploitation of the preference of one or more users, for example:

- Discovering and building preferences that are dynamic and unstable
- Group preference
- Complex structure of criteria
- Non-standard preferences
- Learning in multiobjective optimization (c.f. Seminar 12041)

Focus 2: Complexity in optimization: This topic is mainly concerned with the generation of alternative candidate solutions, given some set of objective functions and feasible space. The following topics are examples for the wide range of issues in this context:

- High-dimensional problems
- Complex optimization problems
- Simulation-based optimization and expensive functions
- Uncertainty and robustness (c.f. Seminar 09041)
- Interrelating decision and objective space information

Focus 3: Complexity in applications: An all-embracing goal is to achieve a better understanding of complexity in practical problems. Many fields in the Social Sciences, Economics, Engineering Sciences are relevant: E-government, Finance, Environmental Assessment, E-commerce, Public Policy Evaluation, Risk Management and Security issues are all examples for areas where the findings of this seminar could apply.

We intend that discussions around these three topics will provide a strong basis for progress in both the theory and practice of handling complexity in multiobjective optimization in all its guises.