Multi-objective optimization  (MOO) or multiple criteria decision making

• The presence of multiple conflicting objectives (e.g., simultaneously minimizing the cost of fabrication and maximizing the product reliability) is natural in many problems.   Since no one solution can be termed as an optimal solution to multiple conflicting objectives, the resulting multi-objective optimization problem resorts to a number of trade-off optimal solutions (Pareto-optimal).
• Multi-objective optimization is an area between cooperative game theory and single-objective optimization.
• A first-cut approach to solving multi-objective optimization problems is to transform multiple objectives into a single objective function by using some user-defined parameters or preference weights.  This approach is called preference-based approach, where the inner product of preference (weight) and the multiple objectives results in a single-objective optimization problem.
  • Limitations
    • In a high dimensional space, the solution may be (very) sensitive to the change of preference weights.  So quantization of the preference weights plays an important role, which is highly undesirable.
    • Determination of the preference weights is subjective.   Even arbitrary preference weights may not lead to all Pareto optimal solutions.   Note that the global optimal solution of each set of preference weights is a Pareto optimal solution.
• Classical optimization methods (e.g., steepest descent and Newton method) can at best find one solution in one simulation run, thereby making these methods inconvenient to solve multi-objective optimization problems.   In contrast, evolutionary algorithms (a randomized algorithm) can find multiple optimal solutions in one single simulation run due to their population-approach; that is, in an evolutionary algorithm, a population of solutions are processed in each iteration, and the outcome of the algorithm is also a population of solutions.  This is different from classical methods, e.g., in the steepest descent algorithm, only one solution is processed in each iteration, and the outcome is also one solution.
• Two kinds of MOO
  • Blind optimization: having no knowledge about the physical meaning of the problem at hand.
  • Physical optimization: having substantive knowledge.
• Goal programming and physical programming
  • Both retain multi-objective nature (not using weighted sum to convert the problem into single objective)
  • Different from goal programming, physical programming allows deliberate imprecision in the statement of the preferences.
• Multiple criteria decision making
  • Multi-attribute decision analysis: deals with a probabilistic setting where the number of decision variables are small.
  • Multiple criteria optimization: deals with a deterministic setting where the number of decision variables are large.

Book Description

The roots of Multiple Criteria Decision Making and Multiple Criteria Optimization were laid by Pareto at the end of the 19th century, and since then the discipline has prospered and grown, especially during the last three decades. Today, many decision support systems incorporate methods to deal with conflicting objectives. The foundation for such systems is a mathematical theory of optimization under multiple objectives. Since its beginnings, there have been a vast number of books, journal issues, papers and conferences that have brought the field to its present state. Despite this vast body of literature, there is no reliable guide to provide an access to this knowledge. Over the years, many literature surveys and bibliographies have been published. With the ever rapidly increasing rate of publications in the area and the development of subfields, these were mostly devoted to particular aspects of multicriteria optimization: Multiobjective Integer Programming, Multi-objective Combinatorial Optimization, Vector Optimization, Multiobjective Evolutionary Methods, Applications of MCDM, MCDM Software, Goal Programming. Hence the need for a comprehensive overview of the literature in multicriteria optimization that could serve as a state of the art survey and guide to the vast amount of publications. Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys is precisely this book. Experts in various areas of multicriteria optimization have contributed to the volume. The chapters in this book roughly follow a thread from most general to more specific. Some of them are about particular types of problems (Theory of Vector Optimization, Nonlinear Multiobjective Programming, Fuzzy Multiobjective Programming, Multiobjective Combinatorial Optimization, Multicriteria Scheduling Problems), while the others are focused on multi-objective methodologies (Goal Programming, Interactive Methods, Evolutionary Algorithms, Data Envelopment Analysis). All contributing authors invested great effort to produce comprehensive overviews and bibliographies and to have references that are as precise as possible.

Many real-world search and optimization problems are naturally posed as non-linear programming problems having multiple objectives. Due to lack of suitable solution techniques, such problems are artificially converted into a single-objective problem and solved. The difficulty arises because such problems give rise to a set of Pareto-optimal solutions, instead of a single optimum solution. It then becomes important to find not just one Pareto-optimal solution but as many of them as possible. Classical methods are not efficient because they require repetitive applications to find multiple Pareto-optimal solutions and in some occasions repetitive applications do not guarantee finding distinct Pareto-optimal solutions. The population approach of genetic algorithms (GAs) allows an efficient way to find multiple Pareto-optimal solutions simultaneously in a single run.
Clearly, GAs (or other evolutionary search methods) have a niche in solving multi-objective optimization problems compared to classical methods. This is why multi-objective optimization using GAs is getting a growing attention in the recent past.

Key Techniques: 

• Evolution programming
• Genetic programming