Nonlinear Multiobjective Optimization
Problems with multiple objectives and criteria are generally known as multiple criteria optimization or multiple criteria decision-making (MCDM) problems. So far, these types of problems have typically been modelled and solved by means of linear programmi
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TERMINOLOGY AND THEORY
1. INTRODUCTION
We begin by laying a conceptual and theoretical basis for the continuation and restrict our treatment to finite-dimensional Euclidean spaces. First, we present the deterministic, continuous problem formulation to be handled and some general notation. Then we introduce several concepts and definitions of multiobjective optimization as well as their interconnections. The concepts and terms used in the field of multiobjective optimization are not completely fixed. The terminology used here is occasionally slightly different from that in general use. In some cases, only one of the existing terms is employed. Somewhat different definitions of concepts are presented, for example, in Zionts (1989). To deepen the theoretical basis, we treat optimality conditions for differentiable and non differentiable multiobjective optimization problems. We also briefly touch on the topics of sensitivity analysis, stability and duality. Throughout the book, even some simple results are proved, for the convenience of the reader (with possible appropriate references), in order to lay firm cornerstones for the continuation. However, to keep the text to a reasonable length, some proofs have been omitted if they can directly be found as such elsewhere. In those cases, appropriate references in the literature are indicated. Multiobjective optimization problems are usually solved by scalarization. Scalarization means that the problem is converted into a single (scalar) or a family of single objective optimization problems. In this way the new problem has a real-valued objective function, possibly depending on some parameters. After the multiobjective optimization problem has been scalarized, the widely developed theory and methods for single objective optimization can be used. Even though multiobjective optimization methods are presented in Part II, we emphasize here at the outset that the methods and the theory of single objective optimization are presumed to be known.
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