The direct method for multicriteria variational problems

In many design problems one is often confronted with several measures of performance that must be simultaneously met by the system under consideration but which are often incompatible. To deal with this situation, designers will usually resort to tradeoff studies which analyze the effect of each parameter on each of the performance measures. These studies generally involve the use of the standard methods of optimization which apply to optimizing a single function at a time. From these methods, the theory of multicriteria optimization has emerged to deal specifically with the question of optimizing multiple functions simultaneously.; In this thesis we present a basic theory for the solution of three classes of multicriteria optimization problems. The first class involves the general problem of simultaneously minimizing several real-valued functions defined on a common domain. Several approaches to this problem are discussed. One of these approaches leads to the second class which is the specific case of the vector optimization problem. This problem is concerned specifically with finding optimizers of vector-valued functions. Part of this theory involves defining appropriate notions of optimality for such functions. We present new existence results and necessary and sufficient conditions for both convex and nonconvex problems.; Finally, we consider the special case of vector-valued integrals. To this third class of problems we apply an extension of the direct method in the calculus of variations to obtain new existence, necessary, and sufficient conditions. We also present new results on the convexity of the range of vector-valued integral functionals. For the second and third classes we present several applications of the theory involving antenna arrays and ship navigation.