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Chapter 2 - Existence Theorems for Minimal Points

In this chapter we investigate a general optimization problem in a real normed space. For such a problem we present assumptions under which at least one minimal point exists. Moreover, we formulate simple statements on the set of minimal points. Finally the existence theorems obtained are applied to approximation and optimal control problems. | Chapter 2 Existence Theorems for Minimal Points In this chapter we investigate a general optimization problem in a real normed space. For such a problem we present assumptions under which at least one minimal point exists. Moreover we formulate simple statements on the set of minimal points. Finally the existence theorems obtained are applied to approximation and optimal control problems. 2.1 Problem Formulation The standard assumption of this chapter reads as follows Let X II II be a real normed space let S be a nonempty subset of X and let f s R be a given functional. 2.1 Under this assumption we investigate the optimization problem min xes 2.2 i.e. we are looking for minimal points of f on s. In general one does not know if the problem 2.2 makes sense because f does not need to have a minimal point on s. For instance for X s R and rr ex the optimization problem 2.2 is not 8 Chapter 2. Existence Theorems for Minimal Points solvable. In the next section we present conditions concerning f and s which ensure the solvability of the problem 2.2 . 2.2 Existence Theorems A known existence theorem is the Weierstrafi theorem which says that every continuous function attains its minimum on a compact set. This statement is modified in such a way that useful existence theorems can be obtained for the general optimization problem 2.2 . Definition 2.1. Let the assumption 2.1 be satisfied. The functional f is called weakly lower semicontinuous if for every sequence n neN in s converging weakly to some X G s we have lim inf f xn z n oo see Appendix A for the definition of the weak convergence . Example 2.2. The functional f K. R with _ i 0 if rr 0 1 J 1 otherwise J is weakly lower semicontinuous but not continuous at 0 . Now we present the announced modification of the Weierstrafi theorem. Theorem 2.3. Let the assumption 2.1 be satisfied. If the set s is weakly sequentially compact and the functional f is weakly lower semicontinuous then there is at least one X E s with f x z .