Introduction to the Theory of Nonlinear Optimization by Steven Kay

By Steven Kay

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Prove that the functional / is quasiconvex if and only if for all x^y E S f{Xx + (1 - X)y) < max{/(x), f{y)} for all A G [0,1]. 4) Prove that every proximinal subset of a real normed space is closed. 4 is solvable. 6) Let C{M) denote the real linear space of continuous real valued functions on a compact metric space M equipped with the maximum norm. Prove that for every n G N and every continuous function x G C{M) there are real numbers a o , . . , 0;^ G R with the property n max I 2_, ^it^ ~~ ^[^)\ teM i=0 30 Chapter 2.

12 we have b f\x){h) - A/,(x(t),^(t),t)/i(t) + /i(x(t),^(t),t)/i(t)]dt a for all heS. 3) For further conclusions in the previous example we need an important result which is prepared by the following lemma. 19. For —oo < a < b < oo let S = {xe C^[a,b] I x{a) = x{b) = 0}. If for some function x G C[a, 6] b / x{t)h{t) dt = 0 for all he then X = constant on [a, 6]. Proof. We define b ^=6^/^(')'\dt S, Chapter 3. Generalized Derivatives 46 and choose especially h E S with t h{t) = / {x{s) - c) ds for all t G [a, h].

If x E X is a minimal point of f on X and f is Gateaux differentiable at x, then it follows f(x){h)=0 for all hex. Proof. Let an element h e X he arbitrarily given. 8, (a) f'{x)ih) > 0, 44 Chapter 3. Generalized Derivatives and for x := —h + x we get f'm-h) > 0. Because of the linearity of the Gateaux derivative the assertion follows immediately. • Finally, we discuss an example from the calculus of variations. 13, holds also for Frechet differentiable functional. 18. We consider a function continuous with respect to all arguments and partial derivatives with respect to the two first let a functional / : C^[a, 6] —> R (with —oo < / : M^ —^ R which is which has continuous arguments.

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