By E. Castagnoli, P. Mazzoleni (auth.), F. H. Clarke, V. F. Demâ€™yanov, F. Giannessi (eds.)

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Extra resources for Nonsmooth Optimization and Related Topics

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Implies [7, Prop. 5) Similarly, the local minimization of fa(P) over C 2 at Per. 6) /J is locally F. p,/ K, (both these sequences being bounded). Letting t and ( denote these respective limits, we see that K;; 1 1(1 + f =lim I'Pal + 1 K, =lim I'Pal + 1' I'Pal+3 a quantity in the interval [1/3, 1]. 6) are precisely the ones in the Lemma, which is therefore proved. 3. A REGULARITY THEOREM IN THE CALCULUS OF VARIATIONS We consider the basic problem in the Calculus of Variations: that of minimizing the functional J(x) := 1b L(t,x(t),x(t))dt over a class of arcs x having prescribed boundary conditions: x(a) = x1, x(b) = Xb.

Pa. +a for some point Pa. in c2. The nature of f implies that Ua. -+ z as a -+ 0. 43 Applications of Proximal Subgradients Let any point u in C 1 be given, as well as any point pin C 2 , and set n' :=u-p. Note then that u belongs to C 1 n (C 2 + n') by construction, whence f(u) 2: V(n'). 4) whenever a'(= u-p) is near n( = Ucr. ), so in particular for all u in cl near Ucr. ) over (u,p) E cl X c2, where Note that the proximal inequality has led to a fully "decoupled" problem. Now note that 1,81 + 1 = 2 is a local Lipschitz constant for f near z (and hence near Ucr.

Therefore g(y) is proper. 0 The Yosida approximates reveal to be an useful tool for computing e-h limits: they were used also in [2] to proof compactness theorems. We examine here some properties of these approximates (in the concave-convex case) in order to obtain conditions which are equivalent to compactness and are expressed by means of Yosida approximates. These conditions refine the ones in [2] and allow us to treat saddle point problems with constraints. Property 2. +. Then L(y) =sup {J(x,y)- ~llxll 2 } is a lsc convex proper function rEX 2 28 E.