By Jean-Pierre Aubin (Eds.)

This publication provides a unified therapy of optimization thought, online game thought and a normal equilibrium thought in economics within the framework of nonlinear sensible research. It not just offers strong and flexible instruments for fixing particular difficulties in economics and the social sciences but in addition serves as a unifying subject within the mathematical concept of those topics in addition to in natural arithmetic itself.

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**Example text**

Taking y = 0, we deduce that 0 < E 6 (p, x). On the other hand, supycp(p, y ) = = 0 since it is bounded above (see Proposition 4). This is the required contradiction. 0 As a consequence, we have the following result. Proposition 8. Let L belong to 2(U,V) and let X be a convex subset of U containing 0. Then (28) L(X)* = L*-1(X#). Furthermore, ifL ( X) is a closed convex subset containing 0, we obtain that (29) 5 L(X)= [(L'-'(X+)]*. 34 MINIMIZATION PROBLEMS AND CONVEXITY [Ch. 4 Proof. e. that L'p belongs to X#, or that p belongs to L'-l(X*).

E. Lx = y). The inf-convolution can be regarded in the following way. e. choose strategies xi such that E=lx’ = y. The inf-convolution Uy=lh associateswith any y the smallest total loss occurred during the sharing. 3. e. and (Q): inf (LS(y)+gW) Y€Y (RCV)) : inf f ( x ) Lx-y for a convenient y. 14 MINIMIZATION PROBLEMS AND CONVEXITY [Ch. 2 holds and the two following statements are equivalent: (1 1) 1 (a) 2 minimizes f ( x ) + g ( L x ) on the set X = L-I(Y) (b) j = LX minimizes Lf ( y ) + g ( y ) on Y and X minimizes f ( x ) on the set L-l(jj).

E. (20) B(J4X)) = (X). Ch. 31 25 MIXED STRATEGIES AND CONVEXITY Thus (21) Xis “convex” if and only i f B maps M ( X )into X . f(Bm) = f ”(4. A function f : U (24) (m,f) = f’(m) - 1- 00, + -1 is “convex” i f and only if V m = ZAiS(xi) E M(U), f (pm) =sf ”(m)= A function f : U --c [- 00, + -[ - n C A’f(xi). i=1 is “concave” i f -f is convex. It is clear that the domain Dom f of a convex function is convex. Also, if X c U is convex and f :X R is convex, its extension fx : U -,1- -, + -1 is convex.