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Example text

The oddness of

X, y) E epi(p). Moreover, for x = y = 0 and 0'. e. (0,0) E epi(p). I> (3) Let X and Y be vector spaces. An operator A : X - Y is called affine (linear) if A is an affine variety (linear subspace) in X x Y (cf. 2). An operator A : X - Y is affine (linear) if and only if 26 Chapter 1 for all Xl, X2 E X and each pair of numbers aI, a2 E JR, al + a2 = 1 (for all Xt,X2 E X and al,a2 E R). We shall conventionally denote the set of all linear operators from X into Y (cf. 6 (3)) by L(X, V). 4 (2) there are simple interrelations between affine and linear operators.

Combining set-theoretic and algebraic operations, we can obtain some more procedures for constructing convex operators. (1) CONVEX HULL OF THE UNION OF EPIGRAPHS. Il>:= co (U{epi(fe): Put eE 3}). ) := inf all> the convex hull of the family (fe). First consider the case of finite set of indices: 3 := {I, ... ,n} and f = co(h,··· ,In). 3(3), the pair (x,e) is contained in Il> if and only if there exist elements Xk E X and numbers Ak E llt+ for which n e~ L Adk(Xk), k=l Thus, Now, take an arbitrary family of convex operators (fe)eES, For a finite set 3 the operator co( 0) is defined by the above formula.

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