By Radu Ioan Bot, Sorin-Mihai Grad, Gert Wanka

This e-book offers basics and finished effects relating to duality for scalar, vector and set-valued optimization difficulties in a common atmosphere. One bankruptcy is solely consecrated to the scalar and vector Wolfe and Mond-Weir duality schemes.

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Consider a function f : X → R. The function co f : X → R, deﬁned by co f (x) := φco(epi f ) (x) = inf{t ∈ R : (x, t) ∈ co(epi f )}, is called the convex hull of f . It is clear from the construction that the convex hull of a function f : X → R is convex and it is the greatest convex function less than or equal to f . Consequently, co f = sup{g : X → R : g(x) ≤ f (x) for all x ∈ X and g is convex}. Thus, f is convex if and only if f = co f . Regarding the convex hull of f we have also the following result.

19. Let be the function h : X → V ∪ {+∞K }. (a) If h is K-lower semicontinuous at x ∈ X, then it is also star K-lower semicontinuous at x. (b) If h is star K-lower semicontinuous, then it is also K-epi closed. 30 2 Preliminaries on convex analysis and vector optimization The following example shows that there are K-epi closed functions which are not star K-lower semicontinuous. 6. Consider the function h : R → R2 ∪ {+∞R2+ }, h(x) = ( x1 , x), if x > 0, +∞R2+ , otherwise. It can be veriﬁed that h is R2+ -convex and R2+ -epi-closed, but not star R2+ lower semicontinuous.

Then (f1 . . fm )∗ = m ∗ i=1 fi . Proof. (a) By deﬁnition there holds for any y ∗ ∈ Y ∗ (Af )∗ (y ∗ ) = sup { y ∗ , y − (Af )(y)} = sup { y ∗ , y − y∈Y y∈Y inf x∈X,Ax=y f (x)} = sup { y ∗ , Ax − f (x)} = sup { A∗ y ∗ , x − f (x)} = (f ∗ ◦ A∗ )(y ∗ ). x∈X x∈X m (b) Taking f : X → R, f (x1 , . . , xm ) = i=1 fi (xi ) and A ∈ L(X m , X), m i A(x1 , . . , xm ) = i=1 x , we have seen that Af = f1 . . fm . Applying the result from (a) we get (f1 . . fm )∗ = f ∗ ◦ A∗ . 2(l) and the fact that A∗ x∗ = (x∗ , .