By Saul I. Gass

Unique, nontechnical creation covers simple techniques of linear programming and its dating to operations examine; geometric interpretation and challenge fixing, resolution suggestions, community difficulties, even more. Appendix bargains specified statements of definitions, theorems, and methods, extra computational techniques. merely high-school algebra wanted. Bibliography.

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17 in the case of minimizing f (w, x) = f1 (x − w) + f2 (w) in w with x as parameter. The symmetry between the roles of f1 and f2 yields the symmetric form of the continuity assertion. Epi-addition is commutative and associative; the formula in the case of more than two functions works out to (f1 f2 ··· fr )(x) = inf x1 +x2 +···+xr =x f1 (x1 ) + f2 (x2 ) + · · · + fr (xr ) . One has f δ{0} = f for all f , where δ{0} is of course the indicator of the singleton set {0}. A companion operation is epi-multiplication by scalars λ ≥ 0; the epi-multiple λ f is deﬁned by (λ f )(x) := λf (λ−1 x) for λ > 0 0 if x = 0, f ≡ ∞, (0 f )(x) := ∞ otherwise.

Yn ). 8 Example (aﬃne functions, half-spaces and hyperplanes). A function f on IRn is said to be aﬃne if it diﬀers from a linear function by only a constant: f (x) = a, x + β for some a ∈ IRn and β ∈ IR. Any aﬃne function is both convex and concave. As level sets of aﬃne functions, all sets of the form x a, x ≤ α and x a, x ≥ α , as well as all those of the form x a, x < α and x a, x > α , are convex in IRn , and so too are all those of the form x a, x = α . For a = 0 and α ﬁnite, the sets in B.

The proper convex functions on IRn are thus the functions obtained by taking a ﬁnite, convex function on a nonempty, convex set C ⊂ IRn and giving it the value ∞ everywhere outside of C. The indicator δC of a set C ⊂ IRn is convex if and only if C is convex. In this sense, convex sets in IRn correspond one-to-one with special convex functions on IRn . On the other hand, convex functions on IRn correspond one-to-one with special convex sets in IRn+1 , their epigraphs. 4 Proposition (convexity of epigraphs).