By Mordecai Avriel, Walter E. Diewert, Siegfried Schaible, Israel Zang

ISBN-10: 0898718961

ISBN-13: 9780898718966

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**Example text**

In a more general case, the relationship between a mathematical program (such as CP, but without the concavity and convexity assumptions) and the associated SP is only one-sided: a solution of SP is optimal for the mathematical program, but not conversely. These relationships were first established by Karush (1939) and Kuhn and Tucker (1951). Let us turn now to concave programs in which the functions involved are differentiable. For these programs, necessary and sufficient optimality conditions involving the Lagrangian function were derived by Karush, and Kuhn and Tucker.

1. 1. The "graph above the chord" property of concave functions. 17 Concavity such that B is between A and C, then B is on or above the line segment (chord) AC. 1. Here are some simple examples of concave functions: 1. f(x) = log x, defined on the open convex set C = {x: x > 0}. 2. f(x) = sin x, defined on the closed convex set C = {x: 0 x <_ IT}. These two functions are actually strictly concave. 3. f(x) = ax + b, defined on the whole real line R. This is an affine function that is both concave and convex (but, clearly, neither strictly concave nor strictly convex).

5. We have then the following proposition. 5 (Fenchel, 1951). Let f be a function defined on a convex set C c R. Then f is concave if and only if its hypograph H(f) is a convex set. Similarly, f is convex if and only if its epigraph P(f) if a convex set. Proof. We prove the proposition for concave functions only. Suppose that f is concave and (x', a'), (x 2 , a 2 ) the two points in H(f). Then for every 0 <— A - I f( Axe+(1—A)x 2 )> Af(x')+(1—A)l(x 2 )> Aa'+(1—A)a 2 . 5. Hypograph of a concave function.

### Generalized Concavity (Classics in Applied Mathematics 63) by Mordecai Avriel, Walter E. Diewert, Siegfried Schaible, Israel Zang

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