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particular equations apply Three segments of slopes are defined, an upper convex ele-ment, a middle straight element, and a lower concave element, where the change in gradient with length is respectively positive, zero, and negative The use of gradient data allows accurate definition of form elements and the intervals over which they exist

1 Theory of convex functions - Princeton University

Convex, concave, strictly convex, and strongly convex functions First and second order characterizations of convex functions Optimality conditions for convex problems 1 Theory of convex functions 1 1 De nition Let’s rst recall the de nition of a convex function De nition 1 A function f: RnRis convex if its domain is a convex set and for

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2 The Concave-Convex Procedure (CCCP) The key results of CCCP are summarized by Theorems 1,2, and 3 Theorem 1 shows that any function, subject to weak conditions, can be expressed as the sum of a convex and concave part (this decomposition is not unique) This implies that CCCP can be applied to (almost) any optimization problem Theorem 1

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f concave if h is concave and h nonincreasing and g is convex nondecreasing or nonincreasing condition on extend value extension of h is fundamental counter example in the book if nondecreasing property holds for h but not for h, the composition no

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Concave, if • Strictly convex, if • Strictly concave, if • Definition A function is Note: is concave if and only if is convex Similarly, is strictly concave if and only if is strictly convex The only functions that are both convex and concave are affine functions; i e , functions of the form: convex