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Definition 1. A function f : S ⊂ Rn → R defined on a convex set S is concave if for any two points x1 x2 ∈ , S and for any λ ∈ [0, 1] we have: λx1 (1 − λ) x2 ≥ λf(x1) (1 − λ)f(x2) + +. is called strictly concave if for any two points x1 , x2 ∈ S and for any λ ∈ (0, 1) we have: λx1 (1 − λ) x2 > λf(x1) (1 − λ)f(x2) + +.
May 22, 2024 · Concavity provides valuable insights into how a function curves, distinguishing between concave upward and concave downward shapes, while points of inflection mark locations where the curvature changes sign.
It is also possible to characterize concavity or convexity of functions in terms of the convexity of particular sets. Given the graph of a function, the hypograph of f,
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This book is about convex optimization, a special class of mathematical optimiza-tion problems, which includes least-squares and linear programming problems. It is well known that least-squares and linear programming problems have a fairly complete theory, arise in a variety of applications, and can be solved numerically very efficiently.
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1 Concave and convex functions. Definition 1 A function f defined on the convex set C ⊂ Rn is called con-cave if for every x1, x2 ∈ C and 0 ≤ t ≤ 1, we have. f(tx1 + (1 − t)x2) ≥ tf(x1) + (1 − t)f(x2). Definition 2 A function f defined on the convex set C ⊂ Rn is called strictly concave if for every x1 6= x2,and 0 < t < 1, we have.
Mar 23, 2012 · In this chapter we introduce convex, concave, strictly convex, and strictly concave functions defined on subsets of R n. Convex and concave functions are extremely important in nonlinear programming because they are among the few functions for which sufficient optimality criteria can be given (Chaps. 5 and 7), and they are the only functions ...
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When you have to compute an angle, you need to divide, or use some approximation (anything involving Pi, for example) or some trigonometric function. When you have to compute an angle in code , you’ll almost always be approximating.
