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f(expr1,expr2,...,expr3) is convex if f is a convex function and for each expri one of the following conditions hold: f is increasing in argument i and expri is convex. f is decreasing in argument i and expri is concave. expri is affine or constant. Similar logic is applied to establishing concavity or affinity of a function. If a function
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) + +.
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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function cvx_optval = g(y) cvx_begin variable t; maximize(t) subject to pos(t) + pow_pos(t, 1.5) + pow_pos(t, 2.5) <= y; cvx_end. use it as an ordinary function, as in g(14.3), or within CVX as a concave function: cvx_begin variables x y; minimize(quad_over_lin(x, y) + 4*x + 5*y) subject to g(x) + 2*g(y) >= 2; cvx_end.
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Convex-concave programming is an or-ganized heuristic for solving nonconvex problems that involve objective and constraint functions that are a sum of a convex and a concave term. DCP is a structured way to define convex optimization problems, based on a family of basic convex and concave functions and a few rules for combining them.
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Anatomy of a Function def main(): mid = average(10.6, 7.2) print(mid) def average(a, b): sum = a + b return sum / 2 Think/Pair/Share: Find the function definition, function name, parameter(s), and return value in average.
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A single variable function f is said to be convex if f λx + (1−λ)y ≤ λf (x) +(1− λ) f (y), (1.1) for 0 < λ < 1. Problem 1.1. Execute the following python script. Is lnx convex or concave? import numpy as np import matplotlib . pyplot as plt # Plotting log ( x ) x = np . linspace (1 ,8 ,50)# points on the x axis f=np . log (x ...
