Convex Optimization by Stephen Boyd, Lieven Vandenberghe

By Stephen Boyd, Lieven Vandenberghe

Convex optimization difficulties come up usually in lots of diversified fields. A finished advent to the topic, this ebook indicates intimately how such difficulties should be solved numerically with nice potency. the focal point is on spotting convex optimization difficulties after which discovering the main acceptable method for fixing them. The textual content includes many labored examples and homework workouts and should attract scholars, researchers and practitioners in fields reminiscent of engineering, laptop technological know-how, arithmetic, information, finance, and economics.

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3 Operations that preserve convexity 41 (θt + (1 − θ)s > 0 is obvious). This follows from θx + (1 − θ)y = µ(x/t) + (1 − µ)(y/s), θt + (1 − θ)s where µ= θt ∈ [0, 1]. θt + (1 − θ)s Linear-fractional functions A linear-fractional function is formed by composing the perspective function with an affine function. 12) where A ∈ Rm×n , b ∈ Rm , c ∈ Rn , and d ∈ R. 13) is called a linear-fractional (or projective) function. If c = 0 and d > 0, the domain of f is Rn , and f is an affine function. So we can think of affine and linear functions as special cases of linear-fractional functions.

Suppose the k + 1 points v0 , . . , vk ∈ Rn are affinely independent, which means v1 − v0 , . . , vk − v0 are linearly independent. The simplex determined by them is given by C = conv{v0 , . . 2 Some important examples 33 where 1 denotes the vector with all entries one. The affine dimension of this simplex is k, so it is sometimes referred to as a k-dimensional simplex in Rn . 5 Some common simplexes. A 1-dimensional simplex is a line segment; a 2-dimensional simplex is a triangle (including its interior); and a 3-dimensional simplex is a tetrahedron.

A set C ⊆ R2 . The dashed line shows the boundary of the domain of the linear-fractional function f (x) = x/(x1 + x2 + 1) with dom f = {(x1 , x2 ) | x1 + x2 + 1 > 0}. Right. Image of C under f . The dashed line shows the boundary of the domain of f −1 . Like the perspective function, linear-fractional functions preserve convexity. , cT x + d > 0 for x ∈ C), then its image f (C) is convex. 12) is convex, and the image of the resulting set under the perspective function P , which yields f (C), is convex.

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