By Stephen Boyd, Lieven Vandenberghe

ISBN-10: 0521833787

ISBN-13: 9780521833783

Convex optimization difficulties come up usually in lots of various fields. A complete advent to the topic, this booklet exhibits intimately how such difficulties will be solved numerically with nice potency. the point of interest is on spotting convex optimization difficulties after which discovering the main acceptable approach for fixing them. The textual content comprises many labored examples and homework routines and may entice scholars, researchers and practitioners in fields resembling engineering, computing device technological know-how, arithmetic, facts, finance, and economics.

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En ∈ Rn . It is the set of vectors that satisfy x 0, 1T x = 1. Vectors in the probability simplex correspond to probability distributions on a set with n elements, with xi interpreted as the probability of the ith element. 6), we proceed as follows. By definition, x ∈ C if and only if x = θ0 v0 + θ1 v1 + · · · + θk vk for some θ 0 with 1T θ = 1. Equivalently, if we define y = (θ1 , . . 8) for some y 0 with 1T y ≤ 1. Now we note that affine independence of the points v0 , . . , vk implies that the matrix B has rank k.

17) These inequalities are infeasible if and only if the (convex) sets C = {b − Ax | x ∈ Rn }, m D = Rm | y ≻ 0} ++ = {y ∈ R do not intersect. The set D is open; C is an affine set. , a nonzero λ ∈ Rm and µ ∈ R such that λT y ≤ µ on C and λT y ≥ µ on D. Each of these conditions can be simplified. The first means λT (b − Ax) ≤ µ for all x. 19) that AT λ = 0 and λT b ≤ µ. The second inequality means λT y ≥ µ for all y ≻ 0. This implies µ ≤ 0 and λ 0, λ = 0. 17) is infeasible if and only if there exists λ ∈ Rm such that λ = 0, λ 0, AT λ = 0, λT b ≤ 0.

1 Generalized inequalities 43 Generalized inequalities Proper cones and generalized inequalities A cone K ⊆ Rn is called a proper cone if it satisfies the following: • K is convex. • K is closed. • K is solid, which means it has nonempty interior. • K is pointed, which means that it contains no line (or equivalently, x ∈ K, − x ∈ K =⇒ x = 0). A proper cone K can be used to define a generalized inequality, which is a partial ordering on Rn that has many of the properties of the standard ordering on R.

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Convex Optimization by Stephen Boyd, Lieven Vandenberghe


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