By Stephen Boyd, Lieven Vandenberghe

Convex optimization difficulties come up often in lots of diversified fields. A finished advent to the topic, this booklet indicates intimately how such difficulties should be solved numerically with nice potency. the point of interest is on spotting convex optimization difficulties after which discovering the main applicable process for fixing them. The textual content includes many labored examples and homework routines and may attract scholars, researchers and practitioners in fields resembling engineering, computing device technology, arithmetic, facts, finance, and economics.

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**Additional resources for Convex Optimization**

**Example text**

The projection of a convex set onto some of its coordinates is convex: if S ⊆ Rm × Rn is convex, then T = {x1 ∈ Rm | (x1 , x2 ) ∈ S for some x2 ∈ Rn } is convex. The sum of two sets is defined as S1 + S2 = {x + y | x ∈ S1 , y ∈ S2 }. If S1 and S2 are convex, then S1 + S2 is convex. To see this, if S1 and S2 are convex, then so is the direct or Cartesian product S1 × S2 = {(x1 , x2 ) | x1 ∈ S1 , x2 ∈ S2 }. The image of this set under the linear function f (x1 , x2 ) = x1 + x2 is the sum S1 + S2 .

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. , 0, e1 , . . , en ∈ Rn . It can be expressed as the set of vectors that satisfy x 0, 1T x ≤ 1.

3], and the survey by Nesterov, Wolkowicz, and Ye [NWY00]. Some notable papers on this subject are Goemans and Williamson [GW95], Nesterov [Nes00, Nes98], Ye [Ye99], and Parrilo [Par03]. Randomized methods are discussed in Motwani and Raghavan [MR95]. Convex analysis, the mathematics of convex sets, functions, and optimization problems, is a well developed subfield of mathematics. Basic references include the books by Rockafellar [Roc70], Hiriart-Urruty and Lemar´echal [HUL93, HUL01], Borwein and Lewis [BL00], and Bertsekas, Nedi´c, and Ozdaglar [Ber03].