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By Chong-Yung Chi, Wei-Chiang Li, Chia-Hsiang Lin

Convex Optimization for sign Processing and Communications: From basics to Applications presents primary historical past wisdom of convex optimization, whereas amazing a stability among mathematical concept and purposes in sign processing and communications.

In addition to complete proofs and point of view interpretations for middle convex optimization conception, this e-book additionally presents many insightful figures, feedback, illustrative examples, and guided trips from concept to state-of-the-art examine explorations, for effective and in-depth studying, in particular for engineering scholars and pros.

With the robust convex optimization thought and instruments, this booklet provide you with a brand new measure of freedom and the potential of fixing difficult real-world medical and engineering problems.

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Extra info for Convex Optimization for Signal Processing and Communications

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N θi s i x= i=1 for all θ ∈ Rn+ , ni=1 θi = 1, and θ = ei for any i. 2, {s1 , s2 } are the extreme points of conv{s1 , s2 } for n=2, and {s1 , s2 , s3 } are those of conv{s1 , s2 , s3 } for n = 3. 4 Let S = {s1 , . . , sn }. The set of extreme points of conv S must be the full set of S when S is affinely independent, otherwise a subset of S. Finding all the extreme points of the convex hull of a finite set C is useful in problem size reduction of an optimization problem with a feasible set being a convex hull of C, especially when |C| (the number of elements in the set C) is large but the total number of the extreme points of conv C is much smaller than |C|.

Xm ] and H = [h1 , . . , hm ] ∈ Rn×m . 56) n×n for some θ1 , θ2 ∈ [0, 1]. 56) are the corresponding second-order Taylor series approximations, if θ1 and θ2 are set to zero. 57) for some θ ∈ [0, 1], which is also the corresponding first-order Taylor series approximation of f (x) if θ is set to zero. 13 The Taylor series of a real-valued function of a complex variable, f : Cn → R, is quite complicated in terms of complex variable x = u + jv ∈ Cn in general, except for the first-order Taylor series approximation which is given by f (x + h) ≃ f (x) + Re ∇f (x)H h .

Sn } ⊆ Rn−1 is affinely independent, then the simplest simplex T = conv{s1 , . . , sn } ⊆ Rn−1 can be reconstructed from the n hyperplanes {H1 , . . , Hn } and vice versa, where Hi aff ({s1 , . . , sn } \ {si }) (cf. 31)). 30), the hyperplane Hi , with affine dimension n − 2, can be parameterized by an “outward-pointing normal vector” bi ∈ Rn−1 such that Convex Sets T ⊂ Hi− (cf. 45) and si ∈ Hi− = x ∈ Rn−1 | bTi x ≤ hi . 46) We will show that T = conv{s1 , . . , sn } = n i=1 Hi− . 47) implies that all the extreme points of T can be obtained from Hi , i = 1, .

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