bv_cvxbook_extra_exercises

bv_cvxbook_extra_exercises - Additional Exercises for...

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Additional Exercises for Convex Optimization Stephen Boyd Lieven Vandenberghe June 15, 2013 This is a collection of additional exercises, meant to supplement those found in the book Convex Optimization , by Stephen Boyd and Lieven Vandenberghe. These exercises were used in several courses on convex optimization, EE364a (Stanford), EE236b (UCLA), or 6.975 (MIT), usually for homework, but sometimes as exam questions. Some of the exercises were originally written for the book, but were removed at some point. Many of them include a computational component using CVX, available at www.stanford.edu/~boyd/cvx/ . Matlab files required for these exercises can be found at the book web site www.stanford.edu/~boyd/cvxbook/ . Some of the exercises require a knowledge of elementary analysis. You are free to use these exercises any way you like (for example in a course you teach), provided you acknowledge the source. In turn, we gratefully acknowledge the teaching assistants (and in some cases, students) who have helped us develop and debug these exercises. Pablo Parrilo helped develop some of the exercises that were originally used in 6.975. Course instructors can obtain solutions by request to [email protected] , or by email to us. In either case please specify the course you are teaching and give its URL. We’ll update this document as new exercises become available, so the exercise numbers and sections will occasionally change. We have categorized the exercises into sections that follow the book chapters, as well as various additional application areas. Some exercises fit into more than one section, or don’t fit well into any section, so we have just arbitrarily assigned these. Stephen Boyd and Lieven Vandenberghe 1
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Contents 1 Convex sets 3 2 Convex functions 5 3 Convex optimization problems 13 4 Duality 26 5 Approximation and fitting 39 6 Statistical estimation 52 7 Geometry 59 8 Unconstrained and equality constrained minimization 74 9 Interior point methods 80 10 Mathematical background 85 11 Circuit design 86 12 Signal processing and communications 93 13 Finance 103 14 Mechanical and aerospace engineering 116 15 Graphs and networks 127 16 Energy and power 135 17 Miscellaneous applications 146 2
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1 Convex sets 1.1 Is the set { a R k | p (0) = 1 , | p ( t ) | ≤ 1 for α t β } , where p ( t ) = a 1 + a 2 t + · · · + a k t k 1 , convex? 1.2 Set distributive characterization of convexity. [7, p21], [6, Theorem 3.2] Show that C R n is convex if and only if ( α + β ) C = αC + βC for all nonnegative α , β . 1.3 Composition of linear-fractional functions . Suppose φ : R n R m and ψ : R m R p are the linear-fractional functions φ ( x ) = Ax + b c T x + d , ψ ( y ) = Ey + f g T y + h , with domains dom φ = { x | c T x + d> 0 } , dom ψ = { y | g T x + h> 0 } . We associate with φ and ψ the matrices bracketleftBigg A b c T d bracketrightBigg , bracketleftBigg E f g T h bracketrightBigg , respectively.
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