solution manual for convex optimization

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solution manual for convex optimization

Convex OptimizationSolutions ManualStephen BoydLieven Vandenberghe38721ExercisesExercisesDefinition of convexity2.1Let G’ r R" be a convex set. with X

solution manual for convex optimization Xi....,Xk G c. and let 01,...,0* «= K satisfy Oi u. 6^1-5---------1- Ok = 1. Show that 1*1X1 H------1- OkXk G c. (The definition of convexity is thatt

his holds for k = 2: you must show it for arbitrary ẨỈ.) Hint. Use induction on k.Solution. This is readily shown by induction from the definition of solution manual for convex optimization

convex set. We illustrate the idea for k = 3, leaving the general case to the reader. Suppose that X1,X2,X3 f c. and 01 +02 +i*3 = 1 with Oi.Oi.Oi > u

solution manual for convex optimization

. We will show that y = 1*1X1 + 02*2 + 03*3 «= c. At least one of the Oi is not equal to one; without loss of generality we can assume that 1*11. Then

Convex OptimizationSolutions ManualStephen BoydLieven Vandenberghe38721ExercisesExercisesDefinition of convexity2.1Let G’ r R" be a convex set. with X

solution manual for convex optimization that P2*2 + P3*3 € c. Since this point and Xi arc in c, y € c.2.2Show that a set is omwx if and only if its intersection with any line is convex. Show

that a set is affine if and only if its intersection with any line is affine.Solution. We prove the first, port. The intersection of two convex set® solution manual for convex optimization

is convex. Therefore if s is a convex set, the intersection of s wit h a line is convex.Conversely. suppose the intcracction of s with any line is con

solution manual for convex optimization

vex. Take any two distinct points Xi and 3’2 € s. The intersection of s with the line through Xi and X2 is convex. Therefore convex combinations of Xi

Convex OptimizationSolutions ManualStephen BoydLieven Vandenberghe38721ExercisesExercisesDefinition of convexity2.1Let G’ r R" be a convex set. with X

solution manual for convex optimization or midpoint (a + 6)/2 is in c. Obviously a convex set is midpoint convex. It can be proved that under mild conditions midpoint convexity implies conv

exity. As a simple case, prove that if c is closed and midpoint convex, then c is convex.Solution. We have to show that Or + (1 - 0)y G c for all 0 G solution manual for convex optimization

[u, 1] and r.y G c. Let o'k) be the binary number of length k. i.c.. a number of tire formo'kì = ci2~' + c-i2~2 + • • • + ck2~kwith e, G {0,1}, cloMMl

solution manual for convex optimization

t4> 0. By midpoint convexity (applied k tiiiHK, recursively), ớ'*>x 1 (I 0)y G c. Because c is clowsl.lim (f*(t>x + (1 - t*

Convex OptimizationSolutions ManualStephen BoydLieven Vandenberghe38721ExercisesExercisesDefinition of convexity2.1Let G’ r R" be a convex set. with X

solution manual for convex optimization conic, or affine, or linear hull of a set 5' is the intersection of all conic sets, or affine sets, or subspaces that contain 5.) Solution. Let. H 1m

v 1.1k; convex hull of s and let D be the inlxTsiM tiiHi of all convex sel« that contain s,V = Q{p I D convex, L> "J S'}.We will show lhal H D by show solution manual for convex optimization

ing that H <_ D and D <- H.First wc show that 11 c 2J. Suppose X G 11. i.e., X is a convex combination of some points Xi,..., x„ G s. .Now let D be an

solution manual for convex optimization

y convex set such that D s. Evidently, we have Xi,..., x„ G D. Since u is convex, and X is a convex combination of X1,..., x„, it follows that X G D.

Convex OptimizationSolutions ManualStephen BoydLieven Vandenberghe38721ExercisesExercisesDefinition of convexity2.1Let G’ r R" be a convex set. with X

solution manual for convex optimization .C., X G T>..Now let us show that V c 11. Since 11 is convex (by definition) and contains s. we must haw II = D for some D in the construction of V. p

roving the claim.2 Convex setsExamples2.5What is the distance between two parallel hypcrplancs {x G R" I it1 r = 01} and {x G R I ứ7 X = Ò2}?Solution. solution manual for convex optimization

The distance between the two hyperplancss is |h] — hjl/llalh. To see this, consider the construction in the figure below.The distance between the two

solution manual for convex optimization

hyperplancs is also the distance between the two points Xi and x-2 where the hypcrplane intersects the line through the origin and parallel to the no

Convex OptimizationSolutions ManualStephen BoydLieven Vandenberghe38721ExercisesExercisesDefinition of convexity2.1Let G’ r R" be a convex set. with X

solution manual for convex optimization in another? Give conditions under which

Convex OptimizationSolutions ManualStephen BoydLieven Vandenberghe38721ExercisesExercisesDefinition of convexity2.1Let G’ r R" be a convex set. with X