# Tut3sol - Tutorial 3 Outline of Solutions Q1(a For each of...

This preview shows pages 1–3. Sign up to view the full content.

Tutorial 3: Outline of Solutions Q1. (a) For each of the following sets, decide whether it is representable as a polyhedron? (i) The set of all ( x, y ) R 2 satisfying the constraints x cos θ + y sin θ 1 θ [0 , π/ 2] x 0 y 0 . (ii) The set of all x R satisfying the constraint x 2 - 8 x + 15 0. Solution : x y (1,0) (0,0) (0,1) x 3 5 The first set is a quarter circle in the positive orthant of radius 1. Since an infinite number of linear constraints are needed to describe (i), it is not a polyhedron. The second set can be expressed as x R satisfying the two linear constraints x 3 , x 5. Hence it is representable as a polyhedron. (b) Let f : R n R be a convex function and let a be some constant. Show that the level set S = { x R n | f ( x ) a } is a convex set. Solution : Given x , y S , we see that for λ [0 , 1] f ( λ x + (1 - λ ) y ) λf ( x ) + (1 - λ ) f ( y ) (From definition of convex function) λa + (1 - λ ) a (Since x , y S ) = a. Hence λ x + (1 - λ ) y S and the set is convex. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Q2. Consider the following polyhedron of an LP problem: 2 x 1 - x 2 + 5 x 3 = - 1 (1) 3 x 2 + x 4 5 (2) 7 x 1 - 4 x 3 + x 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern