Lecture6 - C&O 355 Mathematical Programming Fall 2010...

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
C&O 355 Mathematical Programming Fall 2010 Lecture 6 N. Harvey
Image of page 1

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

View Full Document Right Arrow Icon
Polyhedra Definition: For any a 2 R n , b 2 R , define Def: Intersection of finitely many halfspaces is a polyhedron Def: A bounded polyhedron is a polytope , i.e., P µ { x : -M · x i · M 8 i } for some scalar M So the feasible region of LP is polyhedron Hyperplane Halfspaces
Image of page 2
Convex Sets Def: Let x 1 ,…,x k 2 R n . Let ® 1 ,…, ® k satisfy ® i ¸ 0 for all i and i ® i = 1. The point i ® i x i is a convex combination of the x i ’s. Def: A set C µ R n is convex if for every x, y 2 C and 8 ® 2 [0,1], the convex combination ® x+(1- ® )y is in C. Claim 1 : Any halfspace is convex. Claim 2 : The intersection of any number of convex sets is convex. Corollary : Every polyhedron is convex. Not convex x y Convex x y
Image of page 3

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

View Full Document Right Arrow Icon
Convex Functions Let C µ R n be convex. Def: f : C ! R is a convex function if f( ® x+(1- ® )y) · ® f(x) + (1- ® )f(y) 8 x,y 2 C Claim: Let f : C ! R be a convex function, and let a 2 R . Then { x 2 C : f(x) · a } (the “sub-level set”) is convex. Example: Let . Then f is convex. Corollary: Let B = { x : k x k · 1 }. (The Euclidean Ball) Then B is convex.
Image of page 4
Affine Maps Def: A map f : R n ! R m is called an affine map if f(x) = Ax + b for some matrix A and vector b. Fact: Let C µ R n have defined volume. Let f(x)=Ax + b.
Image of page 5

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

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern