Optimization

Lecture 03 10/25/11 Geometry of LPs

A (linear) subspace of is a subset of points that is closed under vector addition and scalar multiplication. Equivalently,

∈ ⋯ 0,1,…, is the set of points satisfying some set of homogeneous linear equations.

It has dimension

dim rank . Note: dim is the maximum number of linear independent vectors in .

2 Geometry of LPs

An affine subspace of is a linear subspace translated by a vector , formally, ∈ . We define dim dim . Equivalently, ∈ ⋯ , 1, … , Is the set of points satisfying some set of (inhomogenous) linear equations.

The dimension of any subset of is the smallest dimension of any affine subspace containing it.

3 Geometry of LPs

For example, every line segment has dimension 1.

Any set of points, 1, has dimension at most 1.

The feasible region | ,0 of the linear program

max. s.t. , an -matrix, 0, has dimension at most .

4 Convex Polytopes

An affine subspace of of dimension 1 is called a hyperplane . Alternatively, a hyperplane is a set of points satisfying

⋯ , with not all ‘s equal to zero. A hyperplane defines two (closed) halfspaces. These are the sets of points satisfying or ⋯ , respectively. ⋯

5 Convex Polytopes

A subset of is called convex , if for any , ∈ and ∈ 0,1, it holds that 1 ∈ . A halfspace is a convex set. Therefore the intersection of halfspaces is convex.

The intersection of a finite number of halfspaces, when it is bounded and non-empty, is called a (convex) polytope .

We are interested in convex polytopes that are included in the non-negative orthant, i.e., by convention of the halfspace defining the polytope will always be , . 0 1, … ,

6 Convex Polytopes Polytope defined by the constraints of LP-1:

4 2

3 6

, , 0

7 Convex Polytopes

Let be a convex polytope of dimension and a halfspace defined by hyperplane .

Let ∩ . If ⊆ , i.e., when and touch just in their exteriors, we call a face of and is the supporting hyperplane defining . There are three distinguished kinds of faces: • A facet is a face of dimension 1. • A vertex is a face of dimension zero (a point). • An edge is a face of dimension one (a line segment).

8 Convex Polytopes Some facets, edges and vertices of the 3-dimensional polytope defined by the constraints of LP-1.

9 Convex Polytopes

The hyperplane defining a facet corresponds to a defining halfspace of the polytope. The converse is not always true. Vertices are the polytope‘s corners. An edge is a line segment joining two vertices. Not every pair of vertices defines an edge, though! Theorem 2 (a) Every convex polytope is the convex hull of its vertices. (b) Conversely, if is a finite set of points, then the convex hull of is a convex polytope . The set of vertices of is a subset of . What‘s the formal relation between standard-form LPs and polytopes? 10 Standard-Form LPs and Polytopes

Given an LP max. s.t. 0, in standard form, we assume w.l.o.g. that the equations are of the form

, 1,…,. We can always achieve this by finding a basis of , multi- plying by and rearranging so corresponds to the last columns of . 11 Standard-Form LPs and Polytopes

So , 0, is in fact equivalent to the inequalities

0, 1,…, . 0, 1,…, This, however, is an intersection of halfspaces (and bounded by Assumption 3) and, thus, defines a convex polytope ⊂ . Thus, we can view the feasible region ⊂ of a standard- form LP , 0, as a convex polytope ⊂ .

12 Standard-Form LPs and Polytopes

Conversely, let be a polytope in the non-negative orthant of . The halfspaces defining can be expressed as

⋯ 0, 1,…,, where w.l.o.g. the first inequalities are of the form

0, 1,…,. Let be the matrix of coefficients of the remaining inequalities. We introduce slack variables to obtain , … , the equivalent formulation

, 0, with matrix , , and | , … , . ∈ 13 Standard-Form LPs and Polytopes

Thus, we can alternatively view any polytope ⊂ as the feasible region ⊂ of a standard-form LP , 0. The transformation of solutions between these two eqivalent views is as follows: Given a point , we , … , ∈ obtain by defining , … , ∈

, 1,…,. Conversely, can be transformed into , … , ∈ by truncating the last coordinates , … , ∈ of . 14 Standard-Form LPs and Polytopes Theorem 3 Let ∈ be a convex polytope, , 0 ⊂ the corresponding feasible region of a standard-form LP, and . Then the following are , … , ∈ equivalent:

(a) The point is a vertex of . (b) If 1 ′′ , with , ∈ and 0 1, then . (c) The corresponding vector ∈ with ∑ , 1,…, is a basic feasible solution of .