Cashflow

Yinyu Ye, MS&E, Stanford

MS&E310 Lecture Note #05

1

The Simplex Method

Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A.

http://www.stanford.edu/˜yyye
(LY, Chapters 2.3-2.5, 3.1-3.4)

Yinyu Ye, MS&E, Stanford

MS&E310 Lecture Note #05

2

Geometry of linear programming

Consider maximize subject to

x1 x1

+2x2 ≤1 x2 ≤1 ≤ 1.5 ≥ 0. +x2 x2

x1 x1 ,

Yinyu Ye, MS&E, Stanford

MS&E310 Lecture Note #05

3

LP Geometry depicted in two variable space
If the direction of c is contained by the norm cone of a corner point, then the point is optimal.

c a2 a1 a2 a3

a3 a4

Each corner point has a norm direction cone

a4 a5

Objective contour

Yinyu Ye, MS&E, Stanford

MS&E310 Lecture Note #05

4

• solution (decision, point): any specification of values for all decision variables, regardless of whether it is a desirable or even allowable choice

• feasible solution: a solution for which all the constraints are satisfied. • feasible region (constraint set, feasible set): the collection of all feasible solution • interior, boundary, face • extreme or corner or vertex point • objective function contour (iso-profit, iso-cost line) • optimal solution: a feasible solution that has the most favorable value of the objective function

• optimal objective value: the value of the objective function evaluated at an optimal solution

• active constraint (binding constraint)

Yinyu Ye, MS&E, Stanford

MS&E310 Lecture Note #05

5

Formal definition of face and extreme point
Let P be a polyhedron in R n , F is a face of P if and only if there is a vector b for which F is the set of points attaining max {b T y maximum is finite. A polyhedron has only finite many faces; each face is a nonempty polyhedron. A vector y

: y ∈ P } provided this

∈ P is an extreme point or a vertex of P if y is not a convex

combination of more than one distinct points.