Download presentation

1
**Lecture #3; Based on slides by Yinyu Ye**

Geometry of LP: Feasible Regions, Feasible Directions, Basic Feasible Solutions Ashish Goel Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. Lecture #3; Based on slides by Yinyu Ye

2
**LP Feasible Region in the Inequality Form**

x simultaneously satisfy This is the intersection of the m Half-spaces, and it is a convex (polyhedron) set Lecture #3; Based on slides by Yinyu Ye

3
**Lecture #3; Based on slides by Yinyu Ye**

4
**Lecture #3; Based on slides by Yinyu Ye**

5
**Corner or Extreme Points**

Convex Hull: Extreme Points: A point in a set that does not belong to the hull (convex combination) of the other points For LP in inequality form, an extreme point is the intersection of n hyperplanes associated with the inequality constraints. Lecture #3; Based on slides by Yinyu Ye

6
**Lecture #3; Based on slides by Yinyu Ye**

Feasible Direction Direction: A direction is notated by a vector d It is always associated with a “location” or point x Together a point and a direction define a ray: x + ϵd, for all ϵ > 0 where d and kd are considered the same direction for all k > 0 Feasible Direction: A direction, d, is said to be “feasible” (relative to a given feasible point x) if x + ϵd is feasible for some ϵ> 0 For LP, all feasible directions at a feasible point form a convex (cone) set Lecture #3; Based on slides by Yinyu Ye

7
**Extreme Feasible Direction**

A feasible direction d is extreme if d cannot be written as an convex combination of other feasible directions Interior Point is a point x where every direction is feasible Lecture #3; Based on slides by Yinyu Ye

8
**LP Problem in Inequality Form**

Lecture #3; Based on slides by Yinyu Ye

9
**Recall the Production Problem**

Objective contour c Lecture #3; Based on slides by Yinyu Ye

10
**Basic Theorems of Linear Programming**

All LP problems fall into one of three cases: Problem is infeasible: Feasible region is empty. Problem is unbounded: Feasible region is unbounded towards the optimizing direction. Problem is feasible and bounded; and in this case: there exists an optimal solution or optimizer. There may be a unique optimizer or multiple optimizers. All optimizers form a convex set and they are on a face of the feasible region. There is always at least one corner (extreme) optimizer if the feasible region has a corner point. Moreover, Local optimality implies global optimality Lecture #3; Based on slides by Yinyu Ye

11
**Sketch Proof of Local Optimality Implies Global**

(P) minimize f (x) subject to x ∈ Ω ⊂ Rn, is a Convex Optimization problem if f (x) is a convex function over a convex feasible region Ω. Proof by contradiction. Suppose x’ is a local minimizer but not a global minimizer x∗, that is, x’ ∈ Ω and x∗ ∈ Ω but f (x∗) < f (x’). Now the convex combination point αx’ + (1 − α)x∗ must be feasible (why?), and f (αx’+ (1 − α)x∗) ≤ αf (x’)+ (1 − α)f (x∗) < f (x’) for any 0 ≤ α < 1. This contradicts the local optimality as α can be arbitrarily close to 1 so that αx’+ (1 − α)x∗ can be arbitrarily close to x’. Lecture #3; Based on slides by Yinyu Ye

12
**Optimality Certification of the Production Problem**

a4 a5 a1 a2 a3 a4 Objective contour c Lecture #3; Based on slides by Yinyu Ye

13
**Feasible Directions at a Corner**

a4 a5 a1 a2 a3 a4 Objective contour c Lecture #3; Based on slides by Yinyu Ye

14
**How to Certify a Corner being an Optimizer**

Every feasible direction at the corner is a worsening (descent in this case) direction, that is, cTd ≤0 in this case. Or, equivalently, c is a conic combination of the normal directions at the corner point, that is, there are multipliers α1 ≥0 and α2 ≥0, such as c= α1 ai1+ α2 ai2, in the 2-dimensional case. In the n-dimensional case: c= α1 ai1+…+ αn ain where ai1,…, ain are the normal directions associated with the corner point. Lecture #3; Based on slides by Yinyu Ye

15
**LP in Standard (Equality) Form**

Lecture #3; Based on slides by Yinyu Ye

16
**Reduction to Standard Form**

max cTx to min – cTx Eliminating ”free” variables: substitute with the difference of two nonnegative variables x := xl − xll, (xl, xll) ≥ 0. Eliminating inequalities: add a slack variable aT x ≤ b = ⇒ aT x + s = b, s ≥ 0 aT x ≥ b = ⇒ aT x − s = b, s ≥ 0 Lecture #3; Based on slides by Yinyu Ye

17
**Reduction of the Production Problem**

min -x1 -2x2 s.t. x1 +x3 = 1 x2 +x4 +x2 +x5 = 1.5 (x1, x2, x3, x4, x5) ≥ 0 x3, x4,and x5 are called slack variables Lecture #3; Based on slides by Yinyu Ye

18
**Basic and Basic Feasible Solution**

In the LP standard form, select m linearly independent columns, denoted by the variable index set B, from A. Solve AB xB = b for the dimension-m vector xB . By setting the variables, xN , of x corresponding to the remaining columns of A equal to zero, we obtain a solution x such that Ax = b. Then, x is said to be a basic solution to (LP) with respect to the basic variable set B. The variables in xB are called basic variables, those in xN are nonbasic variables, and AB is called a basis. If a basic solution xB ≥ 0, then x is called a basic feasible solution, or BFS. Note that AB and xB follow the same index order in B. Two BFS are adjacent if they differ by exactly one basic variable. Lecture #3; Based on slides by Yinyu Ye

19
**BS of the Production Problem in Equality Form**

x1 x2 x1 +x3 = 1 x2 +x4 +x2 +x5 = 1.5 (x1, x2, x3, x4, x5) ≥ 0 Basis 3,4,5 1,4,5 3,4,1 3,2,5 3,4,2 1,2,3 1,2,4 1,2,5 Feasible? √ x1 , x2 0, 0 1, 0 1.5, 0 0, 1 0, 1.5 .5, 1 1, .5 1, 1 Lecture #3; Based on slides by Yinyu Ye

20
**BFS and Corner Point Equivalence Theorem**

Theorem Consider the feasible region in the standard LP form. Then, a basic feasible solution and a corner (extreme) point are equivalent; the formal is algebraic and the latter is geometric. Algorithmically, we need only to look at BFSs for finding an optimizer. But still too many to look… Lecture #3; Based on slides by Yinyu Ye

Similar presentations

Presentation is loading. Please wait....

OK

Simplex method (algebraic interpretation)

Simplex method (algebraic interpretation)

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google