This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 16.410/413 Principles of Autonomy and Decision Making Lecture 17: The Simplex Method Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 10, 2010 Frazzoli (MIT) Lecture 17: The Simplex Method November 10, 2010 1 / 32 Assignments Readings [IOR] Chapter 4. Frazzoli (MIT) Lecture 17: The Simplex Method November 10, 2010 2 / 32 Outline 1 Geometric Interpretation Geometric Interpretation 2 Algebraic Procedure 3 Intro to interior point methods Frazzoli (MIT) Lecture 17: The Simplex Method November 10, 2010 3 / 32 Geometric Interpretation Consider the following simple LP: max z = x 1 + 2 x 2 = (1 , 2) · ( x 1 , x 2 ) , s.t.: x 1 ≤ 3 , x 1 + x 2 ≤ 5 , x 1 , x 2 ≥ . Each inequality constraint defines a hyperplane, and a feasible halfspace. The intersection of all feasible half spaces is called the feasible region . x 1 x 2 c The feasible region is a (possibly unbounded ) polyhedron. The feasible region could be the empty set: in such case the problem is said unfeasible . Frazzoli (MIT) Lecture 17: The Simplex Method November 10, 2010 4 / 32 Geometric Interpretation (2) Consider the following simple LP: max z = x 1 + 2 x 2 = (1 , 2) · ( x 1 , x 2 ) , s.t.: x 1 ≤ 3 , x 1 + x 2 ≤ 5 , x 1 , x 2 ≥ . The “ c ” vector defines the gradient of the cost. Constantcost loci are planes normal to c . x 1 x 2 c Most often, the optimal point is located at a vertex (corner) of the feasible region. If there is a single optimum, it must be a corner of the feasible region. If there are more than one, two of them must be adjacent corners. If a corner does not have any adjacent corner that provides a better solution, then that corner is in fact the optimum. Frazzoli (MIT) Lecture 17: The Simplex Method November 10, 2010 5 / 32 A na¨ ıve algorithm (1) Recall the standard form: min z = c T x s.t.: Ax = b , x ≥ . Corners of the feasible regions (also called basic feasible solutions ) are solutions of Ax = b ( m equations in n unknowns, n > m ), obtained setting n m variables to zero, and solving for the others (basic variables), ensuring that all variables are nonnegative. Frazzoli (MIT) Lecture 17: The Simplex Method November 10, 2010 6 / 32 A na¨ ıve algorithm (1) Recall the standard form: min z = c T x s.t.: Ax = b , x ≥ . or, really: min z = c T y y + c T s s s.t.: A y y + Is = b , y , s ≥ . Corners of the feasible regions (also called basic feasible solutions ) are solutions of Ax = b ( m equations in n unknowns, n > m ), obtained setting n m variables to zero, and solving for the others (basic variables), ensuring that all variables are nonnegative. This amounts to: picking n y inequality constraints, (notice that n = n y + n s = n y + m )....
View
Full
Document
This note was uploaded on 12/26/2011 for the course SCIENCE 16.410 taught by Professor Prof.brianwilliams during the Fall '10 term at MIT.
 Fall '10
 Prof.BrianWilliams
 Mass

Click to edit the document details