MIT16_410F10_lec17

MIT16_410F10_lec17 - 16.410/413 Principles of Autonomy and Decision Making Lecture 17 The Simplex Method Emilio Frazzoli Aeronautics and

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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 half-space. 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. Constant-cost 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 non-negative. 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 non-negative. This amounts to: picking n y inequality constraints, (notice that n = n y + n s = n y + m )....
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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.

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MIT16_410F10_lec17 - 16.410/413 Principles of Autonomy and Decision Making Lecture 17 The Simplex Method Emilio Frazzoli Aeronautics and

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