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MIT16_410F10_lec17 - 16.410/413 Principles of Autonomy and...

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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
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Assignments Readings [IOR] Chapter 4. Frazzoli (MIT) Lecture 17: The Simplex Method November 10, 2010 2 / 32
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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
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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 0 . 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
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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 0 . 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
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A na¨ ıve algorithm (1) Recall the standard form: min z = c T x s.t.: Ax = b , x 0 . 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
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A na¨ ıve algorithm (1) Recall the standard form: min z = c T x s.t.: Ax = b , x 0 . or, really: min z = c T y y + c T s s s.t.: A y y + Is = b , y , s 0 . 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 ). making them active (or binding), finding the (unique) point where all these hyperplanes meet. If all the variables are non-negative, this point is in fact a vertex of the feasible region. Frazzoli (MIT) Lecture 17: The Simplex Method November 10, 2010 6 / 32
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A na¨ ıve algorithm (2) One could possibly generate all basic feasible solutions, and then check the value of the cost function, finding the optimum by enumeration.
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