Lecture4 - Quotes for today 15.053 February 10, 2011 !...

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1 15.053 February 10, 2011 ! Review of Solving Systems of Equations ! Introduction to the Simplex Algorithm 2 Quotes for today Any impatient student of mathematics or science or engineering who is irked by having algebraic symbolism thrust upon him should try to get along without it for a week. -- Eric Temple Bell To become aware of the possibility of the search is to be onto something. -- Walker Percy Overview ! Solving equations ! The simplex algorithm a clever search technique goes from corner point to corner point until an optimal solution is found one of the most important developments in optimization in the last 100 years 3 4 Preview of the Simplex Method 1 2 3 4 5 6 1 2 3 4 5 K S Start at any feasible corner point. Move to an adjacent corner point with better objective value. Continue until no adjacent corner point has a better objective value. Maximize z = 3 K + 5 S
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5 The simplex algorithm (for max problems) Start with a feasible corner point solution Is it optimal? quit with optimal solution Is the optimum unbounded from above? quit with proof of unboundedness find an improved corner point solution No Yes No Yes 6 Goals for this lecture Major Issues of the Simplex Algorithm 1. Review of solving equations 2. The starting form 3. How does one recognize optimality and unboundedness? 4. How does one move to the next corner point solution? A tableau 7 Review: solving a system of m equations with m variables. 2x 1 + 2x 2 + x 3 = 9 2x 1 - x 2 + 2x 3 = 6 x 1 - x 2 + 2x 3 = 5 x 1 x 2 x 3 RHS Equation 1 2 2 1 = 9 Equation 2 2 -1 2 = 6 Equation 3 1 -1 2 = 5 Elementary Row Operations (EROs) 8 There are three EROs that change the constraints, but do not change the set of solutions to the equations. ERO 1. Multiply a single constraint by a non-zero constant. ERO 2. Add a multiple of one constraint to another constraint. ERO 3. Interchange the order of two constraints.
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9 Review: elementary row operations: EROs x 1 x 2 x 3 RHS Equation 1 2 2 1 = 9 Equation 2 2 -1 2 = 6 Equation 3 1 -1 2 = 5 ERO 1. Multiply Equation 2 by ! . x 1 x 2 x 3 RHS Equation 1 2 2 1 = 9 1 -1/2 1 = 3 Equation 3 1 -1 2 = 5 10 ERO 2: Add a multiple of one constraint to another. x 1 x 2 x 3 RHS Equation 1 2 2 1 = 9 Equation 2 2 -1 2 = 6 Equation 3 1 -1 2 = 5 ERO 2. Subtract Equation 3 from Equation 1. x 1 x 2 x 3 RHS 1 3 -1 = 4 Equation 2 2 -1 2 = 6 Equation 3 1 -1 2 = 5 11 ERO 3: Interchange two constraints x 1 x 2 x 3 RHS Equation 1 2 2 1 = 9 Equation 2 2 -1 2 = 6 Equation 3 1 -1 2 = 5 ERO 3. Interchange Equations 2 and 3 x 1 x 2 x 3 RHS Equation 1 2 2 1 = 9 1 -1 2 = 5 2 -1 2 = 6 12 x 1 x 2 x 3 RHS Equation 1 2 2 1 = 9 Equation 2 2 -1 2 = 6 Equation 3 1 -1 2 = 5 x 1 x 2 x 3 RHS 1 0 0 = 1 0 1 0 = 2 0 0 1 = 3 Canonical form (for linear systems): the coefficients form an identity matrix. How to solve a system of equations Carry out EROs until the equations are in l canonical form. z
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13 More on Canonical Form x 1 x 2 x 3 RHS Equation 1 1 0 0 = 1 Equation 2 0 1 0 = 2 Equation 3 0 0 1 = 3 Canonical form: the columns of the identity matrix do not need to be in order.
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Lecture4 - Quotes for today 15.053 February 10, 2011 !...

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