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Optimization Spring 2015

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Practical information Lecturers: Kristoffer Arnsfelt Hansen and Peter Bro Miltersen. Homepage: → bb.au.dk Exam: Written, 3 hours. Compulsary program: 3 assignments. You can transfer credit from earlier years - see course page. The solution to the compulsory assignments should be handed in at a specific tutorial and given to the instructor in person. Text: Robert J. Vanderbei: Linear Programming – Foundations and Extensions.

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Blackboard intermezzo You need to ”self-enroll” to your correct class! To switch classes: find another student willing to switch and then inform the respective TA’s.

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Compulsary assignments To pass the compulsary program and take the exam you must: Hand in all 3 assignments on time and have them approved by your TA. Start early!

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Linear Programming In a sentence: Optimizing a linear function subject to linear inequalities. We will see today: How to model problems using linear programming. How to solve linear programs (Dantzig’s simplex algorithm).

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Example: Diet Problem Serving Size Energy (kcal) Protein (g) Calcium (mg) Price per serving (cents) Oatmeal28 g110423 Chicken100 g205321224 Eggs2 large160135413 Whole Milk237 cc16082859 Cherry Pie170 g42042220 Pork with Beans260 g260148019 Necessary daily intake: Energy 2000 kcal, Protein 55 g, Calcium 800 mg Compose a diet minimizing price and fulfilling necessary daily intake.

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(August 2014)

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Example: Diet Problem Serving Size Energy (kcal) Protein (g) Calcium (mg) Price per serving (cents) Oatmeal28 g110423 Chicken100 g205321224 Eggs2 large160135413 Whole Milk237 cc16082859 Cherry Pie170 g42042220 Pork with Beans260 g260148019 Necessary daily intake: Energy 2000 kcal, Protein 55 g, Calcium 800 mg Compose a diet minimizing price and fulfilling necessary daily intake.

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LP formulation Minimize: Subject to:

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LP modelling steps

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A little history of linear programming Roots in work on linear inequalities of J. Fourier. L. V. Kantorovitch invents LP in USSR 1939 for optimizing production while working for the Soviet government. G. Dantzig invents the simplex method in 1947 for solving LP problems, working for U.S. Air Force, solving planning problems. T.C. Koopmans applies LP to classical economics same year after meeting with Dantzig.

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Nobel prize 1975 The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1975 was awarded jointly to Leonid Vitaliyevich Kantorovich and Tjalling C. Koopmans “for their contributions to the theory of optimum allocation of resources“. Many were surprised that Dantzig was omitted. Koopmans proposed to Kantorovich to decline the prize, but was pursuaded not to do this.

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Abstract example

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View of production manager

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LP formulation

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LP formulation – matrix form Maximize Subject to

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View of liquidator

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LP formulation

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LP formulation – matrix form Minimize Subject to Next week: The two LP formulations are intimately related (duals).

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The LP problem

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LP terminology An assignment to the decision variables is called a solution. A solution that satisfies all linear inequalities and linear equations is called feasible. A feasible solution that attains the desired minimum or maximum is called optimal.

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Linear Programs, Geometric View

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The set of feasible solutions F is a convex Polyhedron. F

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F

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F

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Linear Programs, Geometric view

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Beware of the intuition…

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Linear Programs in Standard Form

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Exceptions: If no feasible solution exist, report “Infeasible”. If arbitrarily good feasible solutions exist, report “Unbounded”.

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Transforming general Linear programs to standard form.

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The Simplex algorithm (geometric sketch)

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Whiteboard intermezzo

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Equivalent system

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Dictionaries

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Initialization: First dictionary

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Improvement: Pivoting Find entering variable: Pick variable with positive coefficient in objective function. Find leaving variable: Pick variable from basis to preserve nonnegativity of basic variables. Rewrite equation of leaving variable in terms of entering variable. Substitute for entering variable in dictionary.

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Special case If entering variable has nonnegative coefficient in all equations, report Unbounded.

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One phase Simplex method

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Analysis: Partial correctness

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1 Bob and Sue solved this by hand: Maximize x 1 + 2 x 2 subject to 1 x 1 + 1 x 2 ≤ 10 -2 x 1 + 1 x 2 ≤ 4 x 1, x 2 ≥ 0 and their last dictionary was: X1.

1 Bob and Sue solved this by hand: Maximize x 1 + 2 x 2 subject to 1 x 1 + 1 x 2 ≤ 10 -2 x 1 + 1 x 2 ≤ 4 x 1, x 2 ≥ 0 and their last dictionary was: X1.

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