P10311F07187

# P10311F07187 - EE 103 Lecture Notes Fall 2007(SEJ Section...

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EE 103 Lecture Notes, Fall 2007 (SEJ) Section 11 SECTION 11: INTRODUCTION TO LINEAR OPTIMIZATION. ................................................... 160 TANGLETOWN (AN INTEGER LINEAR PROGRAM) ............................................................................... 161 DESK PROBLEM: ................................................................................................................................... 163 AN INTRODUCTION TO THE GEOMETRY OF LINEAR PROGRAMMING: ............................................... 171 ORIGINAL DATA AND A BASIS MATRIX ARE ALL THAT ARE NEEDED: ................................................ 175 MATLAB (AS A CALCULATOR) IMPLEMENTATION FOR THE DESK PROBLEM: ................................... 176 Column Generation Implementation of the Simplex Algorithm: ................................................... 179 Desk Problem Revisited ................................................................................................................... 180 Implementation of Column Generation via Matlab: ...................................................................... 183 UNBOUNDEDNESS AND CONVERGENCE: ............................................................................................... 187

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EE 103 Lecture Notes, Fall 2007 (SEJ) Section 11 160 SECTION 11: INTRODUCTION TO LINEAR OPTIMIZATION (INTRODUCTION TO THE SIMPLEX ALGORITHM) The purpose of this section is to present the simplex algorithm for linear optimization problems, and to relate the latter to linear algebraic concepts that have been developed in this class, slides, and lecture notes. The purpose is not so that you’ll be come experts in linear programming (linear optimization), but rather to show the connection of simple linear algebraic concepts to one of the most often used scientific and mathematical algorithms. Optimization, broadly speaking, is the art and science of selecting the best action (or decision) to select from a set of actions. The word “best” is the key that leads to the notion of mathematical optimization (or mathematical programming). In particular, mathematical optimization assumes that there is an “objective function” that measures the value of the various decisions or actions that can be taken and that it is known that the set of possible (feasible) actions are defined to be in some set, say F. Usually, we also assume that the set of feasible actions (feasible solutions) can be described by some form of mathematical relationships. Therefore, we define the optimization problem as max ( ) xF f x where f is the objective function and F is assumed to be a subset of n R . Note that “maximization” and “minimization” are essentially the same since max ( ) min ( ) f xf x = −− . That is, there is not a substantial difference between maximizing and minimizing; any statements made about one problem will apply to the other if one correctly takes into account the minus signs. When F is a finite set, written || F < ∞ , we usually say the optimization problem is a “combinatorial” optimization problem. Otherwise, we usually say the optimization problem is a “continuous” optimization problem (this terminology is not standard since the word “continuous” is being somewhat abused in this latter statement). As we’ll see, while linear programming problems are continuous optimization problems they can also be thought of as combinatorial optimization
EE 103 Lecture Notes, Fall 2007 (SEJ) Section 11 161 problems. Indeed, the class of linear programs can be thought of as being on the “boundary” between continuous and combinatorial optimization problems.

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P10311F07187 - EE 103 Lecture Notes Fall 2007(SEJ Section...

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