Lecture 23 - Linear Programming

# Lecture 23 - Linear Programming - Lecture 21 DADSS...

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1 Lecture 21 Optimization: Introduction to Linear Programming DADSS

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2 Administrative Details Survey Day is Thursday Bring enough copies for 100 students Attendance is mandatory (equal to HW) In addition to surveys, I will be available all  throughout class to discuss your term project  progress and answer questions Questions from last class?
3 Optimization and Linear Programming “Mathematical Programming” is a class of methods for solving  problems that ask you to optimize:  to maximize, to minimize, to  find the best, the worst, the most, the least, the fastest, the  shortest, etc. Linear programming is the simplest form, suitable only for  problems with linear objective functions and constraints Two of the basic linear programming problems: The Product Mix Problem What quantities of products should be produced to maximize profit? Optimal allocation of scarce resources The Blending Problem What combination of ingredients minimizes cost?

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4 Some Optimization Basics Unconstrained: Unconstrained: In general, what are we doing? Set the derivative to zero and solve Why? ( 29 x x f x = max ( 29 ( 29 x x x f x - = 1 max ( 29 2 1 0 2 1 = = - = x x dx x df Easy: Still Easy:
5 Necessary and Sufficient Conditions Necessity Sufficiency Maximum Minimum ( 29 0 = dx x df ( 29 0 2 < dx x f d ( 29 0 = dx x df ( 29 0 2 dx x f d ( 29 0 = x f ( 29 0 < x f ( 29 0 x f ( 29 0 = x f

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6 Some Complications What about more complicated functions? Calculus still works, but is cumbersome to do by hand What about constraints? Again, calculus works (the Lagrangian) for small  problems, but requires automation for larger problems ( 29 ( 29 3 1 subject to , 1 max - = x x x x f x ( 29 ( 29 ( 29 ( 29 y x z x y y x z y x f z y x - - + - + - = 1 1 1 , , max , ,
7 More Complications Multiple Optima Discontinuous Functions Uncertainty Integer Restrictions x  f ( x )  f ( x )  f ( x )  f ( x ) x x x

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The Need for Algorithmic Methods How can we automate searching for optima? Linear programming (LP) is one such method LP is fast and easy, but it has some important limitations Limitations/Requirements Alternatives 1 Decisions are made under certainty Sometimes expectations are used, but  generally stochastic programming is  required 2 Objective function and constraints  must be linear NLP is tough! No guaranteed global  optimizer other than enumeration. GA,  GP, SA, heuristic search, tabu, adaptive  grid refinement, active sets. 3
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## This note was uploaded on 09/20/2010 for the course SDS 88223 taught by Professor Fischbeck during the Spring '10 term at Carnegie Mellon.

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Lecture 23 - Linear Programming - Lecture 21 DADSS...

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