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# L09-LP - Introduction to Linear and Integer Programming...

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1 Introduction to Linear and Integer Programming Lecture 9: Feb 14

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2 Mathematical Programming Input: An objective function f: R n -> R A set of constraint functions: g i : R n -> R A set of constraint values: b i Goal: Find x in R n which: 1. maximizes f(x) 2. satisfies g i (x) <= b i
3 Linear Programming Input: A linear objective function f: R n -> R A set of linear constraint functions: g i : R n -> R A set of constraint values: b i Goal: Find x in R n which: 1. maximizes f(x) 2. satisfies g i (x) <= b i Integer linear program requires the solution to be in Z n

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4 Perfect Matching (degree constraints) Every solution is a perfect matching!
5 Maximum Satisfiability Goal: Find a truth assignment to satisfy all clauses NP-complete!

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6 Different Forms canonical form standard form The general form (with equalities, unconstrained variables) can be reduced to these forms.
7 Linear Programming Relaxation Replace By By Surprisingly, this works for many problems!

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Geometric Interpretation Linear inequalities as hyperplanes Goal: Optimize over integers ! Objective function is
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L09-LP - Introduction to Linear and Integer Programming...

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