Lecture 9A

Lecture 9A - Linear Optimization (Linear Programming) The...

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EE103 Lec 9A (SEJ) 1 Linear Optimization ( Linear Programming ) The basis for nonlinear and combinatorial optimization EE103 Lec 9A (SEJ) 2 The Optimization Problem max ( ) g x xF , . ., ( , ,..., )' 12 1 : n x Ri e xx x x n n FR gF R ∈= max ( ) min ( ) gx =− 2 2 : , min || || ( ) TT x Ex Linear Least Squares Ax b m n A xb A A x A b =≥ −⇔ = ( ) () T A x b A x b
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EE103 Lec 9A (SEJ) 3 () 112 212 12 : ,,, 0 0 0 n n mn Ex Nonlinear Least Squares fxx x x x = = = # 0 fx ⇔= 2 1 ()() () 0 m T i i gx f x f x f x = == 22 2 1 m i n | | () | | m i n () () m i n n T i xx i x fx fx = The Optimization Problem EE103 Lec 9A (SEJ) 4 The Optimization Problem 21 ,, l l l Approximation Ax b mxn m n = ± 2 2 min n xR A xb 1 1 min min nn m ii i A a = ∈∈ −= 1, , min min max im A a = =− " Ex: Approximation
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EE103 Lec 9A (SEJ) 5 The Optimization Problem x y 1.0000 1.0000 2.0000 1.5000 3.0000 2.0000 4.0000 1.0000 5.0000 5.0000 6.0000 5.5000 Ex: Approximation, Con’t EE103 Lec 9A (SEJ) 6 When F = R n , we say the optimization problem is an “unconstrained” optimization problem. Otherwise, the problem is a “constrained” optimization problem. 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 problems. Indeed, the class of linear programs can be thought of as being on the “boundary” between continuous and combinatorial optimization problems. The Optimization Problem
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EE103 Lec 9A (SEJ) 7 The Optimization Problem , 1 n ax b ij j i j = 12 ,, . . .
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This note was uploaded on 03/30/2009 for the course EE 103 taught by Professor Vandenberghe,lieven during the Winter '08 term at UCLA.

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Lecture 9A - Linear Optimization (Linear Programming) The...

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