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CS149-Notes-4-StdCanon - A First Attempt at an LP Algorithm...

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A First Attempt at an LP Algorithm CS 149 Staff September 27, 2006 Before we start developing an algorithm to solve Linear Optimization Problems, we must first define what it is we are trying to solve. We therefore define the Linear Optimization Problem to be a 2-tuple, ( P, z ), where P R n is the space of feasible solutions, and z : P R is the cost function. The slides define two different kinds of instances of the Linear Optimization Problem, Standard Form , and Canonical Form . If A R m n , b R m , c R n , an instance of the linear optimization problem in standard form has: P := { x R n | Ax = b and x 0 } and z ( x ) := c T x . An instance of the linear optimization problem in canonical form has: P := { x R n | Ax b and x 0 } and z ( x ) := c T x . We would like to be able to solve both of these kinds of problems, without having to write 2 different solvers. To do this, we must first define the relation between the 2 forms. Definition We call 2 optimization instances ( P 1 , z 1 ), ( P 2 , z 2 ) equivalent if and only if there exist polynomial-time computable functions f : P 1 P 2 and g : P 2 P 1 such that: x 1 P 1 , z 2 ( f ( x 1 )) z 1 ( x 1 ) x 2 P 2 , z 1 ( g ( x 2 )) z 2 ( x 2 ) We will need a lemma to make our definition useful: Lemma If ( P 1 , z 1 ) and ( P 2 , z 2 ) are equivalent and x 1 P 1 is optimal for ( P 1 , z 1 ), f ( x 1 ) P 2 is optimal for ( P 2 , z 2 ) and z 1 ( x 1 ) = z 2 ( f ( x 2 )). Proof : Let x 1 P 1 be optimal for ( P 1 , z 1 ), and let y 2 P 2 . z 1 ( x 1 ) z 1 ( g ( y 2 )) Because x 1 is optimal z 2 ( f ( x 1 )) z 1 ( x 1 ) z 1 ( g ( y 2 )) Equivalence z 2 ( f ( x 1 )) z 1 ( x 1 ) z 1 ( g ( y 2 )) z 2 ( y 2 ) Equivalence z 2 ( f ( x 1 )) z 2 ( y 2 ) So we have that f ( x 1 ) has lower cost than any y 2 P 2 , so it is optimal. Fur- ther, since the inequality is true for any y 2 , we can set y 2 = f ( x 1 ), and get: 1
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z 2 ( f ( x 1 )) z 1 ( x 1 ) z 2 ( f ( x 1 )), so z 1 ( x 1 ) = z 2 ( f ( x 2 )).
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