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3 IP and LP_1

3 IP and LP_1 - IEE 598 3(I.1 II.4.2 IP and LP Muhong Zhang...

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IEE 598 - 3. (I.1, II.4.2) IP and LP Muhong Zhang DEPARTMENT OF INDUSTRIAL ENGINEERING Feb. 02, 2010

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Relaxation Definition: ( R ) : z * R = max { c ( x ) : x T } is called a relaxation of ( P ) : z * P = max { f ( x ) : x S } if 1 S T , 2 c ( x ) f ( x ) , x S . Proposition: z * P z * R . Proof. Let x * P be the optimal solution of ( P ) . Then x * P S . Since S T , x * P T . So x * P is feasible for ( R ) . By the definition of optimality, z * P = f ( x * ) z * R . Proposition: Let x * R be an optimal solution for ( R ) . If x * R S and f ( x * R ) = c ( x * R ) then x * R is optimal for ( P ) . Proof. For any feasible solution of P , x S , since S T , x T . Since x * R is optimal for ( R ) , c ( x ) c ( x * R ) . Because x * R is also feasible for ( P ) , by the definition of optimality, x * R is optimal for ( P ) . Zhang IEE 376 Introduction to OR 2 / 9
LP Relaxation and IP Proposition: (i) If an LP relaxation is infeasible, the original problem IP is infeasible. (ii) Let x * LP be an optimal solution of the LP relaxation. If x * LP is feasible for IP, then x * LP is an optimal solution of IP.

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3 IP and LP_1 - IEE 598 3(I.1 II.4.2 IP and LP Muhong Zhang...

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