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3 IP and LP

3 IP and LP - 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. 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. 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. Proof. Q: If LP relaxation is unbounded, what can you say for the IP? Zhang IEE 376 Introduction to OR 3 / 9

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Branch and bound algorithm is the most commonly-used algorithm for solving MIPs. Divide and conquer approach.
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3 IP and LP - IEE 598 3(I.1 II.4.2 IP and LP Muhong Zhang...

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