# CH6_Duality - Chapter 6 LP Duality Given an LP we define...

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1 Chapter 6. LP Duality Given an LP, we define another LP derived from the same data, but with different structure. The purpose is to provide an upper bound (estimate) on the optimal objective value of the given LP without solving it to optimality. Very important concept to understand the properties of the LP and the simplex method.

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2 ex) Lower bound : consider feasible solution (0, 0, 1, 0) z * 5 (3, 0, 2, 0) z * 22 Upper bound : consider inequality obtained by multiplying 0 to the 1 st , 1 to the 2 nd , and 1 to the 3rd constraints and add the l.h.s. and r.h.s. respectively …… (1) Since we multiplied nonnegative numbers, any vector that satisfies the three constraints (including feasible solutions to the LP) also satisfies (1). 3 5 3 2 55 8 3 5 1 3 . . 3 5 4 max 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 x x x x x x x x x x x x t s x x x x 0 , , , 4 3 2 1 x x x x 58 3 6 3 4 4 3 2 1 x x x x An Example
3 From earlier derivation, we have , which must be satisfied by vectors that satisfy the three constraints. Further, any feasible solution to the LP must satisfy since any feasible solution must satisfy nonnegativity on the variables. So 58 is an upper bound on the optimal value of the LP. Now, we may use nonnegative weights y i for each constraint. Objective function of the LP is 58 3 6 3 4 4 3 2 1 x x x x ) 58 ( 3 6 3 4 3 5 4 4 3 2 1 4 3 2 1 x x x x x x x

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