Chapter 4 Duality Theory q Linear Programming 2019

Chapter 4. Duality Theory q Linear Programming 2019 1

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4. 2 The dual problem Linear Programming 2019 7

PRIMAL minimize maximize DUAL constraints variables constraints q Table 4. 1 : Relation between primal and dual variables and constraints q Dual of a maximization problem can be obtained by converting it into an equivalent min problem, and then take its dual. Linear Programming 2019 8

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(Ex 4. 1) Dual Linear Programming 2019 10

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q Thm 4. 2 : If we use the following transformations, the corresponding duals are equivalent, i. e. , they are either both infeasible or they have the same optimal cost. (a) free variable difference of two nonnegative variables (b) inequality by using nonnegative slack var. (c) If LP in standard form (feasible) has redundant equality constraints, eliminate them. Linear Programming 2019 13

4. 3 The duality theorem q Linear Programming 2019 14

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1 Duals of equivalent problems are equivalent Equivalent 2 Duality for standard form problems Fig 4. 1 : Proof of the duality theorem for general LP Linear Programming 2019 16

(D) (P) Finite optimum Unbounded Infeasible Finite optimum Possible Impossible Unbounded Impossible Possible Infeasible Impossible Possible q Table 4. 2 : different possibilities for the primal and the dual Linear Programming 2019 17

q Note: (1) Later, we will show strong duality without using simplex method. (see example 4. 4 later) (2) Basic optimal primal, dual solutions provide “certificate of optimality” certificate of optimality is (short) information that can be used to check the optimality of a given solution in polynomial time. ( Two view points : 1. Finding an optimal solution. 2. Proving that a given solution is optimal. If there exists a polynomial time algorithm for an optimization problem, the algorithm and problem itself provide certificate of optimality. The converse is believed to be true although it is not proven yet. Until 1979 (ellipsoid method), LP was one of the rare exceptions. (3) The nine possibilities in Table 4. 2 can be used importantly in determining the status of primal or dual problem. Linear Programming 2019 18

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Geometric view of optimal dual solution q Linear Programming 2019 21

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q Figure 4. 3 Linear Programming 2019 23

q Figure 4. 4 degenerate basic feasible solution Linear Programming 2019 24

4. 4 Optimal dual var as marginal costs q Linear Programming 2019 25

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4. 5 Dual simplex method q Linear Programming 2019 27

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q Ex 4. 7 : Linear Programming 2019 29

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The geometry of the dual simplex method q Linear Programming 2019 33

4. 6 Farkas’ lemma and linear inequalities q Linear Programming 2019 34

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The duality theorem revisited q Linear Programming 2019 36

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q Figure 4. 2 strong duality theorem Linear Programming 2019 38