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Chap 9. General LP problems: Duality and Infeasi-bility

Extend the duality theory to more general form of LPConsider the following form of LP

maximize subject to

(9.1)

Want to define dual problem for this LP so that dual objective value gives an upper bound on the primal optimal value.

(1)

(2)

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Take linear combination of constraints with multiplier for con-straint . for , unrestricted in sign for doesn’t change the direction of the inequality.

,, unrestricted,

Adding up on both sides

holds for satisfying (1) and , unrestricted . Now

Compare this with primalobjective coefficient

Want this as upper bound

Make , if

, if

, satisfying (2) , satisfying (1), (2) & satisfying the given condi-tions. Gives weak duality relationship.

We want strong bound, hence solvemin s.t. , , (dual problem) (9.9)

, free,

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Primal-Dual Correspondence

Primal Dual

maximize minimize

th constraint

free th constraint

th constraint

th constraint free

The dual of the dual is the primal. Weak duality and strong duality relationship hold for general primal, dual

pair.

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Ex) max max s.t. s.t.

,

Dual problem is min s.t.

If the LP is given in minimization form, present the problem as (9.9) and then write (9.1).

Thm 9.1 (The Duality Theorem): If a linear programming problem has an op-timal solution, then its dual has an optimal solution and the optimal values of the two problems coincide. Pf) proof parallels the idea for standard LP. At the termination of the simplex method, we identify dual vector from and show that it is dual feasible and .

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Consider a special case of the general LPmax s.t. ,

which is used as standard LP problem by some people (maybe in minimiza-tion form). Also it is the augmented form we used when we developed the simplex method in Chapter 2, 3.Its dual is

min s.t. unrestricted

Suppose we solve the above primal problem using simplex method and find optimal basis . Then the updated tableau is expressed the same way as we have seen before.

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Here we don’t have slack variables appearing.

Since is obtained from , the updated objective coefficients in the row can be regarded as for all basic and nonbasic variables.At optimality, we have , or , hence is dual feasible vector. The dual objective function value is , which is the same value as the current primal objective function value . Hence providing the proof that the current solution is opti-mal to primal and is optimal to dual respectively.

Unsolvable Systems of Linear Inequalities and Equa-tions

Consider the following pair of constraints

(9.13)

, whenever , for all (9.16)

Then (9.13) is infeasible if and only if (9.16) is feasible. In other words, ex-actly one of (9.13) and (9.16) has a feasible solution (Theorem 9.2).(called theorem of the alternative, many other versions, very important tool and has many applications.)

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Pf in the text) ) Suppose (9.16) has a feasible solution . We multiply on both sides of constraints in (9.13) ( for ) and add the lhs and rhs, respectively.Then, we obtain . Hence, , which must be satisfied by any feasible to (9.13). Since it is impos-sible to satisfy by any , (9.13) is infeasible.) Consider the linear program

max s.t. (9.18)

with if and if .(9.18) has a feasible solution (with for original variables). Also the upper bound on the optimal value is 0, hence it has finite optimal.

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(continued)The optimal value of (9.18) is 0 if and only if (9.13) has a feasible solution.If (9.13) is unsolvable, then the optimal value of (9.18) is negative. Then du-ality theorem guarantees that the dual of (9.18) has optimal value which is negative.

min s.t.

( i = 1, 2, … , m)

Then the optimal dual solution satisfies (9.16).

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Alternative proof)Consider the following primal-dual pair

(P) max (coefficients of are all 0)

(D) min for all whenever

) Suppose (9.16) has a feasible solution with . Then is feasible to (D) for all . Then as , hence (D) is unbounded. Therefore (P) is infeasible, i.e. (9.13) is infeasible, from the possible primal-dual statuses.

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(pf continued) ) Suppose (9.13) is infeasible, i.e. (P) does not have a feasible solution. Then (D) is either infeasible or unbounded. But is a feasible solution to (D), hence the only remaining possibility is (D) unbounded. Then (9.16) has a feasible solution with .

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Thm 9.3 : If a system of linear equations has a nonnegative solution, then it has a solution with at most variables positive. Pf) If the system

(9.19)

has a solution, then, by Theorem 8.3, there is some set of subscripts such that (i) system (9.19) and

(9.20)

have precisely the same set of solutions and(ii) system (9.20) has a basic feasible solution . Now at most variables are positive at a b.f.s.

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Note that if is considered as a vector in , (9.19) means can be expressed as a nonnegative linear combination of columns of coefficient matrix ( is in a cone generated by the columns of ).

(Caratheodory’s Thm) Theorem 9.3 can be used to the case

(9.19) ,

which means can be expressed as a convex combination of column vectors of . The Theorem can now be said that we need at most variables positive. Caratheodory’s theorem says that, if a vector is in the convex hull of a set , then can be expressed as a convex combination of at most points of .

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Thm 9.4 : Every unsolvable system of linear inequalities in variables con-tains an unsolvable subsystem of at most inequalities.Pf) If unsolvable, then, by Theorem 9.2, there exists which satisfies , , .Denote by (e.g. , note that if is a feasible solution to above, then is also feasible, so the actual value of can be chosen as any negative value.), and consider the system consisting of equations. Since is a nonnegative solu-tion, Theorem 9.3 guarantees the existence of nonnegative solution with at most positive components . The desired subsystem consists of those inequalities for which ; since for all but .This subsystem is unsolvable.

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