Chap 9. General LP problems: Duality and Infeasibility

1. Chap 9. General LP problems: Duality and Infeasibility • Extend the duality theory to more general form of LP Consider 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)

2. Take linear combination of constraints with multiplier for constraint . 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 primal objective coefficient Want this as upper bound

3. Make , if , if  , satisfying (2)  , satisfying (1), (2) & satisfying the given conditions.  Gives weak duality relationship. We want strong bound, hence solve min s.t., , (dual problem) (9.9) , free,

4. Primal-Dual Correspondence • The dual of the dual is the primal. • Weak duality and strong duality relationship hold for general primal, dual pair.

5. 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).

6. Thm 9.1 (The Duality Theorem): If a linear programming problem has an optimal 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 . 

7. Consider a special case of the general LP max s.t. , which is used as standard LP problem by some people (maybe in minimization 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.

8. 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 optimal to primal and is optimal to dual respectively.

9. Unsolvable Systems of Linear Inequalities and Equations • 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, exactly 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.)

10. 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 impossible 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.

11. (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 duality 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). 

12. 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.

13. (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 . 

14. 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.

15. 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’sThm) 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 .

16. Thm 9.4 : Every unsolvable system of linear inequalities in variables contains 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 solution, 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.