1. OR-1 20151 Relation between (P) & (D)

2. OR-1 20152 optimal solution InfeasibleUnbounded Optimal solution OXX Infeasible X( O )O Unbounded XOX (D) (P)

3. OR-1 20153 Complementary Slackness Theorem

4. OR-1 20154  The conditions say that if one of the primal (dual) variable is positive, then the corresponding dual (primal) constraint must hold at equality in an optimal solution. (primal, dual variables must be feasible, respectively)  Similarly, if one of the primal (dual) constraint hold at strict inequality, then the corresponding dual (primal) variable should be zero.

5. OR-1 20155  ex) Negative of dual (structural) variables Negative of dual surplus variables (later)

6. OR-1 20156

7. 7

8. 8 Remarks  CS conditions are necessary and sufficient conditions for optimality. But the condition that the coefficients in the z-row are nonpositive in the simplex tableau is only a sufficient condition for optimality, but not necessary.  CS optimality conditions also hold for more general forms of primal, dual pair of problems if dual is defined appropriately. (more on this later in Chap. 9)  In the CS theorem, the solutions x * and y * need not be basic solutions (in equality form LP). Any primal and dual feasible solutions that satisfy the CS conditions are optimal. CS conditions can be used to design algorithms for LP or network problems.  Most powerful form of interior point method tries to find solutions that satisfy the CS conditions with some modifications. (Logic to derive the conditions is different though)

9. OR-1 20159 Interior point method and CS conditions CS conditions Interior Point Method

10. OR-1 201510  Characteristics of the Interior Point Method  Interior point method finds a solution to the system iteratively with   0. (path following method)  Solve system of nonlinear equations.  Newton’s method is used to find the solution.  Try to find solution that satisfies the equations and positivity of the points examined is always maintained.  Strict positivity is maintained and 0 is obtained as small positive number. Hence obtained solution is not a basic feasible solution (We do not know the basis).  To identify a basic feasible solution, a post-processing stage is needed.  Although we mentioned the path following method, there are other types of the interior point methods.

11. OR-1 201511

12. OR-1 201512

13. OR-1 201513

14. OR-1 201514 Economic significance of dual variables

15. OR-1 201515

16. OR-1 201516

17. OR-1 201517

18. OR-1 201518