MS&E 211 Midterm Review 2007-2008. LP Models. Know how to recognize an LP in verbose or matrix form; standard or otherwise; max or min Know how to set up an LP Understand how to make various conversions between different types of LPs (max/min; < , > , =; > , free; |abs. value|; etc.).
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An LP problem falls in one of three cases:
When the problem is feasible and bounded,
1. The number of variables in the dual problem is equal to the number of constraints in the original (primal) problem. The number of constraints in the dual problem is equal to the number of variables in the original problem.
2. Coefficient of the objective function in the dual problem come from the right-hand side of the original problem.
3. If the original problem is amax model, the dual is aminmodel; if the original problem is amin model, the dual problem is themaxproblem.
4.The coefficient of the first constraint function for the dual problem are the coefficients of the first variable in the constraints for the original problem, and the similarly for other constraints.
5. The right-hand sides of the dual constraints come from the objective function coefficients in the original problem.
6. The sense of the ith constraint in the dual is = if and only if the ith variable in the original problem is unrestricted in sign.
7. If the original problem is man (min ) model, then after applying Rule 6, assign to the remaining constraints in the dual a sense the same as (opposite to ) the corresponding variables in the original problem.
8. The ith variable in the dual is unrestricted in sigh if and only if the ith constraint in the original problem is an equality.
9. If the original problem is max (min) model, then after applying Rule 8, assign to the remaining variables in the dual a sense opposite to (the same as) the corresponding constraints in the original problem.
Max model Min modelxj 0 jth constraint xj≤ 0 jth constraint ≤ xj free jth constraint =
ith const ≤ yi 0 ith const yi ≤ 0 ith const = yi free
Standard Primal Production Form:
Dual Form (rotate primal 90o clockwise):
Standard Primal Production Form:
Standard (Equality) Primal Form:
1. The dual of the dual problem is again the primal problem.
2. Either of the two problems has an optimal solution if and only if the other does; if one problem is feasible but unbounded, then the other is infeasible; if one is infeasible, then the other is either infeasible or feasible/unbounded.
3. Weak Duality Theorem: The objective function value of the primal (dual) to be minimized evaluated at any primal (dual) feasible solution cannot be less than the dual (primal) objective function value evaluated at a dual (primal) feasible solution.
cTx >= bTy (in the standard equality form)
4. Strong Duality Theorem: When there is an optimal solution, the optimal objective value of the primal is the same as the optimal objective value of the dual.
cTx* = bTy*
5. Complementary Slackness Theorem:Consider an inequality constraint (nonnegative variable) in any LP problem. If that constraint is inactive (nonnegative variable is positive) for an optimal solution to the problem, the corresponding dual variable (inequality constraint) will be zero (active) in any optimal solution for the dual of that problem.
y*j (b-Ax*)j = 0, j=1,…,n;
x*i (c-ATy*)i = 0, i=1,…,m;
No share limit, and short is allowed:
P: the (normalized) state prices