Linear Programming Supplements (Optional). Standard Form LP (a.k.a. First Primal Form). Strictly ≤. All x j 's are non-negative. Transforming Problems into Standard Form. Min c T x Max - c T x Max ( c T x + constant) Max c T x
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All xj's are non-negative
Replace x1 by u1– v1
Every primal LP problem in the form
subject to Ax≤b, x ≥ 0
has a corresponding dual problem in the form
subject to ATy ≥ c, y ≥ 0
Theorem on Primal and Dual Problems
If x satisfies the constraints of the primal problem and y satisfies the constraints of its dual, then cTx≤bTy.
Consequently, if cTx=bTy, then x and y are solutions of the primal problem and the dual problem respectively.
If the original problem has a solution x*, then the dual problem has a solution y*; furthermore, cTx*=bTy*.
If the original primal problem contains much more constraints than variables (i.e., m >> n), then solving the dual problem may be more efficient. (Less constraints implies less corner points to check)
The dual problem also offers a different interpretation of the problem (Maximize profit == Minimize cost)
Partial help manual generated by MATLAB:
X=LINPROG(f,A,b) attempts to solve the linear programming problem:
min f'*x subject to: A*x <= b
X=LINPROG(f,A,b,Aeq,beq) solves the problem above while additionally
satisfying the equality constraints Aeq*x = beq.
X=LINPROG(f,A,b,Aeq,beq,LB,UB) defines a set of lower and upper
bounds on the design variables, X, so that the solution is in
the range LB <= X <= UB. Use empty matrices for LB and UB
if no bounds exist. Set LB(i) = -Inf if X(i) is unbounded below;
set UB(i) = Inf if X(i) is unbounded above.
X=LINPROG(f,A,b,Aeq,beq,LB,UB,X0) sets the starting point to X0. This option is only available with the active-set algorithm. The default interior point algorithm will ignore any non-empty starting point.
% Turn into minimization problem
c = [ -150 -175 ]';
A = [ 7 11; 10 8; 1 0; 0 1 ];
b = [77 80 9 6]';
LB = [0 0]';
% There is no equality constraints
xmin = linprog(c, A, b, , , LB)