Linear%20Programming%20Supplements%20(Optional)

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Linear ProgrammingSupplements(Optional)
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Standard Form LP (a.k.a. First Primal Form)

• Strictly

All xj's are non-negative
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Transforming Problems into Standard Form

• Min cTx ? Max -cTx
• Max (cTx constant) ? Max cTx
• Replace a constraint like ?aijxj bi by -?aijxj
  -bi
• Replace a constraint like ?aijxj bi by ?aijxj
  bi and -?aijxj -bi
• If xj is allowed to take on negative value,
  replace xi by the difference of two nonnegative
  variables, says xi ui vi, where ui 0 and
  vi 0.

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Example of transforming a problem into Standard
Form

• Replace x1 by u1 v1

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Dual Problem

• Every primal LP problem in the form
• Maximize cTx
• subject to Ax b, x 0
• has a corresponding dual problem in the form
• Minimize bTy
• 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.
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Dual Problem
Duality Theorem 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 gtgt 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)

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MATLAB LP solver linprog()

• Partial help manual generated by MATLAB
• XLINPROG(f,A,b) attempts to solve the linear
  programming problem
• min f'x subject to Ax lt b
• x
• XLINPROG(f,A,b,Aeq,beq) solves the problem above
  while additionally
• satisfying the equality constraints Aeqx
  beq.
• XLINPROG(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 lt X lt 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.
• XLINPROG(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.

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MATLAB example

• 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)
• Optimization terminated.
• xmin
• 4.8889
• 3.8889

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Integer LP Problem

• If the variables can only take integer values, we
  cannot take the integers closest to the solution
  of the corresponding LP problem as the solution.
• Integer Programming (IP) or Integer Linear
  Programming (ILP) problems are NP-hard problems.
• Some of the algorithm for solving IP problems
  include branch-and-bound, branch-and-cut.