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

subject to
(1)
(2)
Want to define dual problem for this LP so that
dual objective value gives upper bound on the
primal optimal value.
2

• Take linear combination of constraints with
  multiplier yi for constraint i.
• yi ? 0 for i ? I , yi unrestricted in sign for
  i ? E ? doesnt change the direction of the
  inequality.

holds for x satisfying (1) and yi ? 0, i ? I,
yi unrestricted i ? E.
Want this as upper bound
Compare this with primal obj. coeff. cj
3

• Make

We want strong bound, hence solve
s. t.
(Dual problem)
4

• Primal-Dual Correspondence

Primal Dual
maximize minimize
xj ? 0 j th constraint ?
free xj j th constraint
i th constraint ? yi ? 0
i th constraint free yi

• Weak duality and strong duality relationship hold
  for general primal, dual pair.
• We may convert the general primal problem to
  standard form, take dual, then simplify to get
  the same dual problem. (Another way to get dual
  for general LP)

5

• If the LP is given in min form, we may convert
  it to max form and take its dual. Then
  converting the dual to max form gives the dual.
  Or we may find primal form, regarding the min
  form as dual problem and find dual of dual.

• Ex)

s. t.
s. t.
Dual problem is
s. t.
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 y from yB cB and show
  that it is dual feasible and by cx.

7

• Consider the special case of the general LP
• max cx
• s.t. Ax b
• x ? 0,
• which is used as standard LP problem by some
  people (maybe in minimization form).
• Its dual is
• min yb
• s.t. yA ? c
• y unrestricted
• Suppose we solve the above primal problem using
  simplex method and find optimal basis B. Then
  the updated tableau is expressed the same way as
  we have seen before.

8

• Here we dont have slack variables appearing.
• Since y is obtained from yB cB , the updated
  objective coefficients in the z-row can be
  regarded as cj yAj for all basic and nonbasic
  variables.
• At optimality, we have (cj yAj ) ? 0, or yAj
  ? cj , hence y is dual feasible vector. The dual
  objective function value is yb, which is the
  same value as the current primal objective
  function value cBB-1b cBxB . Hence providing
  the proof that the current solution x is optimal
  to primal and y is optimal to dual respectively.

9
Unsolvable Systems of Linear Inequalities and
Equations

• Consider the following pair of constraints
• ?j1n aijxj ? bi ( i? I )
• ?j1n aijxj bi ( i? E )
• yi ? 0 whenever i ? I
• ?i1m aijyi 0 for all j 1, 2, , n
• ?i1m biyi lt 0
• 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.
• (called theorem of the alternatives, many other
  versions, very important tool and has many
  applications.)

(9.13)
(9.16)
10

• Pf) ?) Suppose (9.16) has a feasible solution y.
  We multiply yi on both sides of constraints in
  (9.13) and add the lhs and rhs, respectively
  (with yi ? 0 for i ? I).
• Then, we obtain ?j1n (?i1m aijyi ) xj ?
  ?i1m biyi
• Hence, ?j1n 0?xj ? ?i1m biyi lt 0, which must
  be satisfied by any feasible x to (9.13). Since
  it is impossible to satisfy ?j1n 0?xj lt 0 by any
  x, (9.13) is infeasible
• ?) Consider the linear program
• max ?i1m (-xni )
• s.t. ?j1n aijxj wixni ? bi ( i? I )
• ?j1n aijxj wixni bi ( i? E )
• xnI ? 0 ( i 1, 2, , m)
• with wi 1 if bi ? 0 and wi -1 if bi lt 0.
• (9.18) has a feasible solution (with x 0 for
  original vars). Also the upper bound on optimal
  value is 0, hence it has finite optimal.

(9.18)
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 ?i1m biyi
• s.t. ?i1m aijyi 0 ( j 1, 2, , n)
• wiyi ? -1 ( i 1, 2, , m)
• yi ? 0 ( i ? I )
• Then the optimal dual solution y1, y2, , ym
  satisfies (9.16) ?