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20 CHAPTER 4. PROBLEM FORMULATION AND SOLUTIONS
• l x ∈ R n is the lower limit on the activity for the variables.
• u x ∈ R n is the upper limit on the activity for the variables.
A primal solution (x) is (primal) feasible if it satisfies all constraints in (4.1). If (4.1) has at least one
primal feasible solution, then (4.1) is said to be (primal) feasible.
In case (4.1) does not have a feasible solution, the problem is said to be (primal) infeasible .
4.1.1 Duality for linear optimization
Corresponding to the primal problem (4.1), there is a dual problem
maximize (l c ) T s c l − (u c ) T s c u + (l x ) T s x l − (u x ) T s x u + c f
subject to A T y + s x l − s x u = c,
− y + s c l − s c u = 0,
s c l , s c u, s x l , s x u ≥ 0.
If a bound in the primal problem is plus or minus infinity, the corresponding dual variable is fixed at
0, and we use the convention that the product of the bound value and the corresponding dual variable
is 0. E.g.
(4.2)
l x j = −∞ ⇒ (s x l ) j = 0 and l x j · (s x l ) j = 0.
This is equivalent to removing variable (s x l ) j from the dual problem.
A solution
(y, s c l , s c u, s x l , s x u)
to the dual problem is feasible if it satisfies all the constraints in (4.2). If (4.2) has at least one feasible
solution, then (4.2) is (dual) feasible, otherwise the problem is (dual) infeasible.
4.1.1.1 A primal-dual feasible solution
A solution
(x, y, s c l , s c u, s x l , s x u)
is denoted a primal-dual feasible solution, if (x) is a solution to the primal problem (4.1) and (y, s c l , sc u, s x l , sx u)
is a solution to the corresponding dual problem (4.2).
4.1.1.2 The duality gap
Let
(x ∗ , y ∗ , (s c l ) ∗ , (s c u) ∗ , (s x l ) ∗ , (s x u) ∗ )