A subgradient-based branch-and-bound algorithm for the ...

2 Computation of a lower bound
Dualizing the demand constraints (2) with multipliers λi, i ∈ I, yields the Lagrangean
subproblem
ZD(λ) =
λi + min (cij − λi)xij +
x,y
i∈I
i∈I j∈J
fjyj
j∈J
s.t.: (3), (4), (5), (6), (7).
Let ji ∈ arg min{cij : j ∈ J}. It is straightforward to show that optimal Lagrangean
multipliers can be found in the interval [λ min , λ max ], where
λ min
i
= min
cij : j ∈ J \ {ji}
and λ max
i = max
cij : j ∈ J
. (9)
Moreover, it is well-known that (8) reduces to the binary knapsack problem
where
λ0 = min
y
j∈J
vj = max
x
i∈I
(fj − vj)yj :
(λi − cij)xij :
i∈I
j∈J
(8)
sjyj ≥ d(I) , yj ∈ {0, 1} ∀ j ∈ J , (10)
dixij ≤ sj , 0 ≤ xij ≤ 1 ∀ i ∈ I , j ∈ J .
The Lagrangean function ZD(λ) = λ0 +
i∈I λi and thus also the Lagrangean dual
bound
ZD = max
ZD(λ) : λ ∈ [λ min , λ max ]
can therefore be computed in pseudo-polynomial time (Cornuejols et al., 1991).
A broad range of different methods is available for exactly solving the Lagrangean
dual (11) and obtaining a corresponding primal solution. A stabilized column generation
method for solving the corresponding primal linear master problem is applied
in (Klose and Drexl, 2005) and extended to a branch-and-price algorithm for the
CFLP in (Klose and Görtz, 2007). Alternatively, regularized decomposition or bundle
methods (Lemaréchal, 1989; Ruszczyński, 1995; Ruszczyński and ´ Swi¸etanowski,
1997) may be used for solving (11). The main principle of a bundle method is to keep
an inner polyhedral approximation to the ɛ-subdifferential ∂ɛZD(λ) of the piecewise
linear and concave function ZD(λ). A trial step is then taken into the best direction
found in this set (which requires to solve a quadratic master program). If this gives
a sufficient increase in ZD(λ), a “serious step” is taken to the next iterate; otherwise
a “null step” is performed and the current approximation of ∂ɛZD(λ) improved by
adding further subgradients. The Volume Algorithm of Barahona and Anbil (2000)
can be seen as a heuristic version of a bundle method. The procedure keeps an estimate
of a primal solution to the Lagrangean dual. This estimated solution is an
exponentially weighted average of the solutions obtained to the Lagrangean subproblem
in the course of the algorithm. Furthermore, this estimated primal solution
is also used for determining subgradient-like search directions. If a move into such a
direction gives an ascent, a serious step is taken and the new dual iterate accepted;
3
(11)