Adjustable robustness was first introduced in [13] and is, like recoverable robustness, a robust version of the two-stage approach in stochastic opti- mization. The idea is to distinguish between here and now variables x which have to be calculated in advance and wait and see variables y which can be defined after the scenario is known. The x variables are also called first stage solutions and y are the second stage solutions. The objective is to calculate a solution x in the first stage such that for every possible scenario ξ there exists a solution y such that (x, y) is feasible and such that (x, y) minimizes the worst-case objective value over all scenarios. After the scenario is known the best of the feasible solutions y is chosen. For a deterministic problem of the form
min fξ(x, y)
s.t. (x, y) ∈ Xξ× Yξ
with Xξ ⊂ Rm and Yξ ⊂ Rn the adjustable robust counterpart is the problem
min
x∈ ¯X
max
ξ∈U y:(x,y)∈Xminξ×Yξ
where the feasible set ¯X is defined by ¯ X := ( x ∈ \ ξ∈U Xξ | ∀ξ ∈ U ∃y ∈ Yξ : (x, y) ∈ Xξ× Yξ ) .
In other words ¯X contains all first stage solutions x such that x is feasible for all scenarios ξ and such that for each scenario ξ there exists a second stage solution y such that (x, y) is feasible for the scenario ξ.
Example 3.12. Consider the network design problem where for a given
graph G = (V, E) each edge e ∈ E has a maximal capacity ue with total
price ce. Furthermore dij are the uncertain demands of units which have
to be sent from node i to j. Each unit which is sent from i to j yields a profit of pij. In the first stage a fraction xe of ue can be bought by paying
the price xece. This fraction xeue can then be used in the second stage to
send fe units over the edge where fe ≤ xeue must hold. In the second stage
when the demands dij are known we can calculate the flow which satisfies all
demands and all capacities xeueand which has the maximal profit. Therefore
the problem can be formulated as an adjustable robust problem with first stage variables xe and second stage variables fe [17].
The adjustable robust problem is hard to solve in general, that is why there are several variations and special cases of the problem which are considered in the literature. One approach, which was presented in [13], is called affinely adjustable robustness. In this approach the authors assume the wait and see variables y to be given by an affine function of the uncertainty parameters, i.e. y = a + Aξ with A ∈ Rn×q and a ∈ Rn. We consider the problem
min c>x
s.t. Tξx + W y ≤ hξ,
(3.4) where ξ ∈ U ⊂ Rq, T
ξ ∈ Rl×m, W ∈ Rl×n and hξ ∈ Rl. Here the uncertainty
parameters only occur in Tξ and hξ and we assume that Tξ and hξ are affine
linear in ξ. The authors prove the following theorem:
Theorem 3.13 ([13]). The adjustable robust counterpart of Problem (3.4), where the second stage variables y are affine functions of the uncertain pa- rameters, is equivalent to the strictly robust problem
min c>x s.t. Bξx + bξ ≥ 0 ∀ξ ∈ U for a matrix Bξ ∈ Rl 0×m and bξ ∈ Rl 0
and it has the same complexity as the strictly robust problem for any U .
For combinatorial problems of the form min c>x + d>y
s.t. (x, y) ∈ X × Y
where X ⊆ {0, 1}m and Y ⊆ {0, 1}n without uncertainty in the constraints
and (c, d) ∈ U ⊆ Rm+n, the adjustable robust counterpart reduces to the
min-max-min problem min
x∈X(c,d)∈Umax miny∈Y c >
x + d>y.
Note that Problem (M3) has the same min-max-min structure and will be
analyzed later. In fact (M3) is a special case of the min-max-min problem
discussed in the following section.
3.5.1
K-Adaptability
As mentioned before the adjustable robust counterpart (RCR) is hard to solve in general. One approach to approximate the latter problem is called K-adaptability and was first introduced in [16] for general linear problems with uncertainty in the objective function and in the constraints. The idea of the approach is to choose K second stage solutions y1, . . . , yK in the first
stage and then choose the best of them after the scenario is revealed. The authors analyze the gaps between the adjustable robust counterpart, the K- adaptability problem and the strictly robust problem in [16] and give neces- sary conditions to obtain a certain improvement. They also present a bilinear optimization formulation to solve the 2-adaptability problem and prove that the latter problem is NP -hard. Later the approach of K-adaptability was analyzed for binary problems in [41]. We consider problems of the form
min ξ>Cx + ξ>Qy s.t. T x + W y ≤ h (x, y) ∈ X × Y
where ξ ∈ U = {ξ ∈ Rq | Aξ ≤ b} are the uncertain parameters, X ⊆ Rm is
a bounded polyhedral set, Y ⊆ {0, 1}n, C ∈ Rq×m, Q ∈ Rq×n, T ∈ Rl×m,
W ∈ Rl×n and h ∈ Rl. The K-adaptability robust counterpart is then the
problem min max ξ∈U k=1,...,Kmin ξ > Cx + ξ>Qy(k) s.t. T x + W y(k)≤ h k = 1, . . . , K x ∈ X, y(k)∈ Y k = 1, . . . , K (K-AR)
for a given positive integer K. Besides other results the authors prove the following theorem.
Theorem 3.14 ([41]). For all
K ≥ min {n, rank Q} + 1,
Problem (K-AR) has the same optimal value as the adjustable robust coun- terpart
min
x∈X maxξ∈U y∈Y :T x+W y≤hmin ξ >
Cx + ξ>Qy. (3.5)
Proof. Assume n ≤ rank Q and let K = |Y | < ∞. Then Problem (3.5) and (K-AR) are equivalent. The objective function of (K-AR) can be written as
max ξ∈U λ∈RminK +:e>λ=1 ξ>Cx + K X k=1 λkξ>Qy(k) = max ξ∈U λ∈RminK +:e>λ=1 ξ>Cx + ξ>Q K X k=1 λky(k). Since y := K X k=1 λky(k)∈ conv y(1), . . . , y(K)
and by the Theorem of Caratheodory 2.3 it follows that for any point y it suffices to choose n + 1 solutions in y(1), . . . , y(K)
to describe y as a convex combination. Since all y(i) are chosen in the first stage it suffices to
set K = n + 1 to reach the same optimal value which proves the theorem. The case rank Q < n can be proved analogously by considering the convex combination y := K X k=1 λkQy(k).
Furthermore the authors in [41] provide a mixed-integer linear program for- mulation for Problem (K-AR).
Theorem 3.15 ([41]). The K-adaptability robust counterpart (K-AR) is equivalent to problem min b>α s.t. A>α = Cx + X k=1,...,K Qzk λ>1 = 1 T x + W yk≤ h k = 1, . . . , K zk ≤ yk, zk ≤ λ k1 k = 1, . . . , K zk ≥ (λk− 1) 1 + yk k = 1, . . . , K x ∈ X, α ∈ Rr+, λ ∈ R K + yk ∈ Y, zk ∈ Rn k = 1, . . . , K .
Problem (M3), which will be studied in Chapters 4 and 5, is the special case of the Problem (K-AR) where no first stage exists, i.e. X = {0}. In Section 4.1.1 we will show that for convex U this case is tractable for all tractable com- binatorial problems if we choose K ≥ n + 1. Additionally we show that this problem is NP -hard for any fixed K and for polyhedral uncertainty, which also proves the NP -hardness of Problem (K-AR).