Neighborhood Structures for Improving Primal Solutions

The Lagrangian dual problem is the challenge of finding an optimal vector of La-grange multipliers λ^∗ so that the lower bound obtained by (LR(λ^∗)) becomes as large as possible. As this maximization problem is convex and piecewise linear, sub-gradient algorithms are well suited for this purpose [18]. While different variants of such methods exist, the volume algorithm [13] has proven to be more effective than several alternatives on various occasions [11, 86], and we therefore apply it here.

Also, our preliminary comparisons indicate the superiority of this algorithm over the standard subgradient strategy as described in [18], see also Section 2.1.7.

3.8.1 Theoretical Comparison to the MCF Formulation

For each concrete instantiation of λ, all subproblems obtained by the Lagrangian decomposition are always solved to optimality and integrality. Therefore, f^k ∈ conv(F_k) holds for all k ∈ C, and the abstract constraints (3.96) of our model (3.94)–(3.98) can be regarded as “ideally instantiated”. Assuming we would be able to identify an optimal Lagrange vectorλ^∗, the lower bound obtained by (LR(λ^∗)) is at least as good as the lower bound determined by an LP relaxation of the model.

As already argued before, the MCF formulation from [178] is weaker than an “ideal”

instantiation of the abstract model, compare also Figure 3.11. We therefore conclude that (LR(λ^∗)) is stronger than the LP relaxation of the MCF formulation.

3.9 Neighborhood Structures for Improving Primal Solutions

Our algorithms make use of three types of neighborhood structures. While the first two aim to reduce the cost of a given solution, the last type consisting of two concrete neighborhood structures tries to improve a solution by removing customers from a candidate solution. Therefore, the latter is only applicable to the SST variant.

For our neighborhood structures, next to its node set V⁰ ⊆ V and its edge set E⁰ ⊆ E, each candidate solution G⁰ = (V⁰, E⁰) is further represented by its set of feasibly connected customersC⁰ ⊆C and the corresponding individual connections X⁰ ={E_k⁰ | k ∈ C⁰}, where E_k⁰ denotes the set of edges used to connect customer k. Note that there may exist multiple connections to a single customer node in a solutionG⁰ in which case we store only one of them.

Furthermore, for each connectionE_k⁰ we maintain its internal structure consisting of its branch nodeB(E⁰_k)∈V⁰, edge setsP(E_k⁰), Q(E_k⁰)⊆E⁰ of its two paths between r and B(E⁰_k) and finally the edge set of its branch line L(E_k⁰) ⊆E⁰. Note that we

Chapter 3 The b_max-Survivable Network Design Problem

Figure 3.15: An exemplary candidate solution and the representation of its connec-tions.

assumeB(E_k⁰) =kifb_max(k) = 0 ork∈C₁ as well as defineP(E_k⁰) to be the “first”

path of a connection, i.e. P(E_k⁰) is used for type-1 customers whileQ(E_k⁰) =L(E_k⁰) =

∅ for type-1 customers, see Figure 3.15. Finally, to allow for efficient updates of a solution with respect to connections, we maintain for each edge e∈E⁰ a list of the customers that are connected via this edge: M_e={k∈C⁰|e∈E_k⁰}.

3.9.1 Connection Exchange Neighborhood

The Connection Exchange Neighborhood (CEN) consists of all solutions differing from the current solution G⁰ by exactly one connection E_k⁰, see Algorithm 3.2. To determine the best neighboring solution for a fixed customerk∈C⁰, CEN calculates the saving due to removing the corresponding connectionE_k⁰ (which is the sum of all edge costs exclusively used to connectk). The connection tok leading to minimum additional costs is then determined by calculating the cheapest feasible connection to k in a graph with edge costsc⁰_e = 0, ∀e∈E⁰⁰ =E⁰\ {e∈E_k⁰ |Me ={k}} and c⁰_e=c_e, ∀e∈E\E⁰⁰, see Figure 3.16 for an exemplary move.

For type-1 customer nodes as well as type-2 customer nodes with b_max(k) = 0, the computational complexity of finding this new connection for one specific client node kis bounded byO(|E|+|V|log|V|), whereas the worst case complexity for a single type-2 customerkwithb_max(k)>0 isO(b_max(k)|E(k)|+|E|+|V|log|V|), see

3.9 Neighborhood Structures for Improving Primal Solutions

Algorithm 3.2: Connection Exchange (SolutionG⁰) c⁰_e= 0, ∀e∈E⁰

c⁰_e=ce,∀e∈E\E⁰ d_opt = 0

forall the k∈C⁰ do

E⁰⁰={e∈E_k⁰ |Me={k}}

c⁰_e=ce,∀e∈E⁰⁰ d=P

e∈E⁰⁰c_e

E_k⁰⁰ = shortest connection tok using edge costsc⁰ d=P

e∈E⁰⁰ce−P

e∈E_k⁰⁰c⁰_e if d > dopt then

d_opt=d

store solutionG⁰ withE_k⁰⁰ replacingE_k⁰ as best solution c⁰_e= 0, ∀e∈E⁰⁰

return best solution

k∈C₂

⇒

k∈C₂

Figure 3.16: An exemplary connection exchange for customer k∈C₂.

Chapter 3 The b_max-Survivable Network Design Problem

Section 3.4. Thus, the overall complexity of completely searching CEN is bounded by O(|C|(max_k∈C{b_max(k)|E(k)|}+|E|+|V|log|V|)).

3.9.2 Key-Path Exchange Neighborhood

A key-node of a solution G⁰ is a node v ∈V⁰\C⁰ with node degree deg_G⁰(v) ≥ 3, while a key-path is a path K_P = (V_P, E_P) whose end nodes are either key-nodes, connected customer nodes k ∈ C⁰, or the root node r, while all other nodes are Steiner nodes v ∈V⁰\(C⁰∪ {r}) of degree two, i.e. deg_G⁰(v) = 2. This concept of key-paths is well known for the STP and several metaheuristic methods utilizing a key-path exchange neighborhood have been proposed, see e.g. [137]. TheKey-Path Exchange Neighborhood (KPEN) given in Algorithm 3.3 extends this concept by exchanging key-paths while respecting node- as well as b_max-redundancy, see also Figure 3.17 for a visualization of an exemplary path exchange.

Algorithm 3.3:Key-Path Exchange (Solution G⁰) determine key-paths W

d_opt = 0

forall the key-paths (V_P, E_P)∈W do

// actual key-path connects its end nodes m, n c⁰_e= 0 ∀e∈E⁰\E_P

c⁰_e=c_e ∀e∈E_P∪(E\E⁰) choose e∈EP randomly l_max=∞

forall thek∈M_e⁰ do if e∈P(E_k⁰)then

c⁰_e=∞,∀e∈E incident to a inner node of Q(E_k⁰) else if e∈Q(E_k⁰) then

c⁰_e=∞,∀e∈E incident to a inner node of P(E_k⁰) else if e∈L(E_k⁰) then

lmax=bmax(k)−P

e∈(L(E⁰_k)\EP)le

(V_P⁰, E_P⁰ ) = shortest path from mton usingc⁰_e with max. lengthl_max d=P

e∈EPce−P

e∈E_P⁰ c⁰_e if d > dopt then

d_opt =d

store solutionG⁰ with (V_P, E_P) replacing (V_P, E_P) as best solution return best solution

3.9 Neighborhood Structures for Improving Primal Solutions

⇒

k∈C₂ k∈C₂

Figure 3.17: An exemplary key-path exchange between r and k.

KPEN of a candidate solution G⁰ consists of all feasible solutions that differ from G⁰ by at most one key-path. To ensure feasibility, after exchanging a key-path KP, three relevant cases need to be considered. IfK_Pis used to connect type-1 customers only, it may simply be replaced by any other path, while if it is used in a branch line L(E_k⁰) of a type-2 customerk∈C2, the maximum length of the new path may be at mostb_max(k)−P

e∈(L(E⁰_k)\EP)l_e. Finally, ifK_Pis used in the first pathP(E_k⁰) of aC₂ customerk, all edges incident to “internal” nodes of its second pathQ(E_k⁰) may not be used by the new key-path to guarantee node redundancy (and vice versa for the alternate pathQ(E_k⁰)). All other edges e∈E⁰ are treated as pseudo-infrastructure, i.e. c⁰_e= 0.

It is obvious that the number of key-paths |W| of any solution G⁰ = (V⁰, E⁰) is smaller or equal to |E⁰|. Let B = max{b_max(k) : k ∈ C₂} denote the maximum value ofb_max(k),∀k∈C₂, andE = max{|E(k)|:k∈C₂} be the maximum number of edges in the bmax-neighborhood of a type-2 customer node. Then, the worst case complexity for finding the cheapest feasible path eventually replacing a key-path (V_P, E_P)∈W isO(max{B · E,|E|+|V|log|V|}) and the overall complexity of searching KPEN isO(|E⁰| max{B · E,|E|+|V|log|V|}

).

However, despite its large worst case complexity, KPEN performs well in practice since number of key-paths is usually relatively small in realistic solutions.

3.9.3 Connection Remove Neighborhood

Instead of exchanging a customer’s connection as in CEN, the Connection Remove Neighborhood (CRN) – which is given in Algorithm 3.4 – removes the connection to a single customer nodek∈C⁰.

CRN of a current solution G⁰ therefore consists of all solutions G⁰⁰ where exactly one customer connected in C⁰ is not connected anymore, i.e. C⁰⁰ ( C⁰ such that

Chapter 3 The b_max-Survivable Network Design Problem

Algorithm 3.4:Connection Remove (Solution G⁰) d_opt = 0

forall the k∈C⁰ do d=P

e∈E_k⁰|Me={k}c_e−p_k if d > dopt then

d_opt =d

storeG⁰ without connection to k as best solution return best solution

k∈C₂

⇒

k∈C₂

Figure 3.18: An exemplary move, removing the connection to customerk.

|C⁰ \C⁰⁰|= 1. As a customer’s connection may consist of O(|V|) edges only, CRN consisting of |C⁰| neighboring solutions can be searched in O(|C⁰||V|) time. Fig-ure 3.18 depicts an exemplary move, removing the connection to a type-2 customer k∈C2.

3.9.4 Restricted two Connection Remove Neighborhood

CRN can be easily generalized to simultaneously remove multiple customer nodes.

However, removing the connections to l > 1 customers at once will result in |C⁰|^l neighboring solutions and the computational effort of searching such a neighborhood would beO(|C⁰|^l|V|). To limit the effort, we therefore concentrate on simultaneously removing only pairs of customers i, j ∈ C⁰, i 6= j, that share at least one edge exclusively used by them, i.e. ∃e ∈ E⁰ | M_e = {i, j}; see also Algorithm 3.5.

The Restricted two Connection Remove Neighborhood (R2CRN) can be searched in O(|V|min(|E⁰|,|C⁰|²)); see Figure 3.19 for an exemplary move.

3.9 Neighborhood Structures for Improving Primal Solutions

Algorithm 3.5: Restricted two Connection Remove Neighborhood (Solu-tion G⁰).

P ={{u, v} | ∃e∈E⁰:Me ={u, v}}

d_opt = 0

forall the {u, v} ∈P do d=P

e∈E_k⁰|Me∈{{u},{v},{u,v}}ce−pu−pv

if d > dopt then dopt=d

storeG⁰ without connection to u, vas best solution restore best solution

k∈C

⇒

2

k⁰ ∈C2 k⁰ ∈C2

k∈C2

Figure 3.19: An exemplary move, removing the connections to customers kand k⁰.

Chapter 3 The b_max-Survivable Network Design Problem

3.10 Metaheuristics

In this section we present metaheuristic approaches utilizing the neighborhood struc-tures explained in Section 3.9 to compute feasible solutions. After describing a construction heuristic in Section 3.10.1, we present a variable neighborhood search (VNS) with embedded variable neighborhood descent (VND) in Section 3.10.2 and – as an alternative – a GRASP/VND hybrid in Section 3.10.3.

3.10.1 Minimum Spanning Tree Augmentation Heuristic

We use a three-phase approach called Minimum Spanning Tree Augmentation Heuristic (MSTAH) to construct a feasible solution for a given selection of cus-tomers C⁰⊆C to be connected.

Initially, a Steiner tree is computed using the minimum spanning tree (MST) heuris-tic from [111], see also [138]. This procedure determines a MSTTD on the distance network, which is the complete graphD= (C⁰, C⁰×C⁰) with node set C⁰ and edge costsd(u, v) corresponding to the costs of the cheapest paths between anyu, v∈C⁰ inG. A feasible solution G⁰⁰ to the Steiner tree problem is derived by further com-puting a MST on G(T_D) which is the subgraph of G induced by all edges part of any cheapest path corresponding to an edge in T_D.

In its second phase, MSTAH augmentsG⁰⁰= (V⁰⁰, E⁰⁰) by feasible connections toC₂ customers. Such connections are determined by individually calculating the cheapest feasible connection (compare Section 3.4) for all customersk∈C2 in random order.

All so far selected edges e ∈ E⁰⁰ are considered as pseudo-infrastructure, i.e. have zero costs.

Finally, an edge minimal solution is extracted (i.e. no further edges can be deleted without violating feasibility) by greedily removing unnecessary key-paths in random order. A similar heuristic which does not consider b_max redundancy has been pre-sented in [36]. Similar to MSTAH the heuristic from [36] uses the MST heuristic [111, 138] to compute a Steiner tree. As opposed to MSTAH redundancy for C₂ customers is ensured by adding a redundant route to each type-2 customer avoiding any inner node of the existing primary path using so far selected edge as pseudo-infrastructure.

3.10 Metaheuristics

Algorithm 3.6: Minimum Spanning Tree Augmentation Heuristic (Cus-tomers C⁰)

G⁰⁰= (V⁰⁰, E⁰⁰) = Steiner tree computed by MST heuristic c⁰_e= 0, ∀e∈E⁰⁰

c⁰_e=c_e,∀e /∈E⁰⁰ G⁰ =G⁰⁰

forall the k∈C2∩C⁰ do

f_k= cheapest connection to kusing edge costs c⁰ G⁰ =G⁰∪f_k

updatec⁰ according to f_k make edge minimal

determine solution structure

3.10.2 Variable Neighborhood Search

We use the general VNS scheme with VND as embedded local improvement [82]

as described in Section 2.2.4. In VND, we alternate between CEN, KPEN, CRN, R2CRN in this order, with the latter two considered only in the SST variant.

Our shaking algorithm used to escape local optima modifies a solutionG⁰ by exclud-ing a subset of its Steiner nodes as well as changexclud-ing the set of connected customers C⁰ in the SST variant: A set of l= 1, . . . , l_max = b^|C₂^|c Steiner nodes V_F ⊆V⁰ \C of the current solution G⁰ is randomly chosen for removal. Furthermore, we select a set ofl customer nodes V_C ⊆ C at random. The set of customers C⁰⁰ connected in the new solutionG⁰⁰ is C⁰⁰ =C⁰4V_C, i.e. we add those customers of V_C that are currently unconnected while removing the so far connected ones. Finally, we apply MSTAH using the following adapted edge costsc⁰ with a sufficiently large value for M (M max_e∈Ec_e).

c⁰_e=







M ifeis incident to a nodes v∈V_F,

0 ife∈E⁰ and enot incident to a node v∈V_F, c_e else.

Edge costs c⁰ ensure the creation of a new solutionG⁰⁰ that is in general similar to G⁰ while the Steiner nodes selected for exclusion will not be used unless there is no other option to obtain a feasible solutionG⁰⁰. Algorithm 3.7 summarizes the details of the VND approach.

Chapter 3 The b_max-Survivable Network Design Problem

Algorithm 3.7:VND for b_max-SNDP.

l= 1

while l≤4 do switchl do

case1:

x⁰ =CEN(x) // see Algorithm 3.2 case2:

x⁰ =KP EN(x) // see Algorithm 3.3 case3:

x⁰ =CRN(x) // see Algorithm 3.4 case4:

x⁰ =R2CRN(x) // see Algorithm 3.5 if c(x⁰)< c(x) then

x=x⁰ l= 1 else

l=l+ 1 return x

Im Dokument SolvingTwoNetworkDesignProblemsbyMixedIntegerProgrammingandHybridOptimizationMethods DISSERTATION (Seite 81-91)