Decomposition Methods for the Network Optimization Problem of Simultaneous Routing and Bandwidth Allocation Based on Lagrangian Relaxation
Abstract
:1. Introduction
2. Problem Formulation
- | set of all network nodes and a single node, respectively; | |
- | set of all network arcs and a single arc, respectively; | |
- | network (graph) for which the optimization problem is formulated; | |
- | set of all labeled links and a single labeled link, respectively; | |
- | one-to-one mapping from arcs to links labeled by a single natural number; | |
- | set of all demands and a single demand, respectively; | |
- | source and destination node for the specific demand w, respectively; | |
- | flow rate for the specific demand w, , | |
- | lower and upper bound on the flow rate for the demand w, we assume that ; | |
- | capacity of the link l; | |
- | binary routing decision variable, whether the link l is used by the demand w; | |
- | positive parameter—weight of the QoS part of the objective function; | |
- | positive parameter—weight of the energy usage part of the objective function, |
2.1. Linearized Constraints
2.2. Alternative Problem Formulation
3. Lagrangian Relaxation
4. Decomposition Methods
4.1. Demands Decomposition
4.2. Subnetworks Decomposition
- | division of a given network | |
into S subnetworks that is ; this is not a partition because of common links on the borders (see below), | ||
- | set of all nodes of a subnetwork - this is a partition of the set N that is: , | |
- | set of all arcs of a subnetwork | |
- | set of internal arcs of the subnetwork | |
- | set of all labels of links of a subnetwork | |
- | flow rate for the demand w in the subnetwork | |
- | binary routing variable for the flow w in the subnetwork and link | |
- | set of external arcs (incoming or outgoing) | |
of the subnetwork | ||
- | indices of subnetworks directly connected | |
with | ||
- | set of labels of external links for the subnetwork | |
- | correcting coefficient to avoid adding external arcs twice. |
4.3. Continuous and Binary Variables Decomposition
5. Coordination Algorithms
- 1.
- Simple gradient algorithmsA simple gradient algorithm to update Lagrange multipliers directly uses the gradient of the dual function, in the following manner:
- For multipliers of the relaxed equality constraints:
- For multipliers of the relaxed inequality constraints:
Parameter (k - iteration number) plays a key role in the simple gradient coordination algorithm. It is responsible for the algorithm convergence rate. Some simple methods for parameter computation can be used [45]:- Square summable, but not summable:
- Nonsummable diminishing:
More sophisticated techniques for evaluation can also be used. Some of them utilize knowledge of the optimal objective value (or its estimate). One of the most advanced methods is Goffin and Kiwiel’s algorithm of the simultaneous optimal value estimation and Lagrange multipliers update [46], which guarantees convergence with an acceptable rate and relatively easy tuning of parameters. This very version of the simple gradient coordination algorithm has been used in our numerical tests.The simple gradient algorithm is strongly dependent on its parameters. Quite often, its convergence rate can be unsatisfactory. However, this algorithm is relatively easy to implement and requires rather small computational efforts. Hence, for a large number of the Lagrange multipliers, this coordination method may be the most appropriate. - 2.
- Cutting plane methodCutting plane algorithms also utilize dual function gradients, but in a different way than simple gradient algorithms. Namely, the approximation of the dual function is based on the dual function gradient in the following manner (for simplicity, we assume that the problem has only inequality constraints):The approximated dual problemIn the cutting plane algorithm, dual function approximation is built iteratively. New cuts, defined by linear functions which use dual function gradients, are added in every iteration. The solution of the approximated dual problem is a point where the new cut will be put. It means that, with every iteration, a better piecewise linear approximation of the dual function is achieved. And then, in every iteration of the algorithm, the optimization problem described by (105)–(107) is solved.A similar, but more sophisticated algorithm (which will be used in numerical tests) can also be applied to approximate the dual function and to solve the dual problem [47]. Namely, the dual function can be formulated in the following manner:In the case of the (108) formula, a penalty component was added to keep a stable convergence rate of the algorithm. Two kinds of steps are described in [47] for the algorithm. The first is a significant step: , the second is zero-step: . Zero step still uses new Lagrange multipliers to build a better approximation, but does not update a penalty parameter, so it allows for achieving a better approximation. Meanwhile, a significant step requires significant reduction of the dual value and is performed when:To summarize, cutting plane methods are more complicated and require more computational effort for solving an additional optimization problem in every iteration. The problem considered in this paper may become rather complicated for a greater number of Lagrange multipliers. Fortunately, this method allows for achieving a more precise result and is almost independent of parameters tuning.
6. Numerical Tests
6.1. Simple Heuristic Routes Finding (HRF) Algorithm
6.2. Generation of Networks
6.3. Comparison of Results for Different Methods
Objective | —value of the problem objective function, |
Spent Time | —time spent to retrieve solutions in seconds, |
Feasibility | —solution feasibility validation (for the original problem P); for infeasible solution number of capacity constraints (5) violations—“CCV: <number>”, and number of routing constraints (2) violations |
—“RCV: <number>”, | |
Status | —solution status retrieved from a solver, |
Dual Objective (LB) | —value of the dual problem objective function, which is also a lower bound for the primal problem, |
Heuristic Objective (LB) | —value of the problems objective function retrieved using the HRF Algorithm, |
Reallocated (UB) | —value of a feasible solution restored from a lower bound solution routing variables using solution of the R problem, which is also an upper bound for the primal problem, |
LB Relative Gap | —ratio of the lower bound gap with respect to the optimal objective given in percents () |
UB Relative Gap | —ratio of the upper bound gap with respect to the optimal objective given in percents () |
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Nodes Number | Arcs Number | Demands Number | Flow Rates Bound | Capacities Bound |
---|---|---|---|---|
25 | 82 | 12 | [0.001, 3] | [0.3, 1] |
Nodes Number | Arcs Number | Demands Number | Flow Rates Bound | Capacities Bound |
---|---|---|---|---|
49 | 146 | 32 | [0.001, 3] | [0.3, 1] |
Objective | Spent Time [s] | Feasibility | Status | |
---|---|---|---|---|
Problem | ||||
1 | 89.62 | 4.39 | Feasible | Optimal |
2 | 89.64 | 34.61 | Feasible | Optimal |
3 | 83.68 | 273.25 | Feasible | Optimal |
4 | 84.35 | 271.22 | Feasible | Optimal |
5 | 85.16 | 725.94 | Feasible | Optimal |
Problem | ||||
1 | 89.62 | 12.16 | Feasible | Optimal |
2 | 89.64 | 825.28 | Feasible | Optimal |
3 | 83.68 | 301.64 | Feasible | Optimal |
4 | 84.35 | 348.25 | Feasible | Optimal |
5 | 85.16 | 2000.09 | Feasible | Timeout |
Dual Objective (LB) | Reallocated (UB) | Spent Time [s] | Feasibility | LB Relative Gap [%] | UB Relative Gap [%] | |
---|---|---|---|---|---|---|
Formulation Coordination: Proximal Cutting Planes | ||||||
1 | 86.02 | 92.9 | 21.63 | CCV: 14; RCV: 0 | 4.0170 | 3.6599 |
2 | 87.22 | 101.33 | 27.86 | CCV: 19; RCV: 0 | 2.6997 | 13.0411 |
3 | 77.17 | 84.76 | 26.28 | CCV: 8; RCV: 0 | 7.7796 | 1.2906 |
4 | 80.82 | 90.16 | 21.28 | CCV: 13; RCV: 0 | 4.1849 | 6.8880 |
5 | 83.04 | 91.61 | 26.04 | CCV: 13; RCV: 0 | 2.4894 | 7.5740 |
Formulation Coordination: Proximal Cutting Planes | ||||||
1 | 86.5 | 93.15 | 19.49 | CCV: 13; RCV: 0 | 3.4814 | 3.9389 |
2 | 87.31 | 94.41 | 22.24 | CCV: 19; RCV: 0 | 2.5993 | 5.3213 |
3 | 77.19 | 87.3 | 23.29 | CCV: 10; RCV: 0 | 7.7557 | 4.3260 |
4 | 80.52 | 87.19 | 25.07 | CCV: 8; RCV: 0 | 4.5406 | 3.3669 |
5 | 82.95 | 90.13 | 22.87 | CCV: 11; RCV: 0 | 2.5951 | 5.8361 |
Formulation Coordination: Simple Gradient Level Method | ||||||
1 | 83.14 | 91.23 | 19.77 | CCV: 15; RCV: 0 | 7.2305 | 1.7965 |
2 | 84.62 | 94.75 | 29.56 | CCV: 19; RCV: 0 | 5.6002 | 5.7006 |
3 | 74.52 | 94.37 | 27.19 | CCV: 14; RCV: 0 | 10.9465 | 12.7749 |
4 | 78.14 | 86.57 | 30.81 | CCV: 13; RCV: 0 | 7.3622 | 2.6319 |
5 | 79.49 | 87.9 | 27.82 | CCV: 19; RCV: 0 | 6.6581 | 3.2175 |
Formulation Coordination: Simple Gradient Level Method | ||||||
1 | 84.42 | 92.64 | 16.72 | CCV: 14; RCV: 0 | 5.8023 | 3.3698 |
2 | 84.83 | 95 | 18.23 | CCV: 20; RCV: 0 | 5.3659 | 5.9795 |
3 | 74.97 | 90.07 | 19.66 | CCV: 10; RCV: 0 | 10.4087 | 7.6362 |
4 | 76.96 | 86.86 | 19.12 | CCV: 18; RCV: 0 | 8.7611 | 2.9757 |
5 | 79.52 | 92.72 | 17.32 | CCV: 19; RCV: 0 | 6.6228 | 8.8774 |
Dual Objective (LB) | Spent Time [s] | Feasibility | LB Relative Gap [%] | |
---|---|---|---|---|
Formulation Coordination: Proximal Cutting Planes | ||||
1 | 88.76 | 37.34 | CCV: 18; RCV: 14 | 0.9596 |
2 | 86.26 | 770.63 | CCV: 29; RCV: 24 | 3.7706 |
3 | 80.51 | 4748.93 | CCV: 24; RCV: 18 | 3.7882 |
4 | 80.21 | 82.85 | CCV: 18; RCV: 20 | 4.9081 |
5 | 81.63 | 3767.56 | CCV: 25; RCV: 31 | 4.1451 |
Formulation Coordination: Proximal Cutting Planes | ||||
1 | 88.9 | 40.87 | CCV: 10; RCV: 18 | 0.8034 |
2 | 84.82 | 132.7 | CCV: 20; RCV: 31 | 5.3771 |
3 | 80 | 221.48 | CCV: 14; RCV: 26 | 4.3977 |
4 | 80.7 | 111.24 | CCV: 15; RCV: 30 | 4.3272 |
5 | 82.25 | 328.37 | CCV: 17; RCV: 34 | 3.4171 |
Formulation Coordination: Simple Gradient Level Method | ||||
1 | 88.13 | 35.6 | CCV: 17; RCV: 4 | 1.6626 |
2 | 85.39 | 524.38 | CCV: 26; RCV: 14 | 4.7412 |
3 | 79.26 | 2072.34 | CCV: 25; RCV: 10 | 5.2820 |
4 | 80.08 | 51.44 | CCV: 30; RCV: 16 | 5.0622 |
5 | 79.88 | 597.26 | CCV: 30; RCV: 18 | 6.2001 |
Formulation Coordination: Simple Gradient Level Method | ||||
1 | 87.02 | 25.7 | CCV: 18; RCV: 14 | 2.9011 |
2 | 83.48 | 202.28 | CCV: 23; RCV: 18 | 6.8719 |
3 | 77.94 | 215.1 | CCV: 22; RCV: 23 | 6.8595 |
4 | 80.1 | 106.3 | CCV: 17; RCV: 26 | 5.0385 |
5 | 78.83 | 91.08 | CCV: 22; RCV: 28 | 7.4331 |
Dual Objective (LB) | Reallocated (UB) | Spent Time [s] | Feasibility | LB Relative Gap [%] | UB Relative Gap [%] | |
---|---|---|---|---|---|---|
Formulation Coordination: Proximal Cutting Planes | ||||||
1 | 63.38 | 90.93 | 14.32 | CCV: 19; RCV: 0 | 29.2792 | 1.4617 |
2 | 65.05 | 97.68 | 14.17 | CCV: 18; RCV: 0 | 27.4320 | 8.9692 |
3 | 36.08 | 89.55 | 15.04 | CCV: 19; RCV: 0 | 56.8834 | 7.0148 |
4 | 38.75 | 87.69 | 15.73 | CCV: 29; RCV: 0 | 54.0605 | 3.9597 |
5 | 34.46 | 92.54 | 16.38 | CCV: 32; RCV: 0 | 59.5350 | 8.6660 |
Formulation Coordination: Proximal Cutting Planes | ||||||
1 | 85.61 | - | 14.97 | CCV: 0; RCV: 24 | 4.4744 | - |
2 | 85.12 | - | 14.11 | CCV: 0; RCV: 24 | 5.0424 | - |
3 | 75.35 | - | 13.06 | CCV: 0; RCV: 24 | 9.9546 | - |
4 | 79.31 | - | 14.51 | CCV: 0; RCV: 24 | 5.9751 | - |
5 | 81.44 | - | 15.03 | CCV: 0; RCV: 24 | 4.3682 | - |
Formulation Coordination: Simple Gradient Level Method | ||||||
1 | 53.53 | 97.07 | 4.04 | CCV: 30; RCV: 0 | 40.2700 | 8.3129 |
2 | 48.27 | 98.97 | 4.09 | CCV: 41; RCV: 0 | 46.1513 | 10.4083 |
3 | 24.6 | 90.62 | 4.18 | CCV: 32; RCV: 0 | 70.6023 | 8.2935 |
4 | 29.43 | 92.31 | 4.33 | CCV: 38; RCV: 0 | 65.1097 | 9.4369 |
5 | 20.5 | 94.11 | 4.27 | CCV: 39; RCV: 0 | 75.9277 | 10.5096 |
Formulation Coordination: Simple Gradient Level Method | ||||||
1 | 85.79 | - | 1.86 | CCV: 3; RCV: 24 | 4.2736 | - |
2 | 85.34 | - | 2.02 | CCV: 3; RCV: 26 | 4.7970 | - |
3 | 75.55 | - | 1.87 | CCV: 0; RCV: 24 | 9.7156 | - |
4 | 79.51 | - | 2.13 | CCV: 2; RCV: 22 | 5.7380 | - |
5 | 81.7 | - | 1.95 | CCV: 1; RCV: 30 | 4.0629 | - |
Heuristic Objective (LB) | Reallocated (UB) | Spent Time [s] | Feasibility | LB Relative Gap [%] | UB Relative Gap [%] | |
---|---|---|---|---|---|---|
Dijkstra algorithm based heuristics (HRF) | ||||||
1 | 73.1 | 96.06 | 4.86 | CCV: 14; RCV: 0 | 18.4334 | 7.1859 |
2 | 71.41 | 103.29 | 4.76 | CCV: 14; RCV: 0 | 20.3369 | 15.2276 |
3 | 69.65 | 94.56 | 4.48 | CCV: 15; RCV: 0 | 16.7663 | 13.0019 |
4 | 70.35 | 94.79 | 3.66 | CCV: 16; RCV: 0 | 16.5975 | 12.3770 |
5 | 73.47 | 97.44 | 5.37 | CCV: 13; RCV: 0 | 13.7271 | 14.4199 |
Objective | Spent Time [s] | Feasibility | Status | |
---|---|---|---|---|
Problem | ||||
1 | 247.08 | 478.79 | Feasible | Optimal |
2 | 239.79 | 1111.43 | Feasible | Optimal |
3 | 246.62 | 2000.15 | Feasible | Timeout |
4 | 248.7 | 2000.08 | Feasible | Timeout |
5 | 252.98 | 2000.12 | Feasible | Timeout |
Problem | ||||
1 | 247.08 | 2000.6 | Feasible | Timeout |
2 | 239.79 | 2000.26 | Feasible | Timeout |
3 | 246.62 | 2000.55 | Feasible | Timeout |
4 | 248.66 | 2000.47 | Feasible | Timeout |
5 | 252.98 | 2000.34 | Feasible | Timeout |
Dual Objective (LB) | Reallocated (UB) | Spent Time [s] | Feasibility | LB Relative Gap [%] | UB Relative Gap [%] | |
---|---|---|---|---|---|---|
Problem Coordination: Proximal Cutting Planes | ||||||
1 | 239.03 | 250.01 | 154.07 | CCV: 17; RCV: 0 | 3.2581 | 1.1859 |
2 | 233.17 | 239.79 | 130.02 | CCV: 17; RCV: 0 | 2.7607 | 0.0000 |
3 | 241.33 | 250.82 | 137.3 | CCV: 15; RCV: 0 | 2.1450 | 1.7030 |
4 | 234.93 | 252.54 | 205.87 | CCV: 14; RCV: 0 | 5.5368 | 1.5440 |
5 | 250.49 | 257.5 | 141.15 | CCV: 13; RCV: 0 | 0.9843 | 1.7867 |
Problem Coordination: Proximal Cutting Planes | ||||||
1 | 238.97 | 250.28 | 345.64 | CCV: 16; RCV: 0 | 3.2823 | 1.2951 |
2 | 233.11 | 242.78 | 201.82 | CCV: 14; RCV: 0 | 2.7858 | 1.2469 |
3 | 241.35 | 249.96 | 204.16 | CCV: 14; RCV: 0 | 2.1369 | 1.3543 |
4 | 234.99 | 252.42 | 230.01 | CCV: 17; RCV: 0 | 5.4975 | 1.4958 |
5 | 249 | 255.32 | 225.91 | CCV: 15; RCV: 0 | 1.5732 | 0.9250 |
Problem Coordination: Simple Gradient Level Method | ||||||
1 | 234.52 | 250.47 | 111.44 | CCV: 30; RCV: 0 | 5.0834 | 1.3720 |
2 | 230.12 | 246.14 | 176.46 | CCV: 25; RCV: 0 | 4.0327 | 2.6482 |
3 | 239.22 | 248.44 | 117.68 | CCV: 21; RCV: 0 | 3.0006 | 0.7380 |
4 | 230.37 | 251.91 | 132.05 | CCV: 22; RCV: 0 | 7.3703 | 1.2907 |
5 | 245.62 | 257.05 | 120.96 | CCV: 15; RCV: 0 | 2.9093 | 1.6088 |
Problem Coordination: Simple Gradient Level Method | ||||||
1 | 233.88 | 251.28 | 148.46 | CCV: 30; RCV: 0 | 5.3424 | 1.6999 |
2 | 231.22 | 242.2 | 182.3 | CCV: 24; RCV: 0 | 3.5740 | 1.0050 |
3 | 239.59 | 248.88 | 185.31 | CCV: 19; RCV: 0 | 2.8505 | 0.9164 |
4 | 231.37 | 254.85 | 208.15 | CCV: 19; RCV: 0 | 6.9533 | 2.4729 |
5 | 245.86 | 253.67 | 233.64 | CCV: 36; RCV: 0 | 2.8145 | 0.2727 |
Heuristic Objective (LB) | Reallocated (UB) | Spent Time [s] | Feasibility | LB Relative Gap [%] | UB Relative Gap [%] | |
---|---|---|---|---|---|---|
Dijkstra algorithm based heuristics (HRF) | ||||||
1 | 199.48 | 251.52 | 67.61 | CCV: 28; RCV: 0 | 19.265 | 1.7970 |
2 | 203.27 | 253.23 | 84.56 | CCV: 40; RCV: 0 | 15.2300 | 5.6049 |
3 | 190.52 | 255.81 | 77.33 | CCV: 32; RCV: 0 | 22.7475 | 3.7264 |
4 | 200.47 | 256.75 | 78.81 | CCV: 32; RCV: 0 | 19.3928 | 3.2368 |
5 | 202.66 | 264.61 | 91.64 | CCV: 32; RCV: 0 | 19.8909 | 4.5972 |
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Ruksha, I.; Karbowski, A. Decomposition Methods for the Network Optimization Problem of Simultaneous Routing and Bandwidth Allocation Based on Lagrangian Relaxation. Energies 2022, 15, 7634. https://doi.org/10.3390/en15207634
Ruksha I, Karbowski A. Decomposition Methods for the Network Optimization Problem of Simultaneous Routing and Bandwidth Allocation Based on Lagrangian Relaxation. Energies. 2022; 15(20):7634. https://doi.org/10.3390/en15207634
Chicago/Turabian StyleRuksha, Ihnat, and Andrzej Karbowski. 2022. "Decomposition Methods for the Network Optimization Problem of Simultaneous Routing and Bandwidth Allocation Based on Lagrangian Relaxation" Energies 15, no. 20: 7634. https://doi.org/10.3390/en15207634