Wireless Network Design: Optimization Models and Solution ...

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8 The Design of Partially Survivable Networks 185 and initialize the master problem with every cell being connected to the MTSO. Instead, one could initialize using a feasible solution obtained via a heuristic procedure. Solving the master problem provides dual prices which are used to generate columns by solving a subproblem for each of the cells. We need not go over all the cells in every iteration — we can switch to the master problem as soon as we have columns to add to it; the actual number of columns to add at each iteration can be determined empirically. We iterate between the master and subproblems until no more columns price out, at which point we have the optimal solution to the linear programming relaxation of (M2). 8.3.2.2 Representing the Subproblem One important factor in the success of a column generation based solution procedure is the ease with which the subproblem can be represented and solved. The subproblem gets solved many times in the solution process, and it is critical that the subproblem be easy to solve. A column is attractive to the linear programming relaxation of the master problem if the associated reduced cost is negative. If we associate dual variables π∈ R |N| with the constraints 8.14 and λ∈ RT − with the constraints 8.15, a column s could (potentially) improve our objective function value if and only if � fg − πi − T ∑ λ jb jg j=1 � < 0. Therefore, the key issue becomes one of identifying a schedule with this feature if it exists. At any iteration of the column generation process, we have a subproblem for each cell i that is best described by Figure 8.2. In this figure, every time period j is represented by two sets of nodes R j and Hj, corresponding to the cell being connected to the MTSO and the hub, respectively. In addition, a source node P and a sink node Q are added, with the source node being connected to the nodes associated with time period 1 (edges P → R1 and P → H1) and the sink node being connected to the final period (edges R j → Q and H j → Q). There are edges from every MTSO node R j, j = 1,..., (T − 1) indicating that the cell is connected to the root (R j → R ( j+1)) or the hub (R j → H ( j+1)) in the next period. Similarly, there are edges from every hub node Hj, j = 1,..., (T − 1) indicating that the cell is connected to the root (Hj → R ( j+1)) or the hub (Hj → H ( j+1)) in the next period. The P → R1 edge and the R j → R ( j+1) edges have cost ri1 and r i( j+1) associated with them, respectively; the P → H1 edge and the R j → H ( j+1) edges have cost (hi1 +si1) and (h i( j+1) +s i( j+1)) associated with them, respectively; the Ht → R ( j+1) edges have cost (r i( j+1) + t i( j+1)) associated with them, and the cost on the edges R j → Q and Hj → Q are 0. In addition, there are costs of (−λ j × di j) associated with each of the Hj nodes as a result of the dual multipliers. Given this description, the subproblem becomes one of identifying the lowest cost to send one unit of flow from P to Q. In the absence of the node weights (−λ j × di j), this problem is equivalent to one of finding the shortest path from P to Q, and any shortest path algorithm (e.g., Dijkstra’s) will provide the solution. In the case of the special network of the sub-
186 Syam Menon −λ1di1 −λ2di2 −λT di j ✓✏ ✓✏ ✓✏ H1 H2 H j ✒✑ ✒✑ ✒✑ ✓✏ � ✓✏ ��✒ hi1 h ✲ i j ✲ ❆ ❆ ✁✕ ❅ ❆❆❆❆❆❆❆❆❆❆❆❯ ❅❅❅❅❘ ❅❅❅❅❘ ❆ ✁ ❆ ✁ connection to HUB P connection to MTSO (h i1 + s i1 ) r i1 (r i2 +t i2 ) ✒✑ ✒✑ ❅ ❅❅❅ ❅❘ ✓✏ ✁ ✓✏ ✓✏ r R i2 1 ✲ R2 R j ✒✑ ✒✑ ✒✑ ✁✁✁✁✁✁✁✁✁✁✁✕ ✁ ❆ ✁ ❆ ✁ ❆❯ � ri j ✲ ��✒ (h i2 + s i2 ) Fig. 8.2 Representing the Subproblem (r i j +t i j ) � �� ✒ problem, since we have a graph without cycles and since every period has to be visited exactly once, we can add the node weights to the weights on the edges entering the nodes Hj without loss of generality. Note that this problem can also be represented as a min-cost-flow problem where one unit of flow needs to be sent from P to Q. In our context, the min-cost-flow formulation (without the node weights) is (S1) below. min ∑ (i, k)∈E cikzik s.t. zPR + zPH 1 1 = 1 (8.17) zPR1 − zR1R2 − zR1H2 = 0 (8.18) zPH − zH 1 1R − zH 2 1H2 = 0 (8.19) zR ( j−1) R j + zH ( j−1) R j − zR jR − zR ( j+1) jH ( j+1) = 0 j = 2,...,(T − 1)(8.20) (S1) zR ( j−1) Hj + zH ( j−1) H j − zHjR − zHjH = 0 j = 2,...,(T − 1)(8.21) ( j+1) ( j+1) zR (T −1) RT + zH (T −1) RT − zRT Q = 0 (8.22) zR (T −1) HT + zH (T −1) HT − zHT Q = 0 (8.23) zRT Q + zHT Q = 1 (8.24) zik ∈ {0, 1}∀(i,k) ∈ E (8.25) The cik are the costs on the edges as identified earlier; E is the set of edges in the network; zik is a variable indicating flow between nodes i and k. Constraint 8.17 indicates flow out of P; constraints 8.18 and 8.19 are flow balance constraints on (h i j + s i j ) 0 0 Q
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International Series in Operations
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Editors Jeff Kennington Eli Olinick
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vi Preface assistance. The work of
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viii Contents 3.4.1 Experiments to
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x Contents 8.3.2 The Solution Proce
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xii Contents References . . . . . .
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List of Contributors Khaldoun Al Ag
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List of Contributors xvii Nitin Sal
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Chapter 1 Introduction to Optimizat
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1 Introduction to Optimization in W
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1 Introduction to Optimization in W
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Part I Background
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10 Dinesh Rajan The goal of this ch
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12 Dinesh Rajan The source encoder
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14 Dinesh Rajan ping (FH) CDMA, and
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16 Dinesh Rajan of SNR is given in
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18 Dinesh Rajan 2.3 Information The
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20 Dinesh Rajan Perror ≤ e −nE
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22 Dinesh Rajan For instance, for t
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24 Dinesh Rajan 2.4.1 Block Codes I
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26 Dinesh Rajan B(z) = 1 � (1 + z
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28 Dinesh Rajan layers. Two of the
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30 Dinesh Rajan nel can support a p
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32 Dinesh Rajan 2.5.2 Physical laye
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34 Dinesh Rajan matched filter for
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36 Dinesh Rajan codes) leads to bet
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38 Dinesh Rajan a length N Inverse
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40 Dinesh Rajan the transmission of
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42 Dinesh Rajan setting, a node may
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44 Dinesh Rajan weighting of the si
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46 Dinesh Rajan 29. Oestges, C., Cl
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48 K. V. S. Hari terrain of operati
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50 K. V. S. Hari fading is calculat
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52 K. V. S. Hari terms have been pr
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54 K. V. S. Hari Fig. 3.1 Normalize
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56 K. V. S. Hari Table 3.3 Impulse
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58 K. V. S. Hari where m antennas a
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60 K. V. S. Hari 3.5.2.4 Dominant R
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62 K. V. S. Hari March, 1984-13 Sep
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64 K. V. S. Hari 47. Van Rees, J.:
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66 J. Cole Smith and Sibel B. Sonuc
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Part II Optimization Problems for N
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102 Eli Olinick carrying capacity.
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104 Eli Olinick 93 85 160 16 38 21
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106 Eli Olinick Amaldi et al. [7] p
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108 Eli Olinick levels (possibly le
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110 Eli Olinick gmℓ ∑ ∑ m∈M
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112 Eli Olinick Using bk to denote
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114 Eli Olinick 5.3.5 Additional De
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116 Eli Olinick capacity, provide n
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118 Eli Olinick ficult optimization
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120 Eli Olinick and-bound so that t
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122 Eli Olinick solution times. Cai
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124 Eli Olinick 27. Glover, F.: Tab
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Chapter 6 Optimization Based WLAN M
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6 Optimization Based WLAN Modeling
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10 Integer Programming Models for P
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Chapter 11 Improving Network Connec
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11 Improving Network Connectivity i
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Chapter 12 Simulation-Based Methods
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12 Simulation-Based Methods for Net
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296 Hang Jin and Li Guo formance in
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298 Hang Jin and Li Guo (ISI) as lo
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300 Hang Jin and Li Guo ARQ (HARQ)
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302 Hang Jin and Li Guo 13.3 Radio
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304 Hang Jin and Li Guo Table 13.2
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306 Hang Jin and Li Guo is transmit
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308 Hang Jin and Li Guo uplink powe
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310 Hang Jin and Li Guo The link ad
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312 Hang Jin and Li Guo where ρ mi
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314 Hang Jin and Li Guo 13.4.4.3 Ex
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316 Hang Jin and Li Guo Fig. 13.9 P
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318 Hang Jin and Li Guo 1. Guarante
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Chapter 14 Cross Layer Scheduling i
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14 Cross Layer Scheduling in Wirele
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Chapter 15 Major Trends in Cellular
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15 Major Trends in Cellular Network
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