Resource Allocation in OFDM Based Wireless Relay Networks ...

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1.3 Basics of the Optimization Theory or numerically, since they have a global minima. However, most of the engineering problems are not necessarily convex in nature, therefore some reformulations are done to state the the problem in standard convex form. A common technique to formulate a problem into convex one, is by relaxing the problem from removing some of constraints such that it becomes convex. If the convex optimization problem does not have any constraint function the efficient iterative algorithms have been developed to find the global optimal solution e.g., the gradient descent method, the steepest descent method, and the Newton’s method. The Lagrange duality theory is one of the fundamental tool for solving the constrained convex optimization problems. The basic idea behind the Lagrange duality is to take the constraints of problem (1.1) into account by augmenting the objective function with a weighted sum of constraint functions. The Lagrangian is defined as m∑ p∑ L(x, λ, ν) = f o (x) + λ i f i (x) + ν i h i (x), (1.2) i=1 i=1 where λ = {λ i , ∀i} and ν = {ν i , ∀i} are called the Lagrange multipliers or dual variables. The Lagrange dual function is defined as the minimum value of the Lagrangian over x g(λ, ν) = inf L(x, λ, ν). (1.3) x∈D The dual function is always convex in terms of dual variables regardless of the convexity of the original function. The dual optimization problem is given by max λ,ν subject to g(λ, ν) (1.4) λ i ≥ 0, ν i ≥ 0, ∀i. The duality gap is is defined as d = f o (x ∗ ) − g(λ ∗ , ν ∗ ) ≥ 0, (1.5) where f o (x ∗ ) and g(λ ∗ , ν ∗ ) are the optimal primal and dual values, respectively. It has been shown in [13] that if the original problem is convex, under some constraint conditions, the duality gap reduces to zero (i.e., strong duality) and (1.5) holds with 7
1.3 Basics of the Optimization Theory equality. In other words, both the primal and the dual optimization problems have the same solution. Therefore, one way to solve the original problem is to solve, instead, its associated dual problem. For the optimal primal and dual points (x ∗ , λ ∗ , ν ∗ ) with zero duality gap, the Karush-Kuhn-Tucker (KKT) conditions are defined as [13] f i (x ∗ ) ≤ 0, i = 1, ....., m h i (x ∗ ) = 0, i = 1, ....., p λ ∗ i ≥ 0, i = 1, ....., m ∇f o (x ∗ ) + m∑ λ i ∇f i (x ∗ ) + i=1 λ ∗ i f i (x ∗ ) = 0, i = 1, ....., m p∑ ν i ∇h i (x ∗ ) = 0 i=1 For differentiable f o and f i , ∀i, the KKT conditions are necessary for the optimality, and for the convex primal problem, the KKT conditions are also sufficient for the points to be primal and dual optimal [13]. In summary, for any convex optimization problem with differentiable objective and constraint functions, any points that satisfy the KKT conditions are primal and dual optimal. The dual decomposition technique has been widely used to solve the optimization problems and is appropriate when the optimization problem has a coupling constraint such that, when relaxed, the problem decouples into several sub-problems. The idea behind the technique is to decompose the original complicated problem into solvable simple sub-problems. The KKT conditions are the necessary and sufficient conditions for the convex optimization problems. However it may not be always possible to find the closed form solution from the KKT conditions. Thus, other promising algorithms have been developed for the constrained convex optimization problems for example the interior point methods and the sub-gradient methods [13]. Interior point method is a search algorithm which could be adopted to transform the constrained optimization problem into simplified unconstrained problems. For non differentiable objective functions, the sub-gradient method is used. The sub-gradient method is generally used with decomposition techniques to solve the large optimization problems. 8
Page 1 and 2: Resource Allocation in OFDM Based W
Page 3 and 4: Dedications: To my family
Page 5 and 6: Acknowledgment I am greatly thankfu
Page 7 and 8: Contents 2.3.3 Proposed Low-Complex
Page 9 and 10: Abstract The combination of relay t
Page 11 and 12: List of Tables 3.1 Complexity compa
Page 13 and 14: List of Figures 4.1 System model fo
Page 15 and 16: List of Acronyms IEEE MU RS AWGN OW
Page 17 and 18: 1.1 Overview of Relay Networks wire
Page 19 and 20: 1.2 Overview of OFDM Figure 1.2: Th
Page 21: 1.3 Basics of the Optimization Theo
Page 25 and 26: 1.4 Resource Allocation Problem is
Page 27 and 28: 1.5 Contribution and Organization o
Page 29 and 30: 1.5 Contribution and Organization o
Page 31 and 32: 2.1 Introduction allocated to a nod
Page 33 and 34: 2.1 Introduction The main contribut
Page 35 and 36: 2.2 System Model where h m,k denote
Page 37 and 38: 2.3 Resource Allocation Schemes opt
Page 39 and 40: 2.3 Resource Allocation Schemes ( )
Page 41 and 42: 2.3 Resource Allocation Schemes and
Page 43 and 44: 2.3 Resource Allocation Schemes tha
Page 45 and 46: 2.3 Resource Allocation Schemes 2.3
Page 47 and 48: 2.4 Generalization to Multiple Rela
Page 49 and 50: 2.4 Generalization to Multiple Rela
Page 51 and 52: 2.5 Simulation Results • OptSol/J
Page 53 and 54: 2.5 Simulation Results 0.9 0.85 0.8
Page 55 and 56: 2.6 Summary 2.6 2.4 2.2 2 Rate (bit
Page 57 and 58: 2.6 Summary 2.4 2.2 2 1.8 N=4 N=3 N
Page 59 and 60: 3.1 Introduction A new technology c
Page 61 and 62: 3.1 Introduction • Broadcast Phas
Page 63 and 64: 3.2 System Model MU MU 3 8 A 1 2 7
Page 65 and 66: 3.4 Joint Resource Allocation Schem
Page 67 and 68: 3.4 Joint Resource Allocation Schem
Page 69 and 70: 3.5 Step-Wise Low-Complexity Resour
Page 71 and 72: 3.6 Simulation Results and SNR B m,
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3.6 Simulation Results 3 2.5 2 Uppe
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3.7 Summary 1.8 1.6 1.4 JntOpt SubO
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4.1 Introduction • Non-orthogonal
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4.3 Cooperative Non-Orthogonal Tran
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4.4 Optimization under Orthogonal T
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4.4 Optimization under Orthogonal T
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4.5 Simulations Then, the over all
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4.5 Simulations 40 30 20 Sum−ρ S
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4.6 Summary 0.9 0.8 0.7 JPSC Pow EP
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Chapter 5 Lifetime Maximization in
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5.1 Introduction Figure 5.2: Linear
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5.2 System Model antenna that canno
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5.4 Lifetime Maximization Scheme 5.
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5.4 Lifetime Maximization Scheme wh
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5.5 Simulation Results 150 OPT−SO
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5.6 Summary Lifetime (S) 150 100 50
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Chapter 6 Resource Allocation in Co
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7. Conclusions were considered for
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Bibliography [12] A. Peled, and A.
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Bibliography [36] Z. Luo and S. Zha
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Bibliography [61] D. H. N. Nguyen,
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Bibliography [84] A. Goldsmith, S.
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Bibliography [105] Y. Li, W. Wang,
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Appendix A Derivations of Closed-Fo
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B. Derivations of the Optimal Power
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