Enhanced Inter-Cell Interference Coordination in LTE HetNets
4.3 Interior Point MTS Optimization Algorithm
4.3.1 Basic Idea of the Interior Point Methods
Consider an inequality constrained problem, Problem 2.
minimize f (p)
subject to hi(p) ≤ 0, i = 1, ..., m (4.3) where p is the solution vector of dimension n, and m is the number of constraints.
This problem could be approximately formulated as an unconstrained prob-lem. First, make the inequality constraints implicit in the objective. The inequal-ity constraints are eliminated by placing in the indicator function,
Problem 3.
where I− : R → R is the indicator function for the nonpositive reals, I−(u) =
( 0, u ≤ 0
∞, u > 0 (4.5)
The Problem 3 has no inequality constraints, but its objective function is not differential, so Newton’s method cannot be applied. Therefore, the logarithmic barrier is introduced to handle this situation. The basic idea of the barrier method is to approximate the indicator function I− by the function
Iˆ−(u) = −(1/t)log(−u), dom ˆI− = −R++ (4.6)
4.3 Interior Point MTS Optimization Algorithm
(a) Coverage Area with P7 (b) Coverage Area with P6
(c) Coverage Area with P5 (d) Coverage Area with P4
(e) Coverage Area with P3 (f) Coverage Area with P2
where t > 0 is a parameter that sets the accuracy of the approximation. Like I−, the function ˆI− is convex and nondecreasing, and takes on the value ∞ for u > 0. Unlike ˆI−, however, ˆI− is differentiable and closed: the approximation ˆI−, for several values of t. As t increases, the approximation becomes more accurate.
Substituting ˆI− for I− in the Problem 3 gives the approximation, Problem 4.
minimize f (p) +
m
X
i=1
−(1/t)log(−fi(p)) (4.7) The objective here is convex, since −(1/t)log(−u) is convex and increasing in u, and differentiable. Assuming an appropriate closedness condition holds, Newton’s method can be used to solve it.
One advantage of choosing the logarithmic penalty is that the approximation becomes more accurate as t increases. When t → ∞, the optimal solution of Problem 4 converges to the optimal solution of the original Problem 2. Figure 4.4 shows the approximation accuracy as t increases. Solving Problem 2 is thus equal to solving a series of Problem 4 with increasing t.
Although the basic idea of the interior point methods is simple, it still requires careful considerations to design an suitable mechanism to solve the subproblems at hand. Design interior point methods is still an active research topic.
4.3.2 Detailed Description of Interior Point MTS Opti-mization Algorithm
Although the interior point methods were already proposed for several tens of years, the detailed design and implementation of the interior point methods are still active research areas. It is hard to provide reliable design guidelines that are suitable for all types of problems. Therefore, it is important to customize
4.3 Interior Point MTS Optimization Algorithm
−5 −4 −3 −2 −1 0 1
−5 0 5 10
hi(p)
−(1/t)log(−h i(p))
t = 0.5
t = 1
t = 2
Figure 4.4: Effect of t on approximation accuracy. As t increase, the approxima-tion becomes more accurate.
the solution to reflect the suitability for the problem at hand. In this section, we present the detailed description of the customized interior point algorithm to solve the MTS problem. First, we relax some constraints of the Problem ?? to get the relaxed problem, therefore, the interior point can be used on the MTS optimization. Second, the integer rounding algorithm is proposed to mapping the solution of the relaxed problem into the original problem.
4.3.2.1 The Relaxed Problem
Before solving Problem ??, the first step is to relax some constraints to make the problem solvable within a reasonable amount of time. The second step is to convert the solution obtained in the first step into a feasible solution for Problem
??.
Problem ?? is a mixed-integer nonlinear programming (NLP) problem and is considered to be an NP-hard problem in the literature. Therefore, we consider
an NLP problem by ignoring the association constraints and relaxing the integer constraints in Problem 1. The new NLP problem is shown in the following:
Problem 5 (Relaxed MTS Assignment Problem). Maximise the aggregate log-capacity of UEs in the system, i.e.,
maxN,x,y
2. Macro and Pico Subframes Constraints:
X
Because the association constraints are ignored, each UE can associate with both MBS and PBS at the same time. The second constraints in Problem 1 are no longer integer constraints, so we can take non-integer values into consideration.
The relaxed problem is therefore an NLP problem.
4.3 Interior Point MTS Optimization Algorithm
4.3.2.2 Algorithm for The Relaxed MTS Assignment Problem
Since the constraints of Problem ?? are all linear, the feasible region is a con-vex set, With the concave objective function and the concon-vex feasible region, the relaxed MTS problem sould be a convex optimization problem.
We solve this problem by a primal-dual interior-point algorithm [98]. First, we show the Lagrangian of the relaxed MTS Assignment Problem in the following:
L(N, x, y, λ, ν, α, β) = X where the variables λ, ν, α and β’s are dual variables and also called Lagrange multipliers. For notation simplicity, we use p to denotes the vector of all Lagrange multipliers, i.e., p = (λ, ν, α, β). In the similar way, we use z to denotes the vector of all primal variables, i.e., z = (N, x, y). Therefore, we can rewrite the problem in the following expression:
minimize f (z)
subject to hi(z) ≤ 0, i = 1, ..., c (4.9) where c is the number of all constraints in Problem ??, and the function f (z)
is equal to the objective function of Problem ?? multiply −1. The Lagrangian is
where piis the ith element in vector p, i = 1, ..., c, are the Lagrange multipliers.
There exist primal optimal z∗ and dual optimal p∗which satisfy the Karush-Kuhn-Tucker (KKT) conditions.
We denote the value of the objective function with the optimal solution z∗ as f∗. Combining (4.10) and (4.11), it is found that
which is the logarithmic penalty function for the problem (4.9). The gradient and Hessian of the logarithmic penalty function φ are given by
4.3 Interior Point MTS Optimization Algorithm
which has the same minimizer as (4.7) does. We denote the solution of (4.15) as z∗(t). The first order necessary condition for (4.15) is
0 = ∆f (z∗(t)) + (1/t)∆φ(z∗(t))
where the second equality comes from (4.14). From (4.11) and (4.16), the dual feasible solution is defined as
p∗i(t) = −1/t
hi(z∗(t)), i = 1, ..., c. (4.17) Since t > 0 and hi(z∗) < 0, i = 1, ..., c, it is guaranteed that p∗i ≻ 0, which meets the requirements shown in (4.11).
Applying (4.17) to (4.10),
L(z∗(t), p∗(t)) = f (z∗(t)) +
where the last inequality comes from (4.12). Therefore,
f (z∗(t)) − f∗ ≤ c/t (4.19)
(4.19) means that the primal feasible solution z∗(t) is no more than c/t-suboptimal. The term c/t is called the duality gap. When t → ∞, z∗(t) converges to the optimal solution of the original problem (4.9).
With (4.17), the Karush-Kuhn-Tucker (KKT) conditions shown in (4.11) can be written as rt(z, p) = 0, where the residual rt(z, p) is defined as for t > 0. Here h(z) is the constrain matrix and its derivative matrix Dh(z) are defined as
4.3 Interior Point MTS Optimization Algorithm
If (z, p) satisfies rt(z, p) = 0, then z = z∗(t) and p = p∗(t) are the primal feasible and dual feasible solutions, respectively, and it lead to m/t-suboptimal solution. Next, we apply the Newton method in the following steps to solve the nonlinear equations rt(z, p) = 0, for fixed t, at a point (z, p) which satisfies h(z) ≺ 0, p ≻ 0. The Newton step ∆x = (∆z, ∆p) is ∆x = −Drt(x)−1rt(x). In
Therefore, we have the primal-dual search direction (∆z, ∆p), and we find a suitable step length s by performing a backtracking line search method [99]. We update the next primal-dual point as x = (z + s∆z, p + s∆p). When a pre-defined precision is met, the iteration stops. Boyd and Vandenberghe propose a surrogate duality gap [98], which is defined as
η(z, p) = −h(z)ˆ Tλ (4.24)
The surrogate duality gap is adopted to the algorithm to choose the approx-imation parameter t and to decide the termination condition. The overview of the first step algorithm is shown in Algorithm2.
4.3.2.3 Integer Rounding for Relaxed MTS Assignment
In the second step of the proposed algorithm, the goal is to convert the solution obtained in the first step into a feasible solution for the MTS assignment problem.
In the relaxed problem, each UE can associate with both an MBS and a PBS;
Algorithm 2 Interior point relaxed MTS optimization algorithm.
Input:initial primal point z, p ≻ 0, µ > 1, α, β, ǫf easible > 0, ǫ > 0 Output:optimal solution ˆz for the relaxed problem
repeat
1. Determine t.
t = µc/ˆη
2. Compute the primal-dual search direction ∆x = (∆z, ∆p).
"
3. Determine s by line search.
smax= min {1, min {−λi/∆λ|∆λi < 0}}
s = 0.99smax
if krt(z+, p+)k2 ≥ (1 − αs) krt(z, p)k2 then s = β × s end if
4. Update the primal-dual point.
z+ = z + s∆z.
4.3 Interior Point MTS Optimization Algorithm
however, it can associate with only one of them in the original problem. We must therefore determine the association of each UE to either an MBS or a PBS.
We compute the throughput of each UE from MBS and PBS according to the solution obtained from the first step. The calculation is shown as follows:
Rmacrou = If Rmacrou ≥ Rpicou , UE u would associate with MBS, otherwise, it associates with PBS. Denote U⋆m and U⋆p are the set of UEs which associate MBS m and PBS p after the procedure, respectively.
Next, we need to convert Nml solved in the first step into an integer. The goal of ABS is to protect PUEs, and we leave this property in the proposed algorithm.
Therefore, we convert Nml into the greatest integer lower than Nml . The integer rounding procedure for Nml is shown as follows:
Nm⋆l =
The value of A⋆lp is dependent on Nm⋆l, and the calculation is according to (4.2).
The last step is to find the final value of x⋆lu and yu⋆l. For each MBS m, for all u ∈ U⋆m, the x⋆lu is
x⋆lu = xˆluNm⋆l
Xml (4.27)
where Xml = P
yˆpl. The overview of the second step of the algorithm is shown in Algorithm 3.
4.3.2.4 Proof of Optimality
Theorem 4. A limit point of the sequence {z∗(t)(k)} generated by the interior point relaxed MTS optimisation algorithm is a global minimum of the original optimisation problem (4.9). contradiction occurs. Therefore, it is proved that ¯p∗ is a global minimum of the original optimisation problem.
4.3.2.5 Complexity Analysis
The primal-dual interior point method requires a total of O(√n) iterations [100], where n is the size of the problem. In the relaxed MTS optimisation problem, n is equal to (M × P × L × N). While considering the number of computations in
4.3 Interior Point MTS Optimization Algorithm
Algorithm 3 Round Relaxed MTS Algorithm: Algorithm for Integer Rounding of Output of Algorithm 2
1. UE Association: For all u ∈ U, perform the following steps:
a) Compute the throughput it get From MBS and PBS according to solu-tion obtained from the step 1 as follows:
Rmacrou =
2. MTS Rounding: Compute integral Nm∗l as follows:
Nm⋆l =
exact arithmetic, solving the Newton step requires at most O(n3) computations, although typically the number of computations would be O(n) or O(n2) in a problem in which the Hessian matrix is sparse [101].
4.3.2.6 Summary
In this chapter, we describe the algorithm to solve the MTS assignment problem, and the algorithm is divided into two steps: Relaxed MTS assignment algorithm and integer rounding for relaxed MTS algorithm. The first step is based on inte-rior point method, and we introduce the basic idea of the inteinte-rior point method.
We provide the detailed description of the relaxed MTS assignment algorithm.
The algorithm solves the problem iteratively toward to the desired approximation accuracy. Each subproblem is solved by Newton’s method which in known to be efficient. In the second step, we round the solution obtained from the step one to integers, because the solution of MTS assignment problem is discrete. Besides, we provide the proof of optimality and convexity of the first step algorithm. The optimality proof shows that the algorithm would coverage to the optimal solution and we show that the algorithm has linear convexity. Therefore, the algorithm is shown to be effective and efficient in the desired problem.