# Primal Decomposition-Based Method for Weighted Sum-Rate Maximization in Downlink OFDMA Systems

- Chathuranga Weeraddana
^{1}Email author, - Marian Codreanu
^{1}, - Wei Li
^{1}and - Matti Latva-Aho
^{1}

**2010**:324780

https://doi.org/10.1155/2010/324780

© Chathuranga Weeraddana et al. 2010

**Received: **19 September 2009

**Accepted: **8 March 2010

**Published: **18 April 2010

## Abstract

We consider the weighted sum-rate maximization problem in downlink Orthogonal Frequency Division Multiple Access (OFDMA) systems. Motivated by the increasing popularity of OFDMA in future wireless technologies, a low complexity suboptimal resource allocation algorithm is obtained for joint optimization of multiuser subcarrier assignment and power allocation. The algorithm is based on an approximated primal decomposition-based method, which is inspired from exact primal decomposition techniques. The original nonconvex optimization problem is divided into two subproblems which can be solved independently. Numerical results are provided to compare the performance of the proposed algorithm to Lagrange relaxation based suboptimal methods as well as to optimal exhaustive search-based method. Despite its reduced computational complexity, the proposed algorithm provides close-to-optimal performance.

## Keywords

## 1. Introduction

Orthogonal Frequency Division Multiple Access (OFDMA) plays a major role in the physical layer specifications of future wireless technologies (e.g., 3G-LTE, WIMAX, IMT-A) [1–4]. In OFDMA systems, the transmit power and subcarriers are dynamically assigned to the users, based on their channel state information (CSI) to optimize a certain performance criteria [5–22]. In general, this process requires solving combinatorial optimization problems. Thus, the existing convex optimization techniques cannot be directly applied, and the complexity of the problem increases exponentially with the number of subcarriers.

Two main radio resource allocation (RRA) problems have been addressed in the literature. The first ones, consist of maximizing an increasing function of the user rates [5–13] subject to different power constraints, whilst the second ones consist of minimizing the transmit power subject to constraints on the minimum user rates [13–17]. A suboptimal greedy method for maximizing the smallest rate among the users has been proposed in [5]. A branch-and-bound based algorithm for sum-rate maximization has been proposed in [6]. However, in practice the branch-and-bound method is still too computationally heavy for finding the global solution [23]. Computationally efficient algorithms for maximizing the sum-rate have been developed in [7, 8]. A suboptimal method for characterizing the achievable rate region of the two-users frequency division multiple access (FDMA) channel have been presented in [10]. The general weighted sum-rate maximization problem has been used in [9] to characterize FDMA capacity region for a broadcast channel. Due to the nontractability of the original problem, a modified convex problem formulation, FDMA-time division multiple access (TDMA) was proposed (i.e., time sharing among users). Authors also considered algorithms to obtain optimal and suboptimal solutions to a particular variation of the original problem where the total power is evenly divided among the used set of subcarriers. Lagrangian relaxation-based approaches to obtain suboptimal algorithms for the weighted sum-power minimization problem has been introduced in [13, 14]. A greedy algorithm is proposed in [15] to obtain and approximate solutions for the same problem. Recently, a Lagrangian relaxation-based method has been proposed in [13] for the weighted sum-rate maximization problem. A bisection search method was used to update the dual variable until the algorithm converges. Due to the nonconvexity of the optimization problem the optimality of the algorithm is not guaranteed.

In this paper, we propose an alternative method based on the primal decomposition technique [24, 25]. Using numerical simulations, its performance is compared to Lagrangian relaxation based algorithm [13] as well as to the optimal exhaustive search algorithm. Numerical results show that the proposed algorithm converges very fast. Although, the optimality of the final value cannot be guaranteed due to the nonconvexity of the problem, the simulations show that rate-region achieved by the proposed algorithm exactly matches with the one obtained using optimal exhaustive search algorithm.

The rest of the paper is organized as follows. In Section 2 we present the system model and problem formulation. The proposed algorithm is presented in Section 3 and convergence properties are discussed in Section 4. Section 5 compares the complexity of the proposed algorithm to Lagrangian relaxation-based algorithm in [13] as well as to the optimal exhaustive search based method. The numerical simulation results are presented in Sections 6, and 7 concludes our paper.

Notations

denotes the absolute value of complex number . represents the transpose of vector and denotes the th standard unit vector. stands for circularly symmetric complex Gaussian distribution with mean , variance per dimension. For any real number , denotes .

## 2. System Model and Problem Formulation

where the optimization variables are . The constraints associated with orthogonal subcarrier allocations in problem (2) have been dropped out and the interference among users allocated to the same subcarrier is reflected in the objective.

Here we can make several observations. First, any solution of problem (4) is such that the second constraint in problem (2) is automatically satisfied, for reasons that will be explained in the beginning of Section 3.2. In other words, any solution of problem (4) is feasible for problem (2). Moreover, at any of these solutions the objective function of problem (4) will be exactly the same as the objective function of problem (2). Based on these observations it can be concluded that any solution of the auxiliary problem (4) is a solution for the original problem (2) as well.

The original problem (2) is combinatorial and it requires exponential complexity to find a global optimum. Although problem (4) is still nonconvex it is noncombinatorial. Thus, in the following, we focus on solving problem (4) instead of solving the original problem (2). A similar approach has been used in [7] to solve the (nonweighted) sum-rate maximization problem, that is, for the particular case , . However, the methods proposed there do not apply to the general case of arbitrary weights, for reasons that will become clear in Section 3.3. Due to the nonconvexity of problem (4) finding the global optimum is intractable. Thus, an approximative method inspired from the primal decomposition technique is presented in Section 3.

## 3. Approximated Primal Decomposition- (APD-) Based Algorithm

### 3.1. Primal Decomposition

where variables are and represents the optimal value of subproblem (6) for fixed .

### 3.2. Algorithm Derivation

*maximize*a convex function.). However, its objective function is convex with respect to (w.r.t.) optimization variables , its feasible set is a nonempty convex polyhedral set (i.e., a simplex [27]) and its objective is bounded above on . Thus, by following the approach of [7, Section ], from [28, Corollary ] (If a convex function is bounded above on a convex set , then the maximum of relative to is attained at one of the finitely many extreme points of .) it follows that the solutions of subproblems (6) must be achieved at one of the vertices of the polyhedral set . Consequently, the solutions of the subproblems can be expressed as

Solution (8) confirms that, even though in subproblems (6) all users are allowed to use all subcarriers, the optimal power allocation consists of allocating only one user per subcarrier. This guarantees that solution (8) is feasible for the original problem (2).

We note that the index depends on according to (9) and the function is the pointwise maximum of a set of concave functions. Therefore is not a concave function w.r.t. [27] in general. Thus, standard convex optimization tools (e.g., subgradient-based methods) cannot be directly applied to solve master problem (7).

where denotes the feasible set of the master problem (7). The lower bound is concave w.r.t. and the solution of the approximated master problem can be found by multilevel waterfilling algorithm [9]. The resulting solution is used as subcarrier power allocation for the next iteration. The proposed algorithm can be summarized as follows.

Algorithm 1.

- (1)
- (2)
- (3)
- (4)

### 3.3. Particularization to the Sum-Rate Maximization

The problem of the sum-rate maximization (i.e.,
for all
) in downlink OFDMA systems is solved in [7, Section
]. The solution method is exactly equivalent to only one iteration of the APD algorithm. Unlike the general weighted sum-rate maximization, in which user weights
's are different, in the sum-rate maximization (i.e.,
for all
) the index
w*ill not depend* on
according to (9). Thus, by using (10) and (11) the function
can be found as
which is concave w.r.t.
(recall that the function
is not concave w.r.t.
when the user weights
's are different). As a result, the inequality given in (12) holds with equality and solving problem (7) gives the optimal subcarrier power allocation [7, Section
].

## 4. Convergence Behavior and Exit Criterion

In this section, we start by investigating the monotonicity of the proposed algorithm. Then we provide a specific exit criterion which certifies that algorithm converged to a fixed power and subcarrier allocation followed by a simple graphical illustration.

### 4.1. Monotonic Behavior

The following theorem states the monotonic behavior of the proposed algorithm.

Theorem 1.

that is, the proposed APD method is an ascent algorithm.

Proof.

where the first inequality follows from (13), the second one follows trivially from the maximization over the users, and the last equality follows from (11) and (10), respectively.

### 4.2. Exit Criterion

The exit criterion for such ascent algorithm is typically chosen heuristically, for example, the increasing in the objective between two successive iterations is below a certain predefined threshold. However, for the proposed algorithm we are able to find an exit criterion which certifies that algorithm converged to a fixed power and subcarrier allocation and further improvement is not possible. This is described by the following theorem.

Theorem 2.

That is, the algorithm converges to a fixed power and subcarrier allocation.

Proof.

Since 's are continuous random variables, the probability to have multiple solutions for (9) is zero. Thus, in the following we assume that given by (9) is unique (Equation (9) has multiple solutions if and only if for some . When we assign any arbitrarily user index.).

Note that the objective function of (13) is strictly concave. Thus it has a unique solution [27]. Therefore, for all , implies that . Since (9) has a unique solution as well, further implies that . Since we have shown that implies that , item ( ) follows directly by induction. Furthermore, item ( ) follows from item ( ) by the uniqueness of the solution of problem (13). Finally, item ( ) follows trivially from item ( ).

Thus the exit criterion checks if the subcarrier allocation between two successive iterations remains unchanged. Such point is a local optimum (possible global) in the sense that the objective cannot be increased by changing the power allocation or subcarrier allocation only.

## 5. Complexity Analysis

In this section, we analyze and compare the computational complexity of the proposed APD algorithm to Lagrangian relaxation-based algorithm [13] as well as to the optimal exhaustive search algorithm. With users and subcarriers, altogether we have user-subcarrier combinations. Therefore finding optimal subcarrier and power assignment requires searches. Combined with multilevel waterfilling at each instance of subcarrier assignment, operations are required to find the solution. The algorithm proposed in [13] for the weighted sum-rate maximization problem requires operations to obtain a suboptimal solution. The proposed APD algorithm described in Section 3 requires operations in step ( ) and operations (This is the number of operations required in ordering.) in step ( ). In practice, it is reasonable to assume that (The assumption is reasonable since the number of users simultaneously serviced by the system can be very large. For example, in a Wi-Max system can be up to [29]. However, the value of will not become very large (in a WiMax system at most).). Therefore the complexity of the APD algorithm can be approximated by .

## 6. Numerical Results

The performance of the proposed APD algorithm is compared to the dual decomposition-based algorithm proposed in [13], denoted as WSRmax, as well as to the optimal algorithm based on exhaustive search. The WSRMax algorithm uses a bisection search method to update the dual variable [13, Section IV]. For initializing the bisection search interval, , we exploit the fact that the subgradient of the dual function can be analytically computed. Since the dual function is convex, the sign of its subgradients changes as we pass through the minimum point of the dual function [13, equation ( )]. Therefore, we use a grid search (with step size ) to identify the interval in which the subgradient of dual function changes its sign, and it is used as initial bisection search interval. Thus, the interval is guaranteed to contain the optimal value of the the dual function and the width of the initial interval is one, that is, . The proposed APD algorithm is initialized by allocating equal power to all subcarriers.

the average normalized weighted sum-rate deviation, where is the optimal weighted sum-rate value obtained using optimal exhaustive search, is the estimated objective value from either the APD algorithm or the WSRMax algorithm, and expectation is taken w.r.t. channel realization. An OFDMA system with subcarriers and a uniform power delay profile with channel taps is considered. We assume , , and define SNR per subcarrier as .

the probability of missing the global optimal, where is a small number which quantifies the maximum admissible deviation between and . It is considered that the global optimum is missed if is more than away from .

In the following we compare the behavior of the APD and the WSRmax algorithm for large number of subcarriers and users. Since, for large number of users and subcarriers the complexity of evaluating is prohibitively high, the metrics defined in (20) and (21) are not used. The behavior of the APD algorithm is compared with that of the WSRmax algorithm.

## 7. Conclusions

A joint subcarrier and power allocation algorithm which is inspired from primal decomposition techniques has been proposed for maximizing the weighted sum-rate in multiuser OFDMA downlink systems. Although the original problem is nonconvex, the proposed APD algorithm finds fast a suboptimal, but still very close-to-optimal solution with very high probability (i.e., more than of the time). Unlike the recently proposed WSRMax algorithm [13], the APD algorithm requires no additional precautions in the initialization, and convergence to a suboptimal solution is possible within a very small number of iterations. Although the proposed primal decomposition-based solution method does not rely on zero duality gap for proving the optimality in the case of large number of subcarriers, our computational experience with larger number of subcarriers suggests that the proposed APD algorithm is capable of finding the same solution as the WSRmax algorithm (which is asymptotically optimal when the number of carriers grows to ) even with very few iterations.

## Declarations

### Acknowledgment

This research was supported by the Finnish Funding Agency for Technology and Innovation (Tekes), Academy of Finland, Nokia, Nokia Siemens Networks, Elektrobit, Graduate School in Electronics, Telecommunications and Automation (GETA) Foundations, Nokia Foundation, and Tauno Tönning Foundation.

## Authors’ Affiliations

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