A Stochastic Multiobjective Optimization Framework for Wireless Sensor Networks

The layered architecture approach has achieved great success in traditional wired network design by dividing the whole architecture into several modules, called layers, each of which performs a separate functionality. As each layer design only needs some interface variables from the layer below, the complexity of other layers can be hidden. The layered architecture approach suggests that the network design can be scalable, evolvable, and implementable. However, it may have limitations in improvement of efficiency and fairness, and suffer potential risks of manageability [1], which motivates the optimization of network design. Chiang et al. [1, 2] propose an optimization decomposition technique to systematically understand the network architecture, known as "layering as optimization decomposition". They model the network as an optimization problem and decompose the problem into many subproblems. They classify the decompositions into vertical decomposition and horizontal decomposition. Vertical decomposition layers the network architecture into several modules and horizontal decomposition provides distributed algorithms to fulfill the functionality within the modules. According to the requirements of the applications, the decomposition may be different, yielding different layers and distributed algorithms. There are usually two steps in the process of layering as optimization decomposition: ( ) modeling the network problem as a specific NUM problem, and ( ) exploring the alternative decompositions to design different modules and distributed algorithms. Most existing efforts have been put to the second step and simply assume that the network problems can be modeled by a unified utility function at the first step [3‐6]. However, not all network problems can be modeled by a unified utility function in a tractable way since there may exist many objectives to be achieved, such as guaranteeing fairness, maximizing throughput, reducing packet dropping and delay, prolonging the network lifetime, and so forth. It may not be possible to integrate all these objectives into a single unified utility function, that is, network problems should be formulated as multiobjective optimization problems.

While the performance of the network can be greatly enhanced by adopting the NUM approach, the corresponding cost of algorithm implementation also increases. As we usually design and implement an algorithm for a specific application from scratch, the implementation can hardly be transplantable to other applications. This is especially aggravated in wireless sensor networks (WSNs) due to the application-oriented and infrastructureless nature of these networks. For example, if we design an efficient algorithm for events monitoring, the network lifetime is the main concern and the propagation delay can be tolerant, but it is difficult to apply such algorithm to online query applications, where the query delay is the primary objective.

In this paper, we utilize the concept of multiobjective optimization and provide a general framework for a specific class of applications in WSNs. It is well known that the TCP/IP reference model, one of the most popular layered architectures, divides the whole architecture into five layers (modules), and each layer only communicates with the layers next to it, while recent work on the NUM approach divides the architectures according to applications. We first list all the constraints and objectives which the applications may have. Then the network architecture is divided into modules, each of which has several interface variables with other modules. In this way, we can inherit the advantages of both the layered architectures (as we have fixed modules) and the NUM approach (as each module can communicate with other modules). We illustrate this in Figure 1, in which and are the interface variables (see Section 3 for detailed definition of and ), and is the objective vector function in each module . We transform the objectives of the network into specific modules through the interface variables. Each sensor optimizes its own objective vector function to achieve the global optimal solution of the whole network. In this way, for different requirements from the network, we do not redesign the framework, that is, the modules and interfaces in Figure 1 can be kept unchanged. We only need to introduce multiobjective methods to optimize the vector function in each module. This will greatly simplify network design for WSNs.

In WSNs, some parameters (e.g., the topology of the networks or channel condition) are time-varying. In [7], Lee et al. demonstrated that the state of the network can be more efficiently utilized to improve the performance of the network (e.g., increasing the throughput and reducing packet delay), by appropriately exploiting the variability of the time-varying channels. Also there are measurement errors in the implementation of distributed algorithms, such as the noisy feedback [8] or lossy links [9]. Therefore, we also characterize these random factors in our model. Our contributions in this paper are summarized as follows.

The remainder of the paper is organized as follows: in Section 2, we discuss related work regarding the NUM problem and stochastic network utility maximization (SNUM). We formulate a general mathematical model and design a distributed algorithm to solve the problem in Section 3, and the stability of the algorithm is also discussed. We provide three paradigms in Section 4 to demonstrate the general framework for different applications. Simulation results are given in Section 5. We conclude the paper in Section 6.

There are several research works in the literature studying the NUM problem [4, 11‐14]. Kelly et al. were the first to propose the optimization approach, which provides a mathematical foundation for NUM problem [11]. In [12], Chiang adopted the NUM approach to obtain a cross-layer design including the physical layer and the transport layer. Zhu et al. [13] considered the energy model in the cross-layer design. In [1], Chiang et al. provided a mathematical theory of network architectures. Wang et al. studied joint interference-aware routing and TDMA link scheduling to improve the throughput in multihop wireless networks [15]. Zhang et al. [8] elaborated on the impact of the feedback in the implementation of distributed NUM algorithms. Since feedback is often collected using error-prone measurement mechanisms, for example, biased estimator or unbiased estimator, they adopted the knowledge of stochastic approximation and proved stability of the algorithms of single-time scale and two-time scale. Lee et al. utilized the variation of channels to guide power and rate control in cross-layer design [7]. In this paper, we formulate a more general mathematical model by considering stochastic multiple objectives in objective functions. We apply our approach to rate allocation problem in multipath WSNs with time-varying channels. Rate allocation is a fundamental problem and has been extensively investigated [16‐19]. Low and Lapsley [16] first introduced the Lagrange dual method to decompose the problem and proposed two algorithms under synchronous and asynchronous scenarios. A multipath formulation for rate control in multi-cast networks was proposed in [20], and three distributed algorithms were proposed to solve the problem. The goal is to maximize the aggregate utility. In [6], Srinivasan et al. considered two objectives: utility maximization and guaranteeing prespecified network lifetime for multipath wireless ad hoc networks. In [10], Zhu et al. also focused on the network lifetime and application performance (utility), and employed the linear weighted method from the multiobjective optimization to transform these two objective functions into a single one which was named to the utility-lifetime tradeoff function.

Throughout the paper, we will denote sets by capital letters, variables by lowercase letters, vectors by bold lowercase letters, and matrices by bold capital letters. For a vector , we denote its th component by and its transpose by . We use capital letters for both the sets and the cardinality of sets.

Consider sensing nodes and sink nodes in the region of interest. Let be a probability space with a -algebra of random events, and have a finite set with the corresponding probability . There are objective functions , defined on the subset of a Hilbert space. Let , . Then is the objective vector function of sensing node . Let be the column vector of variables of sensing node , , and a column vector function. We can formulate the primal problem (PP) as follows:

The objective function, , may be a coupled one. In order to design a distributed algorithm, we introduce auxiliary variable to decouple it. Assume that the node set associated with coupled variables of is . Let , which denotes the node set associated with coupled variables of of sensing node , then the decoupled primal problem (DPP) can be given by

where is the th decoupled objective function, and is the corresponding vector of auxiliary variables.

In our formulation, the objective functions are deterministic, taking the advantage that each sensing node can be obtained from the network. The constraint set contains the random factors of the networks, such as message exchange, and environmental effect. If we know the distribution, , , of , we can transform the problem into a deterministic one, by calculating the expectation. However, in WSNs, there is often no prior knowledge about the randomness from the networks themselves and the environmental effect. Therefore, we develop an algorithm without this prior knowledge, which can be achieved by the stochastic quasigradient method [7].

To decompose the problem, we take Lagrange dual approach. The Lagrange function [21] of (3) is given by

where is the objective vector function of sensing node . It is a formal expression which can be transformed into different objective functions for different applications.

We call and decoupled prices ( is used to decouple the coupling of variables and is used to decouple the coupling of objective functions). Since (4) is separable, we exploit the decomposable structure of Lagrangian function and decompose the problem into subproblems. Maximization is achieved in each sensing node , , with the knowledge of local variables ) and the current state , by solving the following optimization problem .

At iteration , each sensing node updates its resource variables and auxiliary variables according to

We proceed to solve the dual problem. Let . Then the dual problem ( ) is given by

At iteration , each sensing node can acquire the state of random variables . The stochastic quasigradient method only needs this current state information of the system and utilizes it to form the stochastic subgradients of at iteration . For the dual problem, , prices are updated according to

where and are the stochastic quasigradients of .

In our algorithm, we set

where is the state of at iteration .

We summarize our algorithm for the general formulation of stochastic multiobjective optimization problem (ASMOP) in the Algorithm 1.

( ) Price update algorithm: At times , decoupled prices are updated according to

           

         

( ) Sensor node 's Algorithm: At time , each sensing node updates its variables

according to

           

To prove that the algorithm can converge to the optimal solution of the primal problem, we make the following assumptions.

Theorem 1.

If (1) hold, then from an arbitrary point of , and , the sequence generated by (7), (9), and (10) converges. Every limit point of the sequence is primal-dual optimal.

Proof.

Let the sequences of iteration and be generated by (9) and (10), respectively. Then to guarantee the convergence of the algorithm, according to [7, 22], the current stepsize and quasigradients , and should be chosen such that

It can be seen that , , satisfy (14). From [22]; we know that and from (12) and (13) also satisfy (15) and (16).

From assumptions (1) and (2), the primal function is concave and the dual function is convex in and for a fixed . From (7), (9), (10), (11), (12), and (13), we can conclude that the sequence converges to the optimal solution by solving the dual problem [22]. As the primal problem is a convex optimization problem, there is no gap between the primal and dual problems. So the sequence generated by the algorithm is primal-dual optimal.

Remarks 3.

Because of multipath routing, the problem, , may not be strictly concave even if is strictly concave. This may lead to oscillation of the sequences generated by the algorithms. There are several ways to cope with this problem. For example, we can first add some augmented variables to and adopt the first-order Lagrangian method to solve it [23].

The main difference of our proposed approach is that we adopt the knowledge of multiobjective optimization and provide some potential interfaces for each layer. In this way, we can take the advantages of both the layered architectures and cross-layer design. In other words, we can implement different algorithms in each module according to specific applications. In Figure 1, and act as the interface variables between different modules and sensor nodes. Through and , the network architectures can be decomposed into different modules and each module fulfills corresponding functionality distributively. From (7), we can transform the multiple objectives of the whole network into the multiple objectives of each sensor node. Optimizing the objective vector function of each sensor node can achieve the global optimal solution. Therefore, it is very convenient to implement algorithms in each module to solve the objective vector function independently according to different requirements.

In the proposed general framework, in is a vector function and can be transformed into a single function according to different requirements. Therefore, solving is application-dependent, which provides the flexibility of solving a class of applications by the general framework.

In this section, we consider the rate allocation problem as an example and show how the general framework works. Rate allocation problem is a well-investigated problem [24], and has different requirements for different applications. There are usually three methods to cope with the requirements: ( ) Constraint Method [6], ( ) Linear Weighted Method [10], and ( ) Hierarchical Sequence Method. While these methods are extensively studied in existing works, we can integrate these methods together into the general framework. Hence, our approach can be applied to a class of applications with different background, which will offer significant convenience to the designers.

In this paper, we have proposed a general stochastic multiobjective optimization framework for WSNs. Our approach inherits advantages of both layered architectures and cross-layer design. Therefore, even the requirements and objectives are changed, it is not necessary to redesign the optimization framework but to have minor modifications of specific modules to meet the corresponding requirements. Although there may be uncertainty in WSNs, our approach can still achieve desired performance. In our future work, we will focus on investigating the general multiobjective optimization problem, instead of transforming the multiple objectives into a single one. We will study the distributed algorithms to optimize the objective vector function under some criteria, for example, Pareto optimality.