A Stochastic Multiobjective Optimization Framework for Wireless Sensor Networks

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Introduction
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: (1) modeling the network problem as a specific NUM problem, and (2) 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][4][5][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 n 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 λ and μ are the interface variables (see Section 3 for detailed definition of λ and μ), and O i is the objective vector function in each module i.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 O i 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 timevarying 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.
(1) We formulate a general multiobjective stochastic optimization problem for WSNs.We decompose the optimization problem through Lagrange dual decomposition and adopt the stochastic quasigradient method to solve the primal-dual problem.In other words, we transform the multiple objectives of WSNs into the multiple objectives of each individual sensor node.The global optimal solution can be obtained when each sensor node maximizes its own objective vector function.Therefore, our approach  provides a general framework for multiobjective optimization for WSNs.
(2) We study the stability of the algorithm by using the knowledge of stochastic programming, and show that our algorithm for stochastic multiobjective optimization problem (ASMOP) can converge to the optimal solution of the primal problem.
(3) We demonstrate how the general framework can be applied to different applications, by considering the rate allocation problem as an example.We introduce three multiobjective optimization methods: (1) constraint method, (2) linear weighted method, and (3) hierarchical sequence method.The three paradigms show that although different requirements may lead to different models [6,10], we can solve them in the general framework.
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.

Related Work
There are several research works in the literature studying the NUM problem [4,[11][12][13][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 crosslayer 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][17][18][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.

General Multiobjective Formulation and Solutions
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 x, we denote its ith component by x i and its transpose by x T .We use capital letters for both the sets and the cardinality of sets.
Consider S sensing nodes and N sink nodes in the region of interest.Let Ω be a probability space with a σalgebra F of random events, and have a finite set {τ m , m = 1, 2, . . ., M} with the corresponding probability p(τ m ) ≥ 0. There are n objective functions P (i) (x), i = 1, . . ., n, defined on the subset χ of a Hilbert space.Let P(x) = (P (1) (x), P (2) ) is the objective vector function of sensing node s.Let x s ∈ χ s be the column vector of variables of sensing node s, x = (x T 1 , . . ., x T S ) T , and g s (•) a column vector function.We can formulate the primal problem (PP) as follows: (1) The objective function, f (i)  s (x), may be a coupled one.In order to design a distributed algorithm, we introduce auxiliary variable y to decouple it.Assume that the node set associated with coupled variables of , which denotes the node set associated with coupled variables of P s (x) of sensing node s, then the decoupled primal problem (DPP) can be given by DPP : P x, y = P 1 x, y , . . ., P n x, y , where is the ith decoupled objective function, and y s = ({y ss } s ∈H (i) (s) ) 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, p(τ m ), m = 1, 2, . . ., M, of τ, 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 L x, y, λ, μ, τ where P s (x s , y s ) is the objective vector function of sensing node s.It is a formal expression which can be transformed into different objective functions for different applications.
We call λ and μ decoupled prices (λ is used to decouple the coupling of variables and μ is used to decouple the EURASIP Journal on Wireless Communications and Networking coupling of objective functions).Since ( 4) is separable, we exploit the decomposable structure of Lagrangian function and decompose the problem into S subproblems.Maximization is achieved in each sensing node s, s ∈ S, with the knowledge of local variables (x s , y s ) and the current state τ, by solving the following optimization problem DP s .DP s : max P s x s , y s − λ T g s (x s , τ) At iteration t, each sensing node s updates its resource variables x s and auxiliary variables y s according to We proceed to solve the dual problem.Let D(λ, μ, τ) = max x,y L(x, y, λ, μ, τ).Then the dual problem (DP) is given by min λ≥0,μ D λ, μ, τ . ( At iteration t, each sensing node can acquire the state of random variables τ.The stochastic quasigradient method only needs this current state information of the system and utilizes it to form the stochastic subgradients of D(λ, μ, τ) at iteration t.For the dual problem, DP, prices are updated according to where ϑ(t) and ν ss (t) are the stochastic quasigradients of D(λ, μ, τ).
In our algorithm, we set where τ(t) is the state of τ at iteration t.We summarize our algorithm for the general formulation of stochastic multiobjective optimization problem (ASMOP) in the Algorithm 1.
To prove that the algorithm can converge to the optimal solution of the primal problem, we make the following assumptions.
(1) f (i)  s (•) as well as the objective vector function P s (•) (which can be transformed into a single function in applications), s ∈ S, i = 1, 2, . . ., n, are twice continuous differentiable concave functions.
From assumptions (1) and ( 2), the primal function is concave and the dual function D is convex in λ and μ for a fixed τ.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 (x * , y * , λ * , μ * ) generated by the algorithm is primal-dual optimal.
Remarks.Because of multipath routing, the problem, DP s , may not be strictly concave even if P s (•) 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 DP s 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, λ and μ act as the interface variables between different modules and sensor nodes.Through λ and μ, 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 i to solve the objective vector function independently according to different requirements.

Paradigms of Objective Optimization in WSNs
In the proposed general framework, P s (x s , y s ) in DP s is a vector function and can be transformed into a single function according to different requirements.Therefore, solving DP s 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: (1) Constraint Method [6], (2) Linear Weighted Method [10], and (3) 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.

Preliminary Knowledge.
In this section, we give a brief introduction to the three multiobjective methods.
(1) The constraint method tries to solve the multiobjective problems by placing the most important one in the objective function, while other objective functions are constrained within the constraint set.
In other words, constraint method can solve the following DP s of each sensing node s. (without loss of generality, we assume that f (1)   s is the most important objective function) where A ( j) s , j = 2, . . ., n, are constant constraints imposed by applications.
(2) The linear weighted method focuses on solving the multiobjective optimization problems by first associating each objective function f i (•) with a weight γ i and then taking the weighted sum as a new objective function.Using linear weighted method to solve DP s , the variables in each sensing node s are updated according to x s , y s − λ T g s (x s , τ) where γ j , j = 1, . . ., n, are weight coefficients.
(3) The hierarchical sequence method is concerned with solving the multiobjective optimization problems sequentially, that is, solving the most important problem first and then the less important problems.Let I = {x | x s ∈ χ s , y ss ∈ χ s } and divide (5) into n functions F j (x s , y s ), j = 1, 2, . . ., n. Maximization is achieved by solving the following n subproblems sequentially.

Rate Allocation Problem under ASMOP Formulation.
In this section, we consider the rate allocation problem with two objectives: (1) maximizing aggregate utility and (2) prolonging the network lifetime.Assume the sensing nodes can transmit their rates to the sink nodes over a set L = {1, 2, . . ., L} of links, each of which has capacity c l , l ∈ L. Each sensing node s can transmit its rate through R(s) ⊆ R of the routes.Each route r ∈ R(s) traverses over a set L(r) ⊆ L of links with a rate x sr .Let x s be the rate vector of sensing node s, S(l) = {s ∈ S | l ∈ L(r), r ∈ R(s)} the set of sensing nodes using link l, and R(s, l) the subset of routes R(s) used by sensing node s to traverse over link l.We denote the set of sensing nodes that use sensing node s as an interim relay node by S(s) (not including the sensing node s itself).Let R(s, s ) be the subset of routes R(s ) which use sensing node s as a relay node and S(r) the relay nodes used by route r.Let M be finite number of state that the channels have and p(τ m ) the probability of the state τ m , m ∈ M. Each sensing node s is characterized by three parameters (U s (•), b s , b s ), where U s : R + → R is a strictly concave utility function, b s ≥ 0 and b s < ∞ which are the required minimum and maximum transmission rates for each sensing node s, respectively.
From [11,16], we know utility maximization can be formulated as follows.
x sr ≤ b s , ∀s ∈, x sr ≥ 0, ∀s ∈ S, r ∈ R(s), (21) where (p(τ 1 ), p(τ 2 ), . . ., p(τ M )) describes the distribution of the states of link channel condition and c m l is the capacity of link l under state τ m .We will establish algorithms that can guarantee convergence without prior knowledge of the underlying probability distribution of the system channel state.
The network lifetime is often defined as the time interval between initialization of the network and the exhaustion of the battery of the first sensing node.The total power dissipation, w s , at sensing node s is equal to where w t sr and w re are the energy consumptions at sensing node s for transmitting or receiving unit data flow over route r, respectively.
Let e s denote the initial energy of sensing node s.Its lifetime T s is T s = e s /w s .Following [10], we have the energy model for the network lifetime: where z s = 1/T s .We can have the multiobjective model for rate allocation problem: PP : max P = P (1) (x), P (2) s.t.constraints ( 21), ( 22), ( 24), where Here, parameter ω scales the values of the two objective functions into the same order of magnitude.

Algorithm Design.
Similar to (3), the decoupled form of (25) is DPP : max P = P (1) (x), P (2) x, y ⎧ ⎨ ⎩ y ss r = x s r , ∀s ∈ S(s), r ∈ R(s, s ), constraints ( 21), ( 22), ( 24), where Then we have the Lagrange function: x sr λ r − μ sr where λ r = l∈L(r) λ l and μ sr = s ∈S(r) u s sr .Notice that for given λ r , μ sr , μ ss r , s ∈ S(s), r ∈ R(s, s ), the update for each sensing node s is deterministic, no matter what the channel condition is.So at iteration t, each sensing node s updates its rate x s and auxiliary variable y s by solving the optimization below x sr ≤ b s , x sr ≥ 0, where P s (x s , y s ) = (U s (x s ), −(ω/(β − 1))z β−1 s ) is the objective vector function for sensing node s.
At iteration t, t = 1, 2, . .., sensing node s first receives λ r , μ sr and μ ss r , s ∈ S(s), r ∈ R(s, s ), from the network, and then updates its rate and auxiliary variables according to Notice that the update of decoupled prices, μ, does not depend on the channel condition.At each iteration t, with x s r , y ss r , s ∈ S(s), r ∈ R(s, s ), being collected, the decoupled price μ is updated according to To update price λ, without prior knowledge about the distribution of the channel state, each sensing node s can measure channel condition and get the link capacity c l (t) for each link l at time t.Then λ can be updated according to So far, we have introduced interface variables (μ and λ) for a fully distributed implementation and provided a framework for rate allocation in WSNs with time-varying channels.Next, we show how to apply the framework to applications under different methods.

Paradigms: Applications under Different Methods. (1)
Constraint Method.To maximize the utility of a network under the condition that the network lifetime exceeds a prespecified threshold time T, the constraint method can be used to solve the optimization problem below of each sensing node s.
(2) Linear Weighted Method.In [10,13], Zhu et al. considered the tradeoff between lifetime and rate allocation.By introducing the weight parameter, γ, to evaluate the importance of the two objectives, they can be combined into a single one.For (29), the desired tradeoff between network utility and lifetime can be achieved by solving the optimization problem below.
(3) Hierarchical Sequence Method.In our rate allocation paradigm, we have two objectives: (1) find a rate allocation strategy to maximize the total utility and (2) prolong the networks lifetime.To the best of our knowledge, this method has not been applied to the rate allocation problem before.We can achieve the two objectives by solving the two subproblems below sequentially.
We have two objectives: maximizing the aggregate utility and the network lifetime.For the utility objective, we set U s (•) = ξ s log(•) for each sensor node s, where ξ = (52, 54, 56, 58, 60, 62, 64, 66, 68).From [10], the function − s (ω/(β − 1))z β−1 s can have a ratio higher than 0.95 to approximate the original lifetime problem T when β ≥ 8.In our simulations, we use β = 9.The link capacities vary from time to time according to a uniform distribution with the expected capacities of links 1-13 to be c = (2000, 2000, 2200, 2000, 2000, 2500, 2800, 3500, 3500,  2000, 3000, 2800) (bit/s).For the energy consumption model, from [10], w re is a constant and w sr = ρ + σd where d sr is the length of the outgoing link of sensing node s for transmitting rate of route r.We set ρ = 50 nJ/bit, σ = 0.0013 pJ/bit/m 4 , η = 4, w re = 50 nJ/bit.The initial energy of the sensing nodes 1-9 is set to be e = (450, 450, 475, 475, 450, 500, 500, 450, 500) (J) and the sink node (node 10) is assumed to have enough energy.The minimum and maximum rates of each sensing node are set to be b s = 100 and b s = 2500, respectively.800 h.We collect the rate of each route r at each iteration.For each sensing node s, the rates are updated according to (37) and the decoupled price μ is updated according to (34).Each link l first collects information about the channel condition and computes its corresponding capacity, then updates its link price according to (35).The results are shown in Figure 3. Since there are 9 sensing nodes and 15 routes in our simulations, due to space limitation, we only show the aggregate rate of each sensing node.In Figure 3, we can be see that the rates first change sharply and then converge to the optimal one, which indicates the effectiveness of our ASMOP algorithm.

Simulation Results for
( Next, we show the simulation results of the ASMOP algorithm using the linear weighted method.Here we set ω = 8.17 × 10 26 and γ = 0.7.At each iteration t, each sensing node s updates its rates according to (39), and the decoupled price λ and μ are updated as the same in paradigm I.The results are shown in Figure 4, and similar conclusions can be made as in the paradigm I.
A similar simulation is performed for the hierarchical sequence method and the corresponding results are shown in Figure 5.We can see that the rates change sharply at the beginning of each iteration, and then converge to the optimal one in Figure 5.
We further set the threshold of the network lifetime to be 800 h in the simulation for the constraint method and γ = 0.7 for linear weighted method.These two methods mainly target the energy-constraint in WSNs.For the hierarchical sequence method, we focus on the utility of the network.From Figures 3, 4, and 5, it can seen that the rates in Figure 5 are much larger than those in Figures 3 and 4. On the other hand, as the rates in each sensor node become large, the energy consumption increases.So the network lifetime under the hierarchical sequence method is less than that under the constraint method or linear weighted method.In addition, different multiobjective methods obtain different network performances.The results of the three simulations also demonstrate the efficiency and convenience of our proposed framework.

Conclusions
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.

Figure 1 :
Figure 1: An illustration of our proposed framework.

Figure 3 :
Figure 3: Convergence of the ASMOP algorithm by using constraint method.

Figure 4 :
Figure 4: Convergence of the ASMOP algorithm by using linear weighted method.

Figure 5 :
Figure 5: Convergence of the ASMOP algorithm by using hierarchical sequence method.