CN102118821B - Wireless sensor network distributed routing method on basis of Lagrange-Newton method - Google Patents

Abstract

The invention relates to a wireless sensor network distributed routing method on the basis of a Lagrange-Newton method, comprising the steps of: firstly switching life cycle maximization routing model of a wireless sensor network into problem with equality constraint convex optimization, and then adopting a sequential unconstrained minimization method to solve; and particularly, in the core step of the sequential unconstrained minimization method on the basis of the Lagrange-Newton method, designing a matrix splitting technology for solving dual Newton increment by distributed iteration, and utilizing a distributed step-length calculation method to guarantee that the flow of any wireless chain is controlled in the allowed maximum flow range, thus achieving the purpose of using the Lagrange-Newton method to solve the wireless sensor network distributed routing problem. In the wireless sensor network distributed routing method, the model is reasonable, the energy confinement is considered, the convergence rate is fast, and the wireless sensor network distributed routing method can be directly applicable to the problem of optimum route for the life cycle in the wireless sensor network.

Description

Wireless-sensor network distribution type method for routing based on the Lagrange-Newton method

Technical field

The present invention relates to the wireless sensor network technology field, especially a kind of wireless-sensor network distribution type method for routing based on the Lagrange-Newton method.

Background technology

A large amount of cheap microsensor nodes form wireless sensor network in the monitored area by being deployed in, a kind of multihop self-organizing network system that forms by communication.Normally, the sensor node volume is small, by carrying the powered battery of finite energy, its disposal ability, storage capacity and communication capacity relatively a little less than.In addition, the sensor node deployment circumstance complication, some zone even personnel can not arrive, and the replacing of sensor node battery can't realize.The effective use energy prolongs the key that network lifecycle is sensor network.Point out in the special report of Deborah Estrin in Mobicom 2002 meetings that most energy consumption of sensor node are at wireless communication module.And the energy consumption of radio communication and communication distance exponentially relation with increase.Therefore, how to design the multi-hop routing mechanism, reduce the single-hop communication distance under the prerequisite of communication connectivity satisfying as far as possible, prolong network lifetime is one of challenge of facing of sensor network.

At present, the route research based on Energy Efficient has obtained certain achievement.In the recent period, document [1]: Huseyin Ozgur Tan, and Ibrahim Korpeo lu, " Power efficient data gathering and aggregation in wireless sensornetworks ", ACM SIGMOD Record, vol.32, issue 4, pp.66-71, Dec.2003. (a kind of wireless sensor network data of Energy Efficient is collected and fusion method), based on minimum spanning tree, PEDAP and PEDAP-PA (PowerEfficient Data gathering and Aggregation Protocol-Power Aware) Routing Protocol is disclosed.This protocol definition link metric makes up minimum spanning tree by the Prim algorithm, and finally each node arrives the Sink node to convergence along minimum spanning tree; Document [2]: Keyhan Khamforoosh, and Hana Khamforoosh, " A New Routing Algorithm forEnergy Reduction in Wireless Sensor Networks ", the 2nd IEEE International Conference onComputer Science and Information Technology, pp.505-509, Aug.2009. (a kind of new wireless sensor network energy-saving routing algorithm), try hard to seek the shortest path from sensor node to the sink node, and utilizing the high characteristics of data redudancy between adjacent node, packet constantly merges in transport process.It is minimum to the path total energy consumption of destination node that above-mentioned these algorithms are all paid close attention to source node, causes the node energy on these paths to exhaust fast, shortens network life thereby destroy network connectivty.Therefore, the method for routing of maximization network life cycle is more favored.Document [3]: Yi-hua Zhu, Wan-deng Wu, Jian Pan, and Yi-ping Tang, " An energy-efficient data gathering algorithm to prolong lifetime ofwireless sensor networks ", Computer Communications, vol.33, issue 5, pp.639-647, March 2010. (a kind of energy of wireless sensor network efficient data collection algorithm of prolong network lifetime), introduce Data Collection sequence (datagathering sequence) and avoid route loop, disclose a kind of mathematical programming model that considers residue energy of node and total energy consumption, and try to achieve optimal solution with genetic algorithm.Yet genetic algorithm is a kind of centralized iterative algorithm, require the sink node constantly with each sensor node radio communication to obtain global information.Document [4]: Joongseok Park, and Sartaj Sahni, " An Online Heuristic for Maximum Lifetime Routing in Wireless Sensor Networks ", IEEETransactions on Computers, vol.55, issue 8, pp.1048-1056, Aug.2006. (a kind of online didactic wireless sensor network life maximization route) discloses as far as possible prolong network lifetime of a kind of online heuristic method for routing.But heuritic approach can not obtain globally optimal solution, and has certain randomness.Document [5]: Jae-Hwan Chang, and L.Tassiulas, " Maximum Lifetime Routing in Wireless Sensor Networks ", IEEE/ACM Transactionson Networking, vol.12, issue 4, pp.609-619, Aug.2004. (wireless sensor network life maximization route) and document [6]: Y B Turkogullar, N Aras, I K Altinel, and C Ersoy, " Optimal placement, scheduling; and routing to maximize lifetime in sensor networks " Journal of the Operational Research Society61, pp.1000-1012, the June 2010. (optimal location of maximum network life, scheduling and routing policy), network life cycle is maximized route be modeled as linear programming problem under the traffic constraints, unfortunately, they all adopt based on didactic method is distributed and find the solution.Document [7]: R.Madan, and S.Lall, " Distributed Algorithms for MaximumLifetime Routing in Wireless Sensor Networks ", IEEE Transactions on Wireless Communications, vol.5, issue 8, pp.2185-2193, Aug.2006. (prolong the wireless-sensor network distribution type routing algorithm of network life), represent optimum routing issue life cycle with linear programming model equally, but it utilizes Subgradient Algorithm distributed earth iterative.In each iterative process, node only needs local information and information of neighbor nodes, and can converge to optimal solution.But the disadvantage of this class subgradient method is that convergence rate is slow, needs continually exchange message between the neighbor node.Recently, document [8] utilizes the network optimization problem of the distributed solve equation constraint of Lagrange-Newton method, and its key technology is the distributed newton's of finding the solution step-length and antithesis newton increment.Document [9]: E Wei, A.Ozdaglar, and A.Jadbabaie, " A distributed newtonmethod for network utility maximization ", LIDS Report 2832, March 2010. (a kind of distributed Newton method of maximization network effectiveness) further studies with the optimized distributed Lagrange-Newton method of the network utility of inequality constraints.Compare with the subgradient method, distributed Lagrange-Newton method convergence rate is faster.Yet, document [8]: A.Jadbabaie, A.Ozdaglar, and M.Zargham, " A Distributed Newton Method for NetworkOptimization ", the 48th IEEE Conference on CDC/CCC, pp.2736-2741, Dec.2009. (the optimized distributed Newton method of a kind of network utility), and the Network Optimization Model in the document [9] all too simply and not considers energy constraint, and the distributed method for solving of newton's step-length and antithesis newton increment all can't be directly applied for optimum routing issue life cycle in the wireless sense network.

Summary of the invention

Existing wireless sensor network routing method is centralized in order to overcome, convergence rate waits deficiency slowly, the invention provides a kind of model rationally, consider energy constraint, can be directly applied in the wireless sense network life cycle optimum routing issue the wireless-sensor network distribution type method for routing based on the Lagrange-Newton method.

The technical solution adopted for the present invention to solve the technical problems is:

For making narration clearer, some sign flags and the related definition used among the present invention are described at first:

x _iI element in the expression vector x, without specified otherwise, vector all refers to column vector.|| x|| represents Euclid norm, namely

A (n) refers to that matrix A is the square formation of n * n dimension, and A (m * n) refers to that the dimension of matrix A is m * n dimension, [A] _IjOr A _IjRepresent the element that i is capable in the matrix A, j is listed as.{ x _iLimited element x of expression _iSet.

With

Represent that respectively n-ary function f (x) is at x ∈ R ⁿThe first derivative vector sum matrix of second derivatives at place,

Represent that then f (x) is to component x _iThe single order local derviation.

The system model of wireless sensor network represents that with non-directed graph G (V, Ψ) wherein V is the set of sensor node, comprises n ordinary node and 1 Sink node, and described ordinary node and described Sink node are referred to as node; Ψ is the set of Radio Link, always has m bar link, and described system model is defined as follows:

1. P _Max: the node maximum transmit power, if carry out the required transmitting power of radio communication between node i and the node j less than P _Max, claim so the Radio Link l that node i flows to node j _Ij∈ Ψ exists, and Radio Link is two-way existence, l _IjWrite a Chinese character in simplified form into l, especially, described Sink node does not have outgoing link;

2. N _i: the neighbor node of node i is gathered, and namely has the node set of Radio Link with node i;

3.

Node i is to the Radio Link l of node j _IjThe maximum information flow rate that allows,

Write a Chinese character in simplified form into γ _l

4. b _i: the initial cells energy of described ordinary node i, the energy of described Sink node is infinitely great;

5. g _i: the information incidence of described ordinary node i is g _i〉=0;

6. network life: first node energy exhausts the time of inefficacy in the network;

7.

Node i is by Radio Link l _IjSend the energy that unit information consumes to neighbor node j;

8.

Node i receives from Radio Link l _IjThe energy that consumes of the unit information of egress j;

A kind of wireless-sensor network distribution type method for routing based on the Lagrange-Newton method, the method may further comprise the steps:

The A wireless sensor network maximizes the protruding optimization problem of equality constraint that route is modeled as approximately equivalent life cycle:

$\min f\left(x\right)=p\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{q}_i^2-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\log \left(y_i\right)-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\underset{j\∈N_i}{\mathrm{\Σ}}\left[\log \left(r_{l_{\mathrm{ij}}}\right)+\log ({\mathrm{\γ}}_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ij}}})\right]$

$s.t.\underset{j\∈N_i}{\mathrm{\Σ}}(r_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ij}}})-g_i=0,$ i＝1，L，n

$\underset{j\∈N_i}{\mathrm{\Σ}}(e_{l_{\mathrm{ij}}}^t{r}_{l_{\mathrm{ij}}}+e_{l_{\mathrm{ji}}}^r{r}_{l_{\mathrm{ji}}})+y_i-q_i{b}_i=0,$ i＝1，L，n

q _i-q _j＝0i＝1，L，n-1， $j=N_{E_i}$

In the formula, x=({ r _Ij, q ₁, L, q _n, y ₁, L, y _n) ^T

It is the link information flow rate that node i arrives node j; F (x) is called target function; q _iThe rate of death that is called ordinary node i, y _iBe called slack variable, Expression ordinary node i only selects neighbours' ordinary node j to be used for q _iConstraint; P is called barrier parameter, when p trend towards+during ∞, described wireless sensor network maximizes routing issue and the protruding optimization problem equivalent of described equality constraint life cycle; The corresponding matrix form of following formula is:

$\min \mathrm{imizef}\left(x\right)=p\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{q}_i^2-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\log \left(y_i\right)-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\underset{j\∈N_i}{\mathrm{\Σ}}[\log \left(r_{l_{\mathrm{ij}}}\right)+\log ({\mathrm{\γ}}_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ij}}})]$

subjectto Fx＝c

In the formula, c=(g ^T, 0) ^T,

Wherein: I is unit matrix; G=(g ₁, g ₂, L, g _n) ^TThe information incidence vector that is called described ordinary node; B (n) is called described ordinary node initial cells energy matrix, and B is diagonal matrix, and B _Ii=-b _i(n * m) is called node-link association matrix to A; (n * m) is called radio communication energy consumption matrix to E; ((n-1) * n) is called described ordinary node rate of death constraint matrix to D; Described matrix A, E, the concrete form of D is respectively:

The protruding optimization problem of the described equality constraint of B is found the solution by the sequential unconstrained minimization technique method, and concrete steps are as follows:

The S1-1 initialization: in the interval range of feasible zone, arbitrary initial variable x ⁰, x ⁰Feasible solution not necessarily; The described barrier parameter p of initialization＞1; Simultaneously initialization constant μ＞1, constant ε＞0;

S1-2 is based on the Lagrange-Newton method, with x ⁰The primary iteration point calculates p to the optimal solution x of the protruding optimization problem of the described equality constraint of timing ^*(p);

S1-3 upgrades x ⁰, i.e. x ⁰=x ^*(p);

S1-4 checks end condition: if p＜ε forwards S1-5 to; Otherwise the sequential unconstrained minimization technique method finishes, x ^*(p)=({ r _Ij ^*, q ₁ ^*, L, q _n ^*, y ₁ ^*, L, y _n ^*) ^TBe the solution of the protruding optimization problem of described equality constraint, make the information flow-rate r of maximized all links of described network life cycle then be

S1-5 upgrades p, and namely p=μ p forwards S1-2 to.

If it is distributed can finding out described step S1-2, that described sequential unconstrained minimization technique algorithm is distributed.

Further, the optimal solution of the protruding optimization problem of equality constraint is distributed described in the described step S1-2, namely utilize the split matrix technology, finding the solution of described Lagrange-Newton method be distributed in each node in the network calculate, each node only needs and the neighbor node exchange message in computational process, and each node is carried out according to following steps:

The S1-2-1 initialization: ordinary node i calculates the energy that sending/receiving unit's bit information need to consume, namely With

J ∈ N _iPreserve simultaneously information incidence g _iWith initial cells energy b _iThen preserve the outgoing link information flow-rate according to described step S1-1 or described step S1-3

J ∈ N _i, slack variable y _i〉=0, rate of death q _i＞0;

S1-2-2 ordinary node i calculates described target function at current outgoing link information flow-rate

J ∈ N _i, rate of death q _i, and slack variable y _iThe single order partial derivative at place

Inverse with second dervative Q _Ii, Y _Ii

S1-2-3 ordinary node i will

J ∈ N _i, q _i, y _iAnd the result of calculation among the described step S1-2-2 is broadcast to neighbor node;

S1-2-4 ordinary node i utilizes local information and information of neighbor nodes to calculate antithesis newton increment ω 1 _i, ω 2 _i, ω 3 _i

Formula below S1-2-5 ordinary node i utilizes calculates respectively link information flow rate initial value

J ∈ N _i, slack variable y _i, rate of death q _iNewton direction:

$\mathrm{\Δ}{r}_{l_{\mathrm{ij}}}=-R_{l_{\mathrm{ij}}{l}_{\mathrm{ij}}}\{\▿f\left(r_{l_{\mathrm{ij}}}\right)+(\mathrm{\ω}{1}_i-\mathrm{\ω}{1}_j)+(e_{l_{\mathrm{ij}}}^t\mathrm{\ω}{3}_i+e_{l_{\mathrm{ji}}}^r\mathrm{\ω}{3}_j)\}$

$\mathrm{\Δ}{q}_i=-Q_{\mathrm{ii}}\{\▿f\left(q_i\right)+(\mathrm{\ω}{2}_i-{\mathrm{\Σ}}_{i=N_{E_j}}\mathrm{\ω}{2}_j)-b_i\mathrm{\ω}{3}_i\}$

$\mathrm{\Δ}{y}_i=-Y_{\mathrm{ii}}\{\▿f\left(y_i\right)+\mathrm{\ω}{3}_i\}$

S1-2-6 ordinary node i at first calculates (Δ y _i/ y _i) ²With J ∈ N _i, then result of calculation is broadcast to neighbours' ordinary node;

S1-2-7 ordinary node i utilizes average coordination approach to calculate newton's step-length s, and the computing formula of described step-length is:

$s=1/\sqrt{d}$

$d={\mathrm{\Σ}}_{i=1}^n{(\mathrm{\Δ}{y}_i/y_i)}^2+{\mathrm{\Σ}}_{i=1}^n{\mathrm{\Σ}}_{j\∈N_i}\max ({(\mathrm{\Δ}{r}_{l_{\mathrm{ij}}}/r_{l_{\mathrm{ij}}})}^2,{(\mathrm{\Δ}{r}_{l_{\mathrm{ij}}}/(R_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ij}}}))}^2);$

S1-2-8 ordinary node i upgrades described outgoing link information flow-rate according to following formula

J ∈ N _i, described slack variable y _i, and described rate of death q _i:

$r_{l_{\mathrm{ij}}}=r_{l_{\mathrm{ij}}}+\mathrm{s\Δ}{r}_{l_{\mathrm{ij}}}$

y _i＝y _i+sΔy _i

q _i＝q _i+sΔq _i

S1-2-9 judges end condition: if Be constant and

, finish; Otherwise forward S1-2-2 to.

As a kind of preferred scheme, described split matrix technology refers to matrix F H ^-1F ^TSplit into two matrix J, K's and, i.e. FH ^-1F ^T=J+K, wherein, J is diagonal matrix, and

Thereby antithesis newton increment can be iterated through following formula to be obtained:

$\mathrm{\ω}(t+1)=-J^{-1}\mathrm{K\ω}\left(t\right)+J^{-1}[(\mathrm{Fx}-c)-{\mathrm{FH}}^{-1}\▿f\left(x\right)]$

Wherein,

With

Represent that respectively n-ary function f (x) is at x ∈ R ⁿThe first derivative vector sum matrix of second derivatives at place;

Among the described step S1-2-4, ordinary node i calculates described antithesis newton increment ω 1 according to the following steps _i, ω 2 _i, ω 3 _i:

S1-2-4-1 is according to matrix J in the described split matrix technology, the definition of K, and matrix J, K is expressed as:

$J=\left[\begin{array}{ccc}J1\left(n\right)& 0& 0\\ 0& J2(n-1)& 0\\ 0& 0& J3\left(n\right)\end{array}\right]$ With $K=\left[\begin{array}{ccc}{K}_{11}\left(n\right)& K_{12}(n\×(n-1))& K_{13}\left(n\right)\\ K_{21}((n-1)\×n)& K_{22}(n-1)& K_{23}((n-1)\×n)\\ K_{31}\left(n\right)& K_{32}(n\×(n-1))& K_{33}\left(n\right)\end{array}\right]$

Then compute matrix J, respective element J1 among the K _Ii, J2 _Ii, J3 _Ii[K _1s] _Ij, [K _2s] _Ij, [K _2s] _Ij, j ∈ N _i∪ j=i, s=1,2,3;

S1-2-4-2 calculates respectively u1 according to following formula _i, u2 _i, u3 _i, i.e. respective component among the vectorial Fx-c:

$u{1}_i=\underset{j\∈N_i}{\mathrm{\Σ}}(r_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ji}}})-g_i$

$u{2}_i=q_i-q_j,j=N_{E_i}$

$u{3}_i=\underset{j\∈N_i}{\mathrm{\Σ}}(e_{l_{\mathrm{ij}}}^t{r}_{l_{\mathrm{ij}}}+e_{l_{\mathrm{ji}}}^r{r}_{l_{\mathrm{ji}}})+y_i-q_i{b}_i$

S1-2-4-3 calculates respectively v1 according to following formula _i, v2 _i, v3 _i, namely vectorial In respective component:

$v{1}_i={\mathrm{\Σ}}_{j\∈N_i}(R_{l_{\mathrm{ij}}{l}_{\mathrm{ij}}}\▿f\left(r_{l_{\mathrm{ij}}}\right)-R_{l_{\mathrm{ji}}{l}_{\mathrm{ji}}}\▿f\left(r_{l_{\mathrm{ji}}}\right))$

$v{2}_i=Q_{\mathrm{ii}}\▿f\left(q_i\right)-Q_{\mathrm{jj}}\▿f\left(q_j\right),j=N_{E_i}$

$v{3}_i={\mathrm{\Σ}}_{l\∈O_i}{R}_{\mathrm{ll}}{e}_l^t\▿f\left(r_l\right)+{\mathrm{\Σ}}_{l\∈I_i}{R}_{\mathrm{ll}}{e}_l^r\▿f\left(r_l\right)-Q_{\mathrm{ii}}\▿f\left(q_i\right)b_i+Y_{\mathrm{ii}}\▿f\left(y_i\right)$

S1-2-4-4 calculates respectively w1 according to following formula _i, w2 _i, w3 _i, namely vectorial

In respective component:

w1 _i＝J1 _ii ^-1(u1 _i-v1 _i)；

w2 _i＝J2 _ii ^-1(u2 _i-v2 _i)；

w3 _i＝J3 _ii ^-1(u3 _i-v3 _i)；

S1-2-4-5 calculates ω 1 _i, ω 2 _i, ω 3 _i: arbitrary initial antithesis newton increment ω 1 at first _i(0), ω 2 _i(0), ω 3 _i(0), then according to following formula difference iterative computation ω 1 _i(t), ω 2 _i(t), ω 3 _i(t), until satisfy designated precision; In the iterative process, need to exchange ω 1 between the neighbor node _i(t), ω 2 _i(t), ω 3 _i(t) value;

Wherein,

As another kind of preferred version, ordinary node i selects neighbours' ordinary node j to be used for q _iThe method of constraint be: during initialization, ordinary node n at first broadcasts a packets of information, after its neighbours' ordinary node i receives this packets of information, selects node n to be used for q _iConstraint.Then ordinary node i continues this packets of information of broadcasting, and its neighbours' ordinary node j then selects ordinary node i to be used for q _jConstraint.The like, until the selected neighbours' ordinary node of all ordinary nodes.In this process, each ordinary node is only processed the packets of information that receives for the first time.Therefore, ordinary node n need not to find the solution u2 among the described step S2-4, v2, w2, ω 2 and matrix J 2, K ₂₁, K ₂₂, K ₂₃In respective element.

Technical conceive of the present invention is: the present invention maximizes route matrix life cycle with wireless sensor network and carries out approximately equivalent and transform, design a kind of split matrix technology and find the solution antithesis newton increment for distributed iterative, and utilize a kind of distributed step size computation method, the flow that guarantees any wireless links all is controlled in the maximum stream flow scope that allows, thereby reaches the purpose that solves the wireless-sensor network distribution type routing issue with the Lagrange-Newton method.

Can find out that from technique scheme beneficial effect of the present invention is mainly manifested in:

1. the present invention analyzes after wireless sensor network life cycle is maximized the protruding optimization problem that route is modeled as equality constraint, obtain the support of mathematical theory, avoid some routing algorithms can not obtain globally optimal solution, and had the limitation of certain randomness.

2. the invention belongs to full distributed method for routing, need not the management node centralized control, the flow of its all outgoing link when the information that each node only need utilize self information and neighbor node just can local computing goes out the network lifecycle maximization.

3. the present invention is based on the distributed wireless sensor network life cycle maximization route of finding the solution of Lagrange-Newton method, compares with subgradient method commonly used, and distributed Lagrange-Newton method convergence rate is faster.

Description of drawings

Fig. 1 is the overall procedure that wireless sensor network maximizes route life cycle of finding the solution of the present invention.

Fig. 2 is the flow process of utilizing the distributed solve equation constrained convex optimal problem of Newton method of the present invention.

Embodiment

For making the purpose, technical solutions and advantages of the present invention clearer, the invention will be further described below in conjunction with embodiment and with reference to accompanying drawing.

A kind of wireless-sensor network distribution type method for routing based on the Lagrange-Newton method sees figures.1.and.2, typical sensor node energy consumption is mainly produced by wireless data transceiving, therefore only consider the energy consumption that radio communication is relevant among the present invention, namely the node sending/receiving k bit information energy that need to consume is respectively:

E _Tx(k，d)＝E _Tx-elec(k)+E _Tx-amp(k，d)＝kE _elec+kε _fsd ^α(1)

E _Rx(k)＝E _Rx-elec(k)＝kE _elec (2)

In the formula, d is the distance between sending node and the receiving node, and α ∈ [2,4] is the path loss coefficient.E _ElecReceive or send the energy that the unit bit information consumes, ε on the indication circuit _Fsd ^αThe expression power amplifier sends the energy that the unit bit information consumes.

At first derive wireless sensor network ox cycle of deposit maximization route matrix of the present invention, and described route matrix is transformed, namely use the described route matrix of matrix representation.Definition

The link information flow rate that node i arrives node j, the information flow-rate of all links in the network

So during given described r, be the life cycle of ordinary node i:

$T_i\left(r\right)=\frac{b_i}{{\mathrm{\Σ}}_{j\∈N_i}(e_{l_{\mathrm{ij}}}^t{r}_{l_{\mathrm{ij}}}+e_{l_{\mathrm{ji}}}^r{r}_{l_{\mathrm{ji}}})}$

In the following formula,

Dactylus point i is by Radio Link l _IjSend the energy that unit information consumes to neighbor node j, Calculate according to described formula (1);

Dactylus point i receives from Radio Link l _JiThe energy that consumes of the unit information of egress i,

Calculate according to described formula (2).According to the definition of network life, network life cycle then is:

$T_{\mathrm{net}}\left(r\right)=\underset{i=1,L,n}{\min }{T}_i\left(r\right)$

Target of the present invention is to find the solution r under the flow equilibrium constraint, makes described network life cycle maximum, that is:

max T _net(r)

$s.t.\underset{j\∈N_i}{\mathrm{\Σ}}(r_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ji}}})=g_i,$ $\∀i\∈V$

$0\≤r_{l_{\mathrm{ij}}}\≤{\mathrm{\γ}}_{l_{\mathrm{ij}}},$ $\∀i\∈V,\∀j\∈N_i$

Notice that the traffic constraints condition of sink node is redundant, therefore, described following formula and following formula are of equal value:

max T _net(r)

$s.t.\underset{j\∈N_i}{\mathrm{\Σ}}(r_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ji}}})=g_i,$ i＝1，L，n

$0\≤r_{l_{\mathrm{ij}}}\≤{\mathrm{\γ}}_{l_{\mathrm{ij}}},$ i＝1，L，n， $\∀j\∈N_i$

Further, introduce variable T, described following formula can equivalence become:

max T

$s.t.\underset{j\∈N_i}{\mathrm{\Σ}}(r_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ji}}})=g_i,$ i＝1，L，n

$0\≤r_{l_{\mathrm{ij}}}\≤{\mathrm{\γ}}_{l_{\mathrm{ij}}},$ i＝1，L，n， $\∀j\∈N_i$

$T\underset{j\∈N_i}{\mathrm{\Σ}}(e_{l_{\mathrm{ij}}}^t{r}_{l_{\mathrm{ij}}}+e_{l_{\mathrm{ji}}}^r{r}_{l_{\mathrm{ji}}})\≤b_i,$ i＝1，L，n

Note T=1/q then obtains linear programming method for expressing of equal value:

min q

$s.t.\underset{j\∈N_i}{\mathrm{\Σ}}(r_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ji}}})=g_i,$ i＝1，L，n

$0\≤r_{l_{\mathrm{ij}}}\≤={\mathrm{\γ}}_{l_{\mathrm{ij}}},$ i＝1，L，n， $\∀j\∈N_i$

$\underset{j\∈N_i}{\mathrm{\Σ}}(e_{l_{\mathrm{ij}}}^t{r}_{l_{\mathrm{ij}}}+e_{l_{\mathrm{ji}}}^r{r}_{l_{\mathrm{ji}}})\≤q{b}_i,$ i＝1，L，n

Further, described linear programming method for expressing and following protruding quadratic form optimization problem are of equal value:

$\min \underset{i=1}{\overset{n}{\mathrm{\Σ}}}{q}_i^2$

$s.t.\underset{j\∈N_i}{\mathrm{\Σ}}(r_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ji}}})=g_i,$ i＝1，L，n

$0\≤r_{l_{\mathrm{ij}}}\≤{\mathrm{\γ}}_{l_{\mathrm{ij}}},$ i＝1，L，n， $\∀j\∈N_i---\left(3\right)$

$\underset{j\∈N_i}{\mathrm{\Σ}}(e_{l_{\mathrm{ij}}}^t{r}_{l_{\mathrm{ij}}}+e_{l_{\mathrm{ji}}}^r{r}_{l_{\mathrm{ji}}})\≤q_i{b}_i,$ i＝1，L，n

q _i＝q _j i＝1，L，n， $j=N_{E_i}$

Wherein, claim q _iRate of death for node i. The expression node i only selects a neighbor node j to be used for q _iConstraint, purpose is to eliminate redundant constraint formula.In fact, n-1 equation can satisfy all q _i, i=1, L, n equates.Among the present invention, during initialization, ordinary node n at first broadcasts a packets of information, after its neighbours' ordinary node i receives this packets of information, selects ordinary node n to be used for q _iConstraint.Then ordinary node i continues this packets of information of broadcasting, and its neighbours' ordinary node j then selects ordinary node i to be used for q _jConstraint.The like, until the selected neighbours' ordinary node of all ordinary nodes.In this process, each ordinary node is only processed the packets of information that receives for the first time.Obviously, described protruding quadratic form optimization problem.Formula exists equality constraint and inequality constraints in (3) simultaneously, need to carry out to this equivalence and transform.Introduce slack variable y=(y ₁, y ₂, L, y _n) ^T, described formula (3) can equivalence convert to:

$\min \underset{i=1}{\overset{n}{\mathrm{\Σ}}}{q}_i^2$

$s.t.\underset{j\∈N_i}{\mathrm{\Σ}}(r_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ji}}})=g_i,$ i＝1，L，n

$0\≤r_{l_{\mathrm{ij}}}\≤{\mathrm{\γ}}_{l_{\mathrm{ij}}},$ i＝1，L，n， $\∀j\∈N_i---\left(4\right)$

$\underset{j\∈N_i}{\mathrm{\Σ}}(e_{l_{\mathrm{ij}}}^t{r}_{l_{\mathrm{ij}}}+e_{l_{\mathrm{ji}}}^r{r}_{l_{\mathrm{ji}}})+y_i=q_i{b}_i,$ i＝1，L，n

y _i≥0， i＝1，L，n

q _i＝q _j i＝1，L，n-1， $j=N_{E_i}$

Further utilize logarithm barrier function described formula (4) approximately equivalent to be become to only have the optimization problem of equality constraint:

$\min f\left(x\right)=p\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{q}_i^2-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\log \left(y_i\right)-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\underset{j\∈N_i}{\mathrm{\Σ}}\left[\log \left(r_{l_{\mathrm{ij}}}\right)+\log ({\mathrm{\γ}}_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ij}}})\right]$

$s.t.\underset{j\∈N_i}{\mathrm{\Σ}}(r_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ji}}})-g_i=0,$ i＝1，L，n

(5)

$\underset{j\∈N_i}{\mathrm{\Σ}}(e_{l_{\mathrm{ij}}}^t{r}_{l_{\mathrm{ij}}}+e_{l_{\mathrm{ji}}}^r{r}_{l_{\mathrm{ji}}})+y_i-q_i{b}_i=0,$ i＝1，L，n

q _i-q _j＝0 i＝1，L，n-1， $j=N_{E_i}$

X=({ r wherein _Ij, q ₁, L, q _n, y ₁, L, y _n) ^TWhen p trend towards+during ∞, described formula (4) and described formula (5) equivalence.Obviously, the target function f (x) in the described formula (5) is protruding and second order can be little.Be convenient and analyze, convert the equality constraint expression formula of described formula (5) to matrix representation forms.At first define following vector sum matrix:

1. the information incidence of node vector g=(g ₁, g ₂, L, g _n) ^T

2. node initial cells energy matrix B (n), B is diagonal matrix, and B _Ii=-b _i

3. node-link association matrix A (n * m), that is:

4. radio communication energy consumption matrix E (n * m), that is:

5. node rate of death constraint matrix D ((n-1) * n), and:

So, described formula (5) then can be with following matrix representation, that is:

minimize f(x)

(6)

subject to Fx＝c

Wherein, c=(g ^T, 0) ^T, $F=\left[\begin{array}{ccc}A& 0& 0\\ 0& D& 0\\ E& B& I\end{array}\right],$ I is unit matrix.

After finishing described wireless sensor network and maximizing the protruding optimization problem of equality constraint that route matrix converts described formula (6) to life cycle, the present invention adopts sequential unconstrained minimization technique (sequential unconstrained minimizationtechnique, SUMT) method to find the solution described formula (6).Fig. 1 is that the SUMT of utilization method of the present invention is found the solution the flow process of described formula (6), and concrete steps are as follows:

The S1-1 initialization: in the interval range of feasible zone, arbitrary initial variable x ⁰, x ⁰Feasible solution not necessarily; The described barrier parameter p of initialization＞1; Simultaneously initialization constant μ＞1, constant ε＞0;

S1-2 is based on the Lagrange-Newton method, with x ⁰The primary iteration point calculates p to the optimal solution x of the protruding optimization problem of the described equality constraint of timing ^*(p);

S1-3 upgrades x ⁰, i.e. x ⁰=x ^*(p);

S1-4 checks end condition: if p＜ε forwards S1-5 to; Otherwise the sequential unconstrained minimization technique method finishes, x ^*(p)=({ r _Ij ^*, q ₁ ^*, L, q _n ^*, y ₁ ^*, L, y _n ^*) ^TBe the solution of the protruding optimization problem of described equality constraint, make the information flow-rate r of maximized all links of described network life cycle then be

S1-5 upgrades p, and namely p=μ p forwards S1-2 to.

If it is distributed can finding out described step S1-2, that described SUMT algorithm is distributed.Therefore, emphasis of the present invention is to analyze p to regularly, how to utilize distributed Lagrange-Newton method to find the solution the optimal solution of described formula (6).

At first analyze and work as p to regularly, use the optimal solution of the described formula of Lagrange-Newton method iterative (6) of separating RegionAlgorithm for Equality Constrained Optimization.x ⁰The primary iteration point among the described step S1-2, x ^kThe solution vector that represents the k time iteration, the Lagrange-Newton method is upgraded iteration point according to following formula:

x ^k+1＝x ^k+s ^kΔx ^k

In the formula, s ^kWith Δ x ^kRespectively newton's step-length (positive number) and newton's increment of the k time iterative process.Wherein, Δ x ^kTry to achieve according to following linear equality system:

$\left(\begin{array}{cc}{\▿}^{2}f\left(x^k\right)& F^T\\ F& 0\end{array}\right)\left(\begin{array}{c}\mathrm{\Δ}{x}^k\\ {\mathrm{\ω}}^k\end{array}\right)=-\left(\begin{array}{c}\▿f\left(x^k\right)\\ F{x}^k-c\end{array}\right)$

In the formula, ω ^kIt is antithesis newton increment.Note

According to described following formula, Δ x ^kFind the solution and can be divided into following two steps:

$\left({\mathrm{FH}}_k^{-1}{F}^T\right){\mathrm{\ω}}^k=({\mathrm{Fx}}^k-c)-{\mathrm{FH}}_k^{-1}\▿f\left(x^k\right)---\left(7\right)$

$\mathrm{\Δ}{x}^k=-H_k^{-1}(\▿f\left(x^k\right)+F^T{\mathrm{\ω}}^k)---\left(8\right)$

Because do not have coupling between each variable of target function f (x) in the described formula (6), the matrix of second derivatives of described target function is a diagonal matrix, so matrix

Also be a diagonal matrix, that is:

$H_k^{-1}=\left[\begin{array}{ccc}R\left(m\right)& 0& 0\\ 0& Q\left(n\right)& 0\\ 0& 0& Y\left(n\right)\end{array}\right]$

Wherein, R, Q, Y are respectively that described target function f (x) is in described link information flow rate Described node rate of death

With described slack variable

The second dervative inverse matrix at place, namely

Q _Ii=1/2p,

Examine described matrix

As described antithesis newton increment ω ^kGive regularly, each node utilization this locality and neighbor information just can be tried to achieve Δ x by distributed earth ^kYet, calculate

Inverse matrix need global information, so given x ^kThe time, according to described formula (7) directly distributed earth find the solution described antithesis newton increment ω ^kAs seen, utilizing the distributed key of finding the solution described formula (6) of Lagrange-Newton method is as where trying to achieve ω ^kIn addition, how distributedly determine described newton's step-length s ^kAnd guarantee that all iteration points all also are difficult points in feasible zone.The below proposes respectively the present invention and finds the solution described antithesis newton increment ω ^kWith described newton's step-length s ^kDistributed method.

The present invention is easier to be understood in order to make, and is necessary to introduce the relevant knowledge that the matrix decomposition method is found the solution system of linear equations.If G is the real symmetric positive definite matrix of n * n dimension, b is that length is the column vector of n, if G can resolve into two matrix M, N's and, i.e. G=M+N, and M-N is positive definite matrix.Make s (0) be one arbitrarily length be the column vector of n, converge on system of linear equations Gs=b and separate through the iterate sequence { s (t) } that obtains of following formula so.

s(t+1)＝-M ^-1Ns(t)+M ^-1b

The present described antithesis newton of the analysis and utilization distributed iterative of described matrix decomposition technology increment ω ^kIf I _iAnd O _iRepresent respectively incoming link and the outgoing link set of node i, can try to achieve intuitively

That is,

And matrix P ₁, P ₃, P ₄Satisfy: 1.

2. when node j is the neighbours of node i:

3. other: [P ₁] _Ij=0, [P ₃] _Ij=0, [P ₄] _Ij=0.

Because P=FH _k ^-1F ^T, its matrix F is nonsingular, so P is real symmetric positive definite matrix.Now define diagonal matrix J (3n-1) and real matrix K (3n-1), wherein,

K=P-J.As seen, except diagonal element, matrix K, other elements of P all equate.If Z=J-K=2J-P, that is:

${\left[Z\right]}_{\mathrm{ij}}=\left\{\begin{array}{cc}2\×\underset{j=1}{\overset{3n}{\mathrm{\Σ}}}\left(\mathrm{abs}{\left[P\right]}_{\mathrm{ij}}\right)-{\left[P\right]}_{\mathrm{ii}}& i=j\\ -{\left[P\right]}_{\mathrm{ij}}& i\≠j\end{array}\right.$

Obviously,

So matrix J-K positive definite.According to above analysis, converge on the solution ω of described formula (7) through the iterate value that obtains of following formula ^k:

$\mathrm{\ω}(t+1)=-J^{-1}\mathrm{K\ω}\left(t\right)+J^{-1}[(F{x}^k-c)-{\mathrm{FH}}_k^{-1}\▿f\left(x^k\right)]$

Examine following formula, each node utilization this locality and neighbours' information just can the distributed earth iteration be tried to achieve ω ^k

Distributedly among subsequent analysis the present invention find the solution described newton's step-length s ^kMethod.Positive newton's step-length s of the k time iterative process ^kShould guarantee x ^K+1Still in feasible zone.Particularly, establish x ^kIn feasible zone, to each information flow-rate, s ^kShould satisfy

Be not difficult to infer,

Wherein

$b_{l_{\mathrm{ij}}}=\left\{\begin{array}{cc}({\mathrm{\&gamma;}}_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ij}}}^k)/\mathrm{\&Delta;}{r}_{l_{\mathrm{ij}}}^k& \mathrm{\&Delta;}{r}_{l_{\mathrm{ij}}}^k>0\\ -r_{l_{\mathrm{ij}}}^k/\mathrm{\&Delta;}{r}_{l_{\mathrm{ij}}}^k& \mathrm{\&Delta;}{r}_{l_{\mathrm{ij}}}^k<0\end{array}\right.$

In like manner, for each slack variable, s ^kSatisfy 0＜s ^k≤ t _i, wherein,

$t_i=\left\{\begin{array}{cc}\mathrm{\&theta;}& \mathrm{\&Delta;}{y}_i^k>0\\ -y_i^k/\mathrm{\&Delta;}{y}_i^k& \mathrm{\&Delta;}{y}_i^k<0\end{array}\right.$

Wherein, θ is fixed constant, general θ=1 of getting.To sum up analyze,

The present invention determines described newton's step-length s according to following formula ^k:

$s^k=1/\sqrt{d}$ $\left(9\right)$

$d={\mathrm{\&Sigma;}}_{i=1}^n{(\mathrm{\&Delta;}{y}_i^k/y_i^k)}^2+{\mathrm{\&Sigma;}}_{i=1}^n{\mathrm{\&Sigma;}}_{j\&Element;N_i}\max ({(\mathrm{\&Delta;}{r}_{l_{\mathrm{ij}}}^k/r_{l_{\mathrm{ij}}}^k)}^2,{(\mathrm{\&Delta;}{r}_{l_{\mathrm{ij}}}^k/(R_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ij}}}^k))}^2)$

Utilize average coordination approach, each node utilization this locality and neighbours' information can be calculated the approximation of trying to achieve d.Newton's step-length computational methods that the below utilizes the present invention of induction proof to propose can guarantee that all iteration points are all in feasible zone.

Proof: when k=0, x ⁰It is any one initial solution vector in the feasible zone.Suppose x ^kIn feasible zone, only need proof x ^K+1Still in feasible zone, get final product.

If row vector So:

So

Namely

Further,

So

Namely guarantee x ^K+1In feasible zone.

Based on above analysis, the detailed process of described step S1-2 is described, namely work as p to timing, the distributed process of finding the solution described formula (6) of each node.For convenient narration, introduce vectorial u, v, w, and u=Fx-c,

W=J ^-1(u-v), obviously, vectorial u, v, the length of w is 3n-1.Again with vectorial u, v, w, ω is write as respectively: u=(u1 ^T, u2 ^T, u3 ^T) ^T, v=(v1 ^T, v2 ^T, v3 ^T) ^T, w=(w1 ^T, w2 ^T, w3 ^T) ^T, ω=(ω 1 ^T, ω 2 ^T, ω 3 ^T) ^T, vectorial u2, v2, w2, the length of ω 2 is n-1, vectorial u1, v1, w1, ω 1 and u3, v3, w3, the length of ω 3 is n.In addition, with the square formation J of 3n-1 dimension, K is write as:

$J=\left[\begin{array}{ccc}J1\left(n\right)& 0& 0\\ 0& J2(n-1)& 0\\ 0& 0& J3\left(n\right)\end{array}\right]$ With $K=\left[\begin{array}{ccc}{K}_{11}\left(n\right)& K_{12}(n\&times;(n-1))& K_{13}\left(n\right)\\ K_{21}((n-1)\&times;n)& K_{22}(n-1)& K_{23}((n-1)\&times;n)\\ K_{31}\left(n\right)& K_{32}(n\&times;(n-1))& K_{33}\left(n\right)\end{array}\right]$

In the distributed Lagrange-Newton method process that the present invention proposes, the element Computation distribution in the above-mentioned vector sum matrix is finished in each node, and namely node utilizes local information and neighbor information to ask vectorial u1, v1, w1, ω 1, u2, v2, w2, ω 2, u3, v3, w3, ω 3 and matrix J, the respective element of each matrix in block form among the K, but the front has not had the constraint of rate of death among the analysis node n, so node n need not to find the solution u2, v2, w2, ω 2 and matrix J 2, K ₂₁, K ₂₂, K ₂₃In respective element, the below does not remake differentiation to this.Fig. 2 is the distributed flow chart of finding the solution described formula (6) of the Lagrange-Newton of utilization method of the present invention, and concrete steps are:

The S1-2-1 initialization: ordinary node i calculates the energy that sending/receiving unit's bit information need to consume, namely

With

J ∈ N _iPreserve simultaneously information incidence g _iWith initial cells energy b _iThen preserve the outgoing link information flow-rate according to described step S1-1 or described step S1-3

J ∈ N _i, slack variable y _i〉=0, rate of death q _i＞0;

S1-2-2 ordinary node i calculates described target function at current outgoing link information flow-rate

J ∈ N _i, rate of death q _i, and slack variable y _iThe single order partial derivative at place Inverse with second dervative Q _Ii, Y _Ii

S1-2-3 ordinary node i will

J ∈ N _i, q _i, y _iAnd the result of calculation among the described step S1-2-2 is broadcast to neighbor node;

The S1-2-4 node i is calculated ω 1 _i, ω 2 _i, ω 3 _i, can be divided into following substep:

S1-2-4-1 is according to described matrix J, and J1 is calculated in the definition of K _Ii, J2 _Ii, J3 _Ii[K _1s] _Ij, [K _2s] _Ij, [K _2s] _Ij, j ∈ N _i∪ j=i, s=1,2,3.

S1-2-4-2 calculates u1 _i, u2 _i, u3 _i, i.e. the left side of equality constraint in the described formula (5).

S1-2-4-3 because

V1 _i, v2 _i, v3 _iCan calculate respectively according to following formula:

$v{1}_i={\mathrm{\&Sigma;}}_{j\&Element;N_i}(R_{l_{\mathrm{ij}}{l}_{\mathrm{ij}}}\&dtri;f\left(r_{l_{\mathrm{ij}}}\right)-R_{l_{\mathrm{ji}}{l}_{\mathrm{ji}}}\&dtri;f\left(r_{l_{\mathrm{ji}}}\right))$

$v{2}_i=Q_{\mathrm{ii}}\&dtri;f\left(q_i\right)-Q_{\mathrm{jj}}\&dtri;f\left(q_j\right),j=N_{E_i}$

$v{3}_i={\mathrm{\&Sigma;}}_{l\&Element;O_i}{R}_{\mathrm{ll}}{e}_l^t\&dtri;f\left(r_l\right)+{\mathrm{\&Sigma;}}_{l\&Element;I_i}{R}_{\mathrm{ll}}{e}_l^r\&dtri;f\left(r_l\right)-Q_{\mathrm{ii}}\&dtri;f\left(q_i\right)b_i+Y_{\mathrm{ii}}\&dtri;f\left(y_i\right)$

S1-2-4-4 calculates w1 _i, w2 _i, w3 _i, that is:

w1 _i＝J1 _ii ^-1(u1 _i-v1 _i)；

w2 _i＝J2 _ii ^-1(u2 _i-v2 _i)；

w3 _i＝J3 _ii ^-1(u3 _i-v3 _i)。

S1-2-4-5 calculates ω 1 _i, ω 2 _i, ω 3 _iArbitrary initial antithesis newton increment ω 1 at first _i(0), ω 2 _i(0), ω 3 _i(0), then according to following formula difference iterative computation ω 1 _i(t), ω 2 _i(t), ω 3 _i(t), until satisfy designated precision.In the iterative process, need to exchange ω 1 between the neighbor node _i(t), ω 2 _i(t), ω 3 _i(t) value.

Wherein,

From 5 top sub-steps, be not difficult to find out that node i utilizes local information and information of neighbor nodes can calculate ω 1 _i, ω 2 _i, ω 3 _i

Formula below S1-2-5 ordinary node i utilizes calculates respectively link information flow rate initial value J ∈ N _i, slack variable y _i, rate of death q _iNewton direction:

$\mathrm{\&Delta;}{r}_{l_{\mathrm{ij}}}=-R_{l_{\mathrm{ij}}{l}_{\mathrm{ij}}}\{\&dtri;f\left(r_{l_{\mathrm{ij}}}\right)+(\mathrm{\&omega;}{1}_i-\mathrm{\&omega;}{1}_j)+(e_{l_{\mathrm{ij}}}^t\mathrm{\&omega;}{3}_i+e_{l_{\mathrm{ji}}}^r\mathrm{\&omega;}{3}_j)\}$

$\mathrm{\&Delta;}{q}_i=-Q_{\mathrm{ii}}\{\&dtri;f\left(q_i\right)+(\mathrm{\&omega;}{2}_i-{\mathrm{\&Sigma;}}_{i=N_{E_i}}\mathrm{\&omega;}{2}_j)-b_i\mathrm{\&omega;}{3}_i\}$

$\mathrm{\&Delta;}{y}_i=-Y_{\mathrm{ii}}\{\&dtri;f\left(y_i\right)+\mathrm{\&omega;}{3}_i\}$

S1-2-6 ordinary node i at first calculates (Δ y _i/ y _i) ²With J ∈ N _i, then result of calculation is broadcast to neighbours' ordinary node;

S1-2-7 ordinary node i utilizes average coordination approach to calculate newton's step-length s, and the computing formula of described step-length is:

$s=1/\sqrt{d}$

$d={\mathrm{\&Sigma;}}_{i=1}^n{(\mathrm{\&Delta;}{y}_i/y_i)}^2+{\mathrm{\&Sigma;}}_{i=1}^n{\mathrm{\&Sigma;}}_{j\&Element;N_i}\max ({(\mathrm{\&Delta;}{r}_{l_{\mathrm{ij}}}/r_{l_{\mathrm{ij}}})}^2,{(\mathrm{\&Delta;}{r}_{l_{\mathrm{ij}}}/(R_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ij}}}))}^2);$

S1-2-8 ordinary node i upgrades described outgoing link information flow-rate according to following formula

J ∈ N _i, described slack variable y _i, and described rate of death q _i:

$r_{l_{\mathrm{ij}}}=r_{l_{\mathrm{ij}}}+\mathrm{s\&Delta;}{r}_{l_{\mathrm{ij}}}$

y _i＝y _i+sΔy _i

q _i＝q _i+sΔq _i

S1-2-9 judges end condition: if

Be constant and

Finish; Otherwise forward S1-2-2 to.

Claims (3)

1. wireless-sensor network distribution type method for routing based on the Lagrange-Newton method, the system model of described wireless sensor network non-directed graph G (V, Ψ) expression, wherein V is the set of sensor node, comprise n ordinary node and 1 Sink node, described ordinary node and described Sink node are referred to as node; Ψ is the set of Radio Link, always has m bar link, and described system model is defined as follows:

1. P _Max: the node maximum transmit power, if carry out the required transmitting power of radio communication between node i and the node j less than P _Max, claim so the Radio Link l that node i flows to node j _Ij∈ Ψ exists, and Radio Link is two-way existence, l _IjWrite a Chinese character in simplified form into l, especially, described Sink node does not have outgoing link;

2. N _i: the neighbor node of node i is gathered, and namely has the node set of Radio Link with node i;

3.

Node i is to the Radio Link l of node j _IjThe maximum information flow rate that allows,

Write a Chinese character in simplified form into γ _l

4. b _i: the initial cells energy of described ordinary node i, the energy of described Sink node is infinitely great;

5. g _i: the information incidence of described ordinary node i, g _i〉=0;

6.

Node i is by Radio Link l _IjSend the energy that unit information consumes to neighbor node j;

7.

Node i receives from Radio Link l _JiThe energy that consumes of the unit information of egress j;

It is characterized in that: described wireless-sensor network distribution type method for routing may further comprise the steps:

The A wireless sensor network maximizes the protruding optimization problem of equality constraint that route is modeled as approximately equivalent life cycle:

$\min f\left(x\right)=p\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{q}_i^2-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\log \left(y_i\right)-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j\&Element;N_i}{\mathrm{\&Sigma;}}[\log \left(r_{l_{\mathrm{ij}}}\right)+\log ({\mathrm{\&gamma;}}_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ij}}})]$

s.t. $\underset{j\&Element;N_i}{\mathrm{\&Sigma;}}(r_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ji}}})-g_i=0,$ i=1,…,n

$\underset{j\&Element;N_i}{\mathrm{\&Sigma;}}(e_{l_{\mathrm{ij}}}^t{r}_{l_{\mathrm{ij}}}+e_{l_{\mathrm{ji}}}^r{r}_{l_{\mathrm{ji}}})+y_i-q_i{b}_i=0,$ i=1,…,n

q _i-q _j=0

In the formula, x=({ r _Ij, q ₁..., q _n, y ₁..., y _n) ^T

It is the link information flow rate that node i arrives node j; F (x) is called target function; q _iThe rate of death that is called ordinary node i, y _iBe called slack variable,

Expression ordinary node i only selects neighbours' ordinary node j to be used for q _iConstraint; P is called barrier parameter, when p trend towards+during ∞, described wireless sensor network maximizes routing issue and the protruding optimization problem equivalent of described equality constraint life cycle; The corresponding matrix form of following formula is:

minimize $f\left(x\right)=p\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{q}_i^2-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\log \left(y_i\right)-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j\&Element;N_i}{\mathrm{\&Sigma;}}[\log \left(r_{l_{\mathrm{ij}}}\right)+\log ({\mathrm{\&gamma;}}_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ij}}})]$

subject to Fx=c

In the formula, $c={(g^T,0)}^T,F=\begin{array}{}\end{array}\left[\begin{array}{ccc}A& 0& 0\\ 0& D& 0\\ E& B& I\end{array}\right],$ Wherein: I is unit matrix; G=(g ₁, g ₂..., g _n) ^TThe information incidence vector that is called described ordinary node; B (n) is called described ordinary node initial cells energy matrix, and B is diagonal matrix, and B _Ii=-b _i(n * m) is called node-link association matrix to A; (n * m) is called radio communication energy consumption matrix to E; ((n-1) * n) is called described ordinary node rate of death constraint matrix to D; Described matrix A, E, the concrete form of D is respectively:

The protruding optimization problem of the described equality constraint of B is found the solution by the sequential unconstrained minimization technique method, and concrete steps are as follows:

The S1-1 initialization: in the interval range of feasible zone, arbitrary initial variable x ⁰, x ⁰Feasible solution not necessarily; The described barrier parameter p of initialization〉1; Simultaneously initialization constant μ〉1, constant ε〉0;

S1-2 is based on the Lagrange-Newton method, with x ⁰The primary iteration point calculates p to the optimal solution x of the protruding optimization problem of the described equality constraint of timing ^*(p);

S1-3 upgrades x ⁰, i.e. x ⁰=x ^*(p);

S1-4 checks end condition: if p＜ε forwards S1-5 to; Otherwise the sequential unconstrained minimization technique method finishes, x ^*(p)=({ r _Ij ^*, q ₁ ^*..., q _n ^*, y ₁ ^*..., y _n ^*) ^TBe the solution of the protruding optimization problem of described equality constraint, make the information flow-rate r of maximized all links of described network life cycle then be

S1-5 upgrades p, and namely p=μ p forwards S1-2 to.

2. the wireless-sensor network distribution type method for routing based on the Lagrange-Newton method according to claim 1, it is characterized in that: the optimal solution of the protruding optimization problem of equality constraint is distributed described in the described step S1-2, namely utilize the split matrix technology, finding the solution of described Lagrange-Newton method be distributed in each node in the network calculate, each node only needs and the neighbor node exchange message in computational process, and each node is carried out according to following steps:

The S1-2-1 initialization: ordinary node i calculates the energy that sending/receiving unit's bit information need to consume, namely

With

J ∈ N _iPreserve simultaneously information incidence g _iWith initial cells energy b _iThen preserve the outgoing link information flow-rate according to described step S1-1 or described step S1-3

J ∈ N _i, slack variable y _i〉=0, rate of death q _i0;

S1-2-2 ordinary node i calculates described target function at current outgoing link information flow-rate

J ∈ N _i, rate of death q _i, and slack variable y _iThe single order partial derivative at place

Inverse with second dervative

S1-2-3 ordinary node i will

J ∈ N _i, q _i, y _iAnd the result of calculation among the described step S1-2-2 is broadcast to neighbor node;

S1-2-4 ordinary node i utilizes local information and information of neighbor nodes to calculate antithesis newton increment ω 1 _i, ω 2 _i, ω 3 _i

Formula below S1-2-5 ordinary node i utilizes calculates respectively link information flow rate initial value J ∈ N _i, slack variable y _i, rate of death q _iNewton direction:

$\mathrm{\&Delta;}{r}_{l_{\mathrm{ij}}}=-R_{l_{\mathrm{ij}}{l}_{\mathrm{ij}}}\{\&dtri;f\left(r_{l_{\mathrm{ij}}}\right)+(\mathrm{\&omega;}{1}_i-\mathrm{\&omega;}{1}_j)+(e_{l_{\mathrm{ij}}}^t\mathrm{\&omega;}{3}_i+e_{l_{\mathrm{ji}}}^r\mathrm{\&omega;}{3}_j)\}$

$\mathrm{\&Delta;}{q}_i=-Q_{\mathrm{ii}}\{\&dtri;f\left(q_i\right)+(\mathrm{\&omega;}{2}_i-{\mathrm{\&Sigma;}}_{i=N_{E_j}}\mathrm{\&omega;}{2}_j)-b_i\mathrm{\&omega;}{3}_i\}$

$\mathrm{\&Delta;}{y}_i=-Y_{\mathrm{ii}}\{\&dtri;f\left(y_i\right)+\mathrm{\&omega;}{3}_i\}$

S1-2-6 ordinary node i at first calculates (Δ y _i/ y _i) ²With $\max ({({\mathrm{\&Delta;r}}_{l_{\mathrm{ij}}}/r_{l_{\mathrm{ij}}})}^2,{({\mathrm{\&Delta;r}}_{l_{\mathrm{ij}}}/({\mathrm{\&gamma;}}_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ij}}}))}^2),$ J ∈ N _i, then result of calculation is broadcast to neighbours' ordinary node;

S1-2-7 ordinary node i utilizes average coordination approach to calculate newton's step-length s, and the computing formula of described step-length is:

$s=1/\sqrt{d}$

$d={\mathrm{\&Sigma;}}_{i=1}^n{({\mathrm{\&Delta;y}}_i/y_i)}^2+{\mathrm{\&Sigma;}}_{i=1}^n{\mathrm{\&Sigma;}}_{j\&Element;N_i}\max ({({\mathrm{\&Delta;r}}_{l_{\mathrm{ij}}}/r_{l_{\mathrm{ij}}})}^2,({\mathrm{\&Delta;r}}_{l_{\mathrm{ij}}}/{(R_{l_{\mathrm{ij}}}-r_{l_{\mathrm{ij}}})}^2));$

S1-2-8 ordinary node i upgrades described outgoing link information flow-rate according to following formula J ∈ N _i, described slack variable y _i, and described rate of death q _i:

$r_{l_{\mathrm{ij}}}=r_{l_{\mathrm{ij}}}+\mathrm{s\&Delta;}{r}_{l_{\mathrm{ij}}}$

y _i=y _i+sΔy _i

q _i=q _i+sΔq _i

S1-2-9 judges end condition: if Δ q _i/ q _i＜θ, θ are constant and θ＜＜1, finish; Otherwise forward S1-2-2 to.

3. the wireless-sensor network distribution type method for routing based on the Lagrange-Newton method according to claim 1 and 2 is characterized in that: ordinary node i selects neighbours' ordinary node j to be used for q _iThe method of constraint be: during initialization, ordinary node n at first broadcasts a packets of information, after its neighbours' ordinary node i receives this packets of information, selects node n to be used for q _iConstraint, then ordinary node i continues this packets of information of broadcasting, its neighbours' ordinary node j then selects ordinary node i to be used for q _jConstraint, the like, until the selected neighbours' ordinary node of all ordinary nodes.

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