A Recurrent Neural Network for Nonlinear Fractional Programming

This paper presents a novel recurrent time continuous neural network model which performs nonlinear fractional optimization subject to interval constraints on each of the optimization variables. The network is proved to be complete in the sense that the set of optima of the objective function to be minimized with interval constraints coincides with the set of equilibria of the neural network. It is also shown that the network is primal and globally convergent in the sense that its trajectory cannot escape from the feasible region and will converge to an exact optimal solution for any initial point being chosen in the feasible interval region. Simulation results are given to demonstrate further the global convergence and good performance of the proposing neural network for nonlinear fractional programming problems with interval constraints.


Introduction
Compared with the well-known applications of nonlinear programming to various branches of human activity, especially to economics, the applications of fractional programming are less known until now.Of course, the linearity of a problem makes it easier to tackle and hence contributes its wide recognition.However, it is certain that not all real-life economic problems can be described by linear models and hence are not likely applications of linear programming.Fractional programming is a nonlinear programming method that has known increasing exposure recently and its importance in solving concrete problems is steadily increasing.Moreover, it is known that the nonlinear optimization models describe practical problems much better than the linear optimization models do.
The fractional programming problems are particularly useful in the solution of economic problems in which various activities use certain resources in various proportions, while the objective is to optimize a certain indicator, usually the most favorable return-onallocation ratio subject to the constraint imposed on the availability of goods.The detailed descriptions of these models can be found in Charnes et al. 1 , Patkar 2 , and Mjelde 3 .Besides the economic applications, it was found that the fractional programming problems also appeared in other domains, such as physics, information theory, game theory, and others.Nonlinear fractional programming problems are, of course, the dominant ones for their much widely applications, see Stancu-Minasian 4 in details.
As it is known, conventional algorithms are time consuming in solving optimization problems with large-scale variables and so new parallel and distributed algorithms are more competent then.Artificial neural networks RNNs governed by a system of differential equations can be implemented physically by designated hardware with integrated circuits and an optimization process with different specific purposes could be conducted in a truly parallel way.An overview and paradigm descriptions of various neural network models for tackling a great deal of optimization problems can be found in the book by Cichocki and Unbehauen 5 .Unlike most numerical algorithms, neural network approach can handle, as described in Hopfield's seminal work 6, 7 , optimization process in real-time on line and hence to be the top choice.
Neural network models for optimization problems have been investigated intensively since the pioneer work of Wang et al., see 8-14 .Wang et al. proposed several different neural network models for solving convex programming problems 8 , linear programming 9, 10 , which, proved to be globally convergent to the problem's exact solutions.Kennedy and Chua 11 developed a neural network model for solving nonlinear programming problems where a penalty parameter needed to tune in the optimization process and hence only approximate solutions were generated.Xia and Wang 12 gave a general neural network model designing methodology which put together many gradient-based network models for solving the convex programming problems under this framework with globally convergent stability.Neural networks for the quadratic optimization and nonlinear optimization with interval constraints were developed by Bouzerdorm and Pattison 13 and Liang and Wang 14 , respectively.
All these neural networks can be classified into the following three types: 1 the gradient-based models 8-10 and its extension 12 ; 2 the penalty-function-based model 11 ; 3 the projection based models 13, 14 .Among them the first was proved to have the global convergence 12 ; the third quasi-convergence 8-10 only when the optimization problems are convex programming problems.The second could only be demonstrated to have local convergence 11 and more unfortunately, it might fail to find exact solutions, see 15 for a numerical example.Because of this, the penalty-function-based model has little applications in practice.As it is known, nonlinear fractional programming does not belong to convex optimization problems 4 and how to construct a good performance neural network model to solve this optimization problem becomes a challenge now since.Motivated by this idea, a promising recurrent continous-time neural network model is going to be proposed in the present paper.The proposing RNN model has the following two most important features.1 The model is complete in the sense that the set of optima of the nonlinear fractional programming with interval constraints coincides with the set of equilibria of the proposing RNN model. 2 The RNN model is invariant with respect to the problem's feasible set and has the global convergence property in the sense that all the trajectories of the proposing network converge to the exact solution set for any initial point starting at the feasible interval region.These two properties demonstrate that the proposing network model is quite suitable for solving nonlinear fractional programming problems with interval constraints.
Remains of the paper are organized as follows.Section 2 formulates the optimization problem and Section 3 describes the construction of the proposing RNN model.Complete property and global convergence of the proposing model are discussed in Sections 4 and 5, respectively.Section 6 gives some typical application areas of the fractional programming.Illustrative examples with computational results are reported in Section 7 to demonstrate further the good performance of the proposing RNN model in solving the intervalconstrained nonlinear fractional programming problems.Finally, Section 8 is a conclusion remark which presents a summary of the main results of the paper.

Problem Formulation
The study of the nonlinear fractional programming with interval constraints is motivated by the study of the following linear fractional interval programming: where: Accordingly, constraints a ≤ Ax ≤ b can always be transformed into a ≤ x ≤ b by change of variable method without changing the programming's format, see 13 for quadratic programming and 17 for linear fractional interval programming.So, it is necessary to pay our attention on problem 2.1 only.As the existing studies on problem 2.1 , see 16-19 , focused on the classical method which is time consuming in optimization computational aspects, it is sure that the neural network method should be the top choice to meet the real-time computation requirement.To reach this goal, the present paper is to construct a RNN model that is available both for solving nonlinear fractional interval programming and for linear fractional interval programming problem 2.1 as well.
Consider the following more general nonlinear fractional programming problem: where g x , h x are continuously differentiable function defined on an open convex set O ⊆ R n which contains the problem's feasible set W {x | a ≤ x ≤ b} and x, a, b the same as in problem 2.1 , see the previous v -vi .Similarly, we suppose the objective function's dominator g x always keeps a constant sign, say g x > 0. As the most fractional programming problems arising in real-life world associate a kind of generalized convex properties, we suppose the objective function F x to be pseudoconvex over O.There are several sufficient conditions for the function F x g x /h x being pseudoconvex, two of which, see 20 , are 1 g is convex and g ≥ 0, while h concave and h > 0; 2 g is convex and g ≤ 0, while h is convex and h > 0. It is easy to see that the problem 2.1 is a special case of problem 2.3 .
We are going to state the neural network model which can be employed for solving problem 2.3 and so for problem 2.1 as well.Details are described in the coming section.

The Neural Network Model
Consider the following single-layered recurrent neural network whose state variable x is described by the differential equation: where ∇ is the gradient operator and f W : R n → W is the projection operator defined by  For the interval constrained feasible set W, the operator f W can be expressed explicitly as

3.3
The activation function to one node of the neural network model 3.1 is the typical piecewise linear f W i x i which is visibly illustrated in Figure 1.
To make a clear description of the proposed neural network model, we reformulate the compact matrix form 3.1 as the following component ones: When the RNN model 3.1 is employed to solve optimization problem 2.3 , the initial state is required to be mapped into the feasible interval region W.That is, for any x 0 x 0 1 , x 0 2 , . . ., x 0 n ∈ R n , the corresponding neural trajectory initial point should be chosen as x 0 f W x 0 , or in the component form, Accordingly, the architecture of the proposed neural network model 3.4 is composed of n integrators, n processors for F x , 2n piece-wise linear activation functions, and 2n summers.Let the equilibrium state of the RNN model 3.1 be Ω e which is defined by the following equation:

3.5
The relationship between the minimizer set Ω * of problem 2.3 and the equilibrium set Ω e is explored in the following section.It is guaranteed that the two sets coincide exactly and, this case is the most available expected one in the neural network model designs.

Complete Property
As proposed for binary-valued neural network model in 21 , a neural network is said to be regular or normal if the set of minimizers of an energy function is a subset or superset of the set of the stable states of the neural network, respectively.If the two sets are the same, the neural network is said to be complete.The regular property implies the neural network's reliability and normal effectiveness, respectively, for the optimization process.Complete property means both reliability and effectiveness and it is the top choice in the neural network designing.
Here, for the continuous-time RNN model 3.1 , we say the model to be regular, normal, and complete respectively if three cases of Ω * ⊆ Ω e , Ω e ⊆ Ω * , and Ω * Ω e occur, respectively.The complete property of the neural network 3.1 is stated in the following theorem.

Stability Analysis
First, it can be shown that the RNN model 3.1 has a solution trajectory which is global in the sense that its existence interval can be extended to ∞ on the right hand for any initial point in W.
The continuity of the right hand of 3.1 means, by Peano's local existence theorem, see 24 , that there exists a solution x t; x 0 for t ∈ 0, t max with any initial point x 0 ∈ W, here t max is the maximal right hand point of the existence interval.The following lemma states that this t max to be ∞.
Lemma 5.1.The solution x t; x 0 of RNN model 3.1 with any initial point x 0; x 0 x 0 ∈ W is bounded and so, it can be extended to ∞.
Proof.It is easy to check that the solution x t x t; x 0 for t ∈ 0, t max with initial condition x 0; x 0 x 0 is given by x t e −t x 0 e −t t 0 e s f W x s − ∇F x s ds.

5.1
Obviously, mapping f W is bounded, that is f W ≤ K for some positive number K > 0, where • is the Euclidean 2 norm.It follows from 5.1 that ≤ max x 0 , K .

5.2
Thus, solution x t is bounded and so, by the extension theorem for ODEs, see 24 , it can be concluded that t max ∞ which completes the proof of this lemma.Now, we are going to show another vital dynamical property which says the set W is positive invariant with respect to the RNN model 3.1 .That is, any solution x t starting from a point in W, for example, x 0 ∈ W, it will stay in W for all time t elapsing.Additionally, we can also prove that any solution starting from outside of W will either enter into the set W in finite time elapsing and hence stay in it for ever or approach it eventually.Theorem 5.2.For the neural dynamical system 3.1 , the following two dynamical properties hold: Proof.Method to prove this theorem can be found in 14 and for the purpose of completeness and readability, here we give the whole proof as follows again.
Suppose that, for i 1, . . ., n, W i {x i ∈ R | a i ≤ x i ≤ b i } and x 0 i x i 0; x 0 ∈ W i .We first prove that for all i 1, 2, . . ., n, the ith component x i t x i t; x 0 belongs to W i , that is, x i t ∈ W i for all t ≥ 0. Let Noting that x i t ∈ W for t ∈ 0, t * i and assumption 5.4 implies x i t * i a i , so, by 5.6 , we get This is in contradiction with the assumption 5.4 .So, t * i ∞, that is x i t ∈ W i for all t ≥ 0. This means W is positive invariant and hence a is guaranteed.
Second, for some i, suppose x 0 i x i 0; x 0 / ∈ W i .If there is a t * i > 0 such that x t * i ∈ W i , then, according to a , x i t will stay in W i for all t ≥ t * i .That is x i t will enter into W i .Conversely, for all t ≥ 0, suppose x i t / ∈ W i .Without loss of generality, we assume that x i t < a i .It can be guaranteed by a contradiction that sup{x i t | t ≥ 0} a i .If it is not so, note that x i t < a i , then sup{x i t | t ≥ 0} m < a i .It can be followed by 3.4 that Integrating 5.8 gives us which is a contradiction because of x i t < a i .Thus, we obtain sup{x i t | t ≥ 0} a i .This and the previous argument show that, for x 0 / ∈ W, either x t enters into W in finite time and hence stays in it for ever or ρ t dist x t , W → 0, as t → ∞.
We can now explore the global convergence of the neural network model 3.1 .To proceed, we need an inequality result about the projection operator f W and the definition of convergence for a neural network.Definition 5.3.Let x t be a solution of system ẋ F x .The system is said to be globally convergent to a set X with respect to set W if every solution x t starting at W satisfies ρ x t , X −→ 0, as t −→ ∞, 5.10 here ρ x t , X inf y∈X x − y and x 0 x 0 ∈ W.
Definition 5.4.The neural network 3.1 is said to be globally convergent to a set X with respect to set W if the corresponding dynamical system is so.
Theorem 5.6.The neural network 3.1 is globally convergent to the solution set Ω * with respect to set W.
Proof.From Lemma 5.5, we know that

5.12
Let v x − ∇F x and u x, then

5.14
Define an energy function F x , then, differentiating this function along the solution x t of 3.1 gives us

5.15
According to 5.14 , it follows that

Mathematical Problems in Engineering 11
It means the energy of F x is decreasing along any trajectory of 3.1 .By Lemma 5.1, we know the solution x t is bounded.So, F x is a Liapunov function to system 3.1 .Therefore, by LaSalle's invariant principle 25 , it follows that all trajectories of 3.1 starting at W will converge to the largest invariant subset Σ of set E like However, it can be guaranteed from 5.16 that dF/dt 0 only if f W x − ∇F x − x 0, which means that x must be an equilibrium of 3.1 or, x ∈ Ω.Thus, Ω is the convergent set for all trajectories of neural network 3.1 starting at W. Noting that Theorem 4.1 tells us that Ω * Ω and hence, Theorem 5.6 is proved to be true then.
Up to now, we have demonstrated that the proposed neural network 3.1 is a promising neural network model both in implementable construction sense and in theoretic convergence sense for solving nonlinear fractional programming problems and linear fractional programming problems with bound constraints.Certainly, it is also important to simulate the network's effectiveness by numerical experiment to test its performance in practice.In next section, we will focus our attention on handling illustrative examples to reach this goal.

Typical Application Problems
This section contributes to some typical problems from various branches of human activity, especially in economics and engineering, that can be formulated as fractional programming.
We choose three problems from information theory, optical processing of information and macroeconomic planning to identify the various applications of fractional programming.

Information Theory
For calculating maximum transmission rate in an information channel Meister and Oettli 26 , Aggarwal and Sharma 27 employed the fractional programming described briefly as follows.
Consider a constant and discrete transmission channel with m input symbols and n output symbols, characterized by a transition matrix P p ij , i 1, . . ., m, p ij ≥ 0, i p ij 1, where p ij represents the probability of getting the symbol i at the output subject to the constraint that the input symbol was j.The probability distribution function of the inputs is denoted by x x j , and obviously, x j ≥ 0, j x j 1. Define the transmission rate of the channel as: T x i j x j p ij log p ij / k x k p ik j t j x j . 6.1 The relative capacity of the channel is defined by the maximum of T x , and we get the following fractional programming problem: With the notations:

Optical Processing of Information
In some physics problems, fractional programming can also be applied.In spectral filters for the detection of quadratic law for infrared radiation, the problem of maximizing the signalto-noise ration appears.This means to maximize the filter function φ x a x 2 x Bx β 6.5 on the domain S {x ∈ R n , 0 ≤ x i ≤ 1, i 1, . . ., n} in which a and β are strict positive vector, and constant, respectively, B is a symmetric and positive definite matrix, a x represents the input signal, and x Bx β represents the variance in the background signal.The domain of the feasible solutions S illustrates the fact that the filter cannot transmit more than 100% and less than 0% of the total energy.The optical filtering problems are very important in today's information technology, especially in coherent light applications, and optically based computers have already been built.

Macroeconomic Planning
One of the most significant applications of fractional programming is that of dynamic modeling of macroeconomic planning using the input-output method.Let Y t be the national income created in year t.Obviously, Y t i Y i t .If we denote by C ik t the consumption, in branch k, of goods of type i that were created in branch i and by I ik the part of the national income created in branch i and allocated to investment in branch k, then the following repartition equation applies to the national income created in branch i: Mathematical Problems in Engineering 13 The increase of the national income in branch k is function of the investment made in this branch where I k i I ik .In these conditions, the macroeconomic planning leads to maximize the increase rate of the national income:

Illustrative Examples
We give some computational examples as simulation experiment to show the proposed network's good performance.
Example 7.1.Consider the following linear fractional programming: This problem has an exact solution x * 0, 0 T with the optimal value F x * 1/3.The gradient of F x can be expressed as and pay attention to 7.2 , we get

7.4
The dynamical systems are given by

7.5
Various combinations of 7.5 formulate the proposed neural network model 3.1 to this problem.Conducted on MATLAB 7.0., by ODE 23 solver, the simulation results are carried out and the transient behaviors of the neural trajectories x 1 , x 2 starting at x 0 0.4, 1 T , which is in the feasible region W, are shown in Figure 3.It can be seen visibly from the figure that the proposed neural network converges to the exact solution very soon.Also, according to b of Theorem 5.2, the solution may be searched from outside of the feasible region.Figure 4 shows this by presenting how the solution of this problem is located by the proposed neural trajectories from the initial point x 0 0.5, 3 T which is not in W. Example 7.2.Consider the following nonlinear fractional programming:

7.6
This problem has an exact solution x * 1, 1 T with the optimal value F x * 1/13.The gradient of F x can be expressed as The dynamical systems are given by Figure 6 presents a trajectory from outside of W, here 4, 3 , it can be seen clearly from this that the solution of this problem is searched by the proposed neural trajectory soon.

Conclusions
In this paper, we have proposed a neural network model for solving nonlinear fractional programming problems with interval constraints.The network is governed by a system of differential equations with a projection method.
The stability of the proposed neural network has been demonstrated to have global convergence with respect to the problem's feasible set.As it is known, the existing neural network models with penalty function method for solving nonlinear programming problems may fail to find the exact solution of the problems.The new model has overcome this stability defect appearing in all penalty-function-based models.Certainly, the network presented here can perform well in the sense of real-time computation which, in the time elapsing sense, is also superior to the classical algorithms.Finally, numerical simulation results demonstrate further that the new model can act both effectively and reliably on the purpose of locating the involved problem's solutions.

Figure 1 :
Figure 1: The activation function f i x i of the neural network model 3.1 .

Figure 2 :
Figure 2: Functional block diagram of the neural network model 3.1 .

2 Figure 3 :
Figure 3: Transient behaviors of neural trajectories x 1 , x 2 from the inside of W.

2 Figure 4 :
Figure 4: Transient behaviors of neural trajectories x 1 , x 2 from the outside of W.
on MATLAB 7.0., by ODE 23 solver, the transient behaviors of the neural trajectories x 1 , x 2 from inside of the feasible region W, here x 0 2, 3 , are depicted in Figure5which shows the rapid convergence of the proposed neural network.

2 tFigure 6 :
Figure 6: Transient behaviors of neural trajectories x 1 , x 2 from the outside of W.
Charnse et al. gave a different method which transformed the fractional interval problem into an equivalent problem like 2.1 by using the generalized inverse of A, and the explicit solutions were followed then.Also, B ühler 19 transformed the problem into another equivalent one of the same format, to which Mathematical Problems in Engineering he associated a linear parametric program used to obtain solution for the original interval programming problem.
Theorem 4.1.The RNN model 3.1 is complete, that is, Ω * Ω e .Proof.See 20, Lemma 11.4.1 .Let F x : R n → R be a differentiable pseudoconvex function on an open set Y ⊆ R n , and W ⊆ Y any given nonempty and convex set.Then x * is an optimal solution to the problem of minimizing F x subject to x ∈ W if and only if x − x * T ∇F x * ≥ 0 for all x ∈ W.
I ik t 6.8 subject to the constraints C k t ≥ max C k , C k 0 , where C k t i C ik t , I k 0 ≤ I k t ≤ I k max , and C k represents minimum consumption attributed to branch k whereas I k max is the maximum level of investments for branch k.
Transient behaviors of neural trajectories x 1 , x 2 from the inside of W.