A Modified Liu-Storey Conjugate Gradient Projection Algorithm for Nonlinear Monotone Equations

In this paper, a modified Liu-Storey (LS) conjugate gradient projection algorithm is proposed for solving nonlinear monotone equations based on a hyperplane projection technique. The proposed method is a derivative-free method and can be applied to solving large-scale nonsmooth equations for its lower storage requirement. We can establish its global convergence results under some suitable conditions. Numerical results show that this algorithm is efficient and promising. Mathematics Subject Classification: 65K05, 90C26


Introduction
Considering the square nonlinear system of nonlinear monotone equations where g : n → n is continuous and monotone, i.e. g(x)−g(y), x−y ≥ 0 for ∀x, y ∈ n .Nonlinear monotone equations have many practical background such as the first order necessary condition of the unconstrained convex optimization problem, the subproblems in the generalized proximal algorithms with Bregman distances [4], and economic equilibrium problems [2].Different methods have been developed for nonlinear systems of monotone equations.Newton's method and quasi-Newton methods [9,15] are particularly welcomed for their local superlinear convergence property.However, they are typically unattractive for large-scale nonlinear systems of equations because they need to solve a linear system by using the Jacobian matrix or an approximation of it.The method in [16] is based on the Barzilai-Borwein steplength.Li et al. [6] proposed derivative-free approaches based on modified PRP conjugate gradient techniques for solving (1).A prominent feature of these methods is that the search direction does not need gradient information, therefore these methods can be applied for nonsmooth equations.Recently, many numerical methods [3,11,13,14] for nonlinear equations have been presented.
In this paper, we propose a modified Liu-Storey conjugate gradient projection algorithm for nonlinear monotone equations (1).This paper is organized as follows.In the next section, we discuss the modified LS conjugate gradient projection algorithm for nonlinear monotone equations and prove its global convergence.Some preliminary numerical results are presented in Section 3. Finally, we have a conclusion section.

Algorithm
First we recall the Liu-Storey(LS) conjugate gradient method [7] for unconstrained minimization problem: where f : n → is continuously differentiable.The iterative formula is often used in the CG method, where x k is the kth iteration point, α k > 0 is the steplength, and d k is the kth search direction defined by where g k+1 is gradient of f (x) at the point x k+1 and β k ∈ is a scalar.The LS conjugate gradient formula is where y k = g k+1 − g k .Based on this method, we will present the following modified LS formula for nonlinear monotone equations ( 1) where γ > 0 is a constant.It is easy to see that the given algorithm can be reduced to a standard LS method if exact line search is used.
In general, an iterative method generates the next iteration by (3).For monotone equations, it is desirable to accelerate the iteration process by exploring the monotonicity of the equation.Let With the monotonicity of g, the hyperplane strictly separates the current iteration x k from the solution set of (1).Observing this fact, Solodov & Svaiter [9] advised letting the next iteration x k+1 be the projection of z k onto this hyperplane.Specifically, x k+1 is determined by We will adopt this projection strategy to propose a modified LS conjugate gradient projection algorithm for solving nonlinear monotone equations (1).The steps of our model method are stated as follows.
Before we proceed to study some properties of Algorithm 2.1, we make the following assumption.Assumption A (i) The solution set of the problem (1) is nonempty.(ii) g(x) is Lipschitz continuous on n , i.e. there exists a positive constant L such that g The following lemma shows that Algorithm 2.1 has the sufficiently descent property.
Lemma 2.2.Let Assumption A holds.Then Algorithm 2.1 will produce an iteration z k = x k + α k d k in a finite number of backtracking steps.
Proof.Suppose that g(x k ) → 0 does not hold, or Algorithm 2.1 stops.Then there exists a constant 0 > 0 satisfying We prove this lemma by contradiction.Suppose that the condition (7) does not hold for some iteration indexes k * .Let α By Assumption A and (9) in Lemma 2.1, we have By Assumption A (ii), we conclude that there exists a constant M > 0 satisfying Then it follows from (10) that Thus, for ∀ m ≥ 0, we have which contradicts with the definition of α (m) k * .Consequently, the line search (7) can attain a positive steplength α k in a finite number of backtracking repetitions.
The above lemma shows that the line search of Algorithm 2.1 is reasonable, namely, the given MLS conjugate gradient projection algorithm is well defined.Similar to Lemma 2.3 in [12] and Lemma 2.1 in [9], it is not difficult to get the following lemma, so we omit the proof.Lemma 2.3.Let Assumption A holds and the sequence {x k , z k } be generated by Algorithm 2.1.Suppose that x * is a solution of problem (1) with g(x * ) = 0. Then In particular, both {x k } and {z k } are bounded.Furthermore, Theorem 2.4.Let Assumption A holds and the sequence {x k } be generated by Algorithm 2.1.Then we have Proof.Suppose that (17) is not true.Let 0 > 0 be a constant such that g k ≥ 0 .This together with (9) implies that By the relation of ( 5), ( 10) and ( 13), we obtain which implies that the sequence { d k } is bounded.Then there exists an infinite index set N 1 and an accumulation point d satisfying By the boundedness of {x k } in Lemma 2.3, we can deduce that there exists an accumulation x and an infinite index set By Lemma 2.2 and Lemma 2.3, we get this together with (19), we obtain lim k→∞ α k = 0.By (7), we have where α k = α k ρ .Therefore, taking the limit as k → ∞ in both sides of (20) for all k ∈ N 2 generates g(x) T d > 0.
On the other hand, by taking the limit as k → ∞ in both sides of (9) for all k ∈ N 2 , we have g(x) T d ≤ 0, which generates a contradiction.The proof is complete.

Numerical Experiments
In this section, we do some numerical experiments to test the performance of the MLS conjugate gradient projection method and compare it with the modified Polak-Ribière-Polyak (MPRP) projection method in [6] and the spectral gradient (SG) projection method in [16].All of the algorithms are coded in MATLAB R2010a and run on a personal computer with Intel Core 2 Duo CPU at 2.8 GHz and 2 GB of memory.The test problems are listed as g(x) = (g 1 (x), g 2 (x), • • • , g n (x)) T , which have the associated initial guess x 0 .Problem 1.The function is taken from [5].
Initial guess: The example is taken from [12].
Initial guess: For MLS projection method, the parameters are specified as σ = 0.01, s = 1, ρ = 0.6, γ = 0.5, and = 10 −4 .For MPRP projection method in [6] and SG projection method in [16], we use the default parameters there.For all methods, we adopt the termination condition g(x k ) ≤ .The columns of Tables 3.1 and 3.2 have the following meanings: Dim: the dimension of the problem.NI: the total number of iterations.
NG: the number of the function evaluations.Time: CPU time in seconds.GF: the final function norm evaluations when the program is terminated.
We compared these three methods on problems 1-5.The results are reported in Tables 3.1 and 3.2.Table 3.1 shows the results of the methods with given initial points for problems 1-5.Table 3.2 lists the results obtained by the methods with random initial points generated by Matlab's code rand(n, 1).We see from Tables 3.1 and 3.2 that, all of these three methods terminate successfully at a solution of the problem starting from any initial point.Moreover, Tables 3.1 and 3.2 also indicate that the dimensions of these problems do not affect the number of iterations of the algorithm.However, for high dimension case, the computing time is relatively large.Taking everything together, the numerical experiments show that the proposed method performs well for solving the given systems of monotone equations.In order to analyze the efficiency of these three methods, we also use the tool of Dolan and Moré [1].Figures 3.1 and 3.2 show the performance of these three methods relative to NG and CPU time of Tables 3.1-3.2,respectively.These two figures show that all of these three methods have good performance for all problems.And the proposed method is more competitive than the MPRP and SG projection methods as MLS projection method can get the solution of all the test problem at a smaller horizontal axis.

CONCLUSION
In this paper, we develop a modified LS conjugate gradient projection method for solving nonlinear monotone equations.The proposed method is particularly suitable for large-scale monotone equations because of low memory requirement.The global convergence of the given algorithm is established under suitable conditions.The preliminary numerical results show that our method is promising.
these three methods (NG).