A Spectral Projected Gradient-Newton Two Phase Method for Constrained Nonlinear Equations

In this paper, we proposed a spectral gradient-Newton two phase method for constrained semismooth equations. In the first stage, we use the spectral projected gradient to obtain the global convergence of the algorithm, and then use the final point in the first stage as a new initial point to turn to a projected semismooth asymptotically newton method for fast convergence.


Introduction
In this paper, we consider the constrained nonlinear semismooth equations problem: finding a vector x * ∈ Ω such that where : n f R R → is continuously differentiable and its gradient denoted by ( ) f x ∇ . Many researchers have studied constrained optimization problems such as (2) and given many effective algorithms. For example, a new class of adaptive non-monotone spectral gradient method is given in reference [3], an active set projected trust region algorithm in [4]. The methods of optimization problems involve the first-order methods and the second-order methods. Classical firstorder algorithms include gradient method, sub-gradient method, conjugate gradient method, etc. The main advantage of first-order method is its small storage, which is particularly suitable for large-scale problems. However, the disadvantage of first-order method is that its convergence speed is at most linear, and it can not meet the requirements of high precision. For the second-order method, it has the advantage of fast convergence speed. Under certain conditions, it can achieve superlinear convergence or even quadratic convergence. But its disadvantage is that it needs a good initial point, sometimes it even needs the initial point to approach the local optimal point. Motivated by this, in this paper, we combine the advantages of the first-order method with those of the second-order method. We will consider the two-stage combination algorithm to solve the optimization problem. First, we use the first-order method to obtain the global convergence of the algorithm, and then use the final point obtained by the first-order method as the new initial point to turn to the second-order method to obtain the fast convergence speed. At the same time, we use projection technology to solve the constrained conditions.

Preliminaries
In this section, we present some definitions and theorems that are useful to our main result.

Suppose
: where the 0 C denotes the convex hull of a set, where 0 1 p < ≤ , then we call H is p-order semismooth at x. Lemma Lemma 2.8 [7]: The projection operator Lemma 2.9 [8]: Given n x ∈ » and n d ∈ » , the function ξ defined by is nonincreasing. Lemma 2.9 actually implies that if x X ∈ is a stationary point of (2), then

Algorithm
In order to obtain the global convergence of the algorithm, in the first stage, we adopt the non-monotone spectral projection gradient method of the first-order Journal of Applied Mathematics and Physics method. The one-dimensional search procedure of Algorithm 3.1 will be called SPG1 from now on and Algorithm 3.2 will be called SPG2 in the rest of the paper.
Step 2.2 Set Step 3. compute Step 2. (Backtracking) Step 2.1. Compute Step 2.2. Set Step 0. Choose constants Step 1. Choose Step 2. If k x is a stationary point, stop. Otherwise let Step 3. If the linear system then use the direction Step 4. Let k m be the smallest nonnegative integer m satisfying where for any the optimal solution is Step 5. Let : 1 k k = + , and go to step 1.

Convergence Analysis
Theorem 4.1 [9]: Algorithm SPG1 is well defined, and any accumulation point of the sequence { } k x that is generates is a constrained stationary point.

Application
Many practical problems can be solved by transforming them into constrained semi-smooth equations. For example, mixed complement problem (MCP): Fischer-Burmeister function introduced by Chen et al. [11] and has the form: Here, » is an NCP function, which is given by The mixed complement problem can be transformed into a semi-smooth system of equations by the above functions. Then we can use the two phase method to solve this problem.

Conclusion
In this paper, we proposed a two-phase method for the constrained equations.
We can also combine other first-order and second-order methods. In this paper, the iteration complexity analysis of the first-order method is a meaningful work, and we will do further research.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.