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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.

In this paper, we consider the constrained nonlinear semismooth equations problem: finding a vector x ∗ ∈ Ω such that

H ( x ) = 0 , x ∈ Ω : = { x ∈ ℝ n | l ≤ x ≤ u } , (1)

where

Ω : = { x ∈ ℝ n | l ≤ x ≤ u } , l i ∈ ℝ ∪ { − ∞ } , u i ∈ ℝ ∪ { − ∞ } , l i < u i , i = 1 , ⋯ , n

H : ℝ n → ℝ n is a semismooth mapping. The notation of semismoothness was introduced for the functionals by Mifflin [

Systems of constrained semismooth equations arise in various application, for instance complementarity problems, the box constrained variational inequality problems, the KKT system of variational inequlity problems and so on. The solution of nonlinear equations can be transformed into solving the following constrained optimization problem:

min f ( x ) = 1 2 ‖ H ( x ) ‖ 2 s .t . x ∈ Ω (2)

where f : R n → 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 [

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.

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

Suppose H : R n → R n is a locally Lipschitzian function, according to Rademacher theorem, H is differentiable almost everywhere. Denote the set of points at which H is differentiable by D H . We write H ′ ( x k ) for the usual n × m Jacobian matrix of partial derivatives whenever x is a point at which the necessary partial derivatives exists. Let ∂ H ( x ) be the generalized Jacobian defined by Clarke in [

∂ H ( x ) = C 0 ( ∂ B H ( x ) ) , (3)

where the C 0 denotes the convex hull of a set, ∂ B H ( x ) = { lim x j → x x j ∈ D H H ′ ( x j ) } .

Definition2.1 [

lim V ∈ ∂ H ( x + t h ′ ) h ′ → h , t ↓ 0 { V h ′ } (4)

exists for any h ∈ R n .

Lemma 2.2 [

1) H is semismooth at x;

2) For any V ∈ ∂ H ( x + h ) , h → 0 ,

V h − H ′ ( x ; h ) = ο ( ‖ h ‖ ) , (5)

H ( x + h ) − H ( x ) − V h = ο ( ‖ h ‖ ) . (6)

Lemma 2.3 [

Definition 2.4 [

V h − H ′ ( x ; h ) = Ο ( ‖ h ‖ 1 + p ) , (7)

where 0 < p ≤ 1 , then we call H is p-order semismooth at x.

Lemma 2.5 [

Lemma 2.6 [

Lemma 2.7 [

‖ H ( x ) ‖ ≥ k ‖ x − x ∗ ‖ . (8)

Lemma 2.8 [

1) For any x ∈ X , [ Π X ( z ) − z ] T [ Π X ( z ) − x ] ≤ 0 for all z ∈ ℝ n .

2) ‖ Π X ( y ) − Π X ( z ) ‖ ≤ ‖ y − z ‖ for all y , z ∈ ℝ n .

Lemma 2.9 [

ξ ( λ ) = ‖ ∏ X ( x + λ d ) − x ‖ / λ , λ ≥ 0 (9)

is nonincreasing.

Lemma 2.9 actually implies that if x ∈ X is a stationary point of (2), then

d ¯ G ( λ ) = Π X [ x + λ d G ] − x = 0 , ∀ λ ≥ 0 (10)

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 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.

Given z ∈ ℝ n , we define P ( z ) as the orthogonal projection on Ω , denote g ( x ) = ∇ f ( x ) . x 0 ∈ Ω , integer M ≥ 1 , a small parameter α min > 0 , a large parameter α max > α min , sufficient decrease parameter γ ∈ ( 0 , 1 ) , 0 < σ 1 < σ 2 < 1 , initially α 0 ∈ [ α min , α max ] , x 0 ∈ Ω .

Algorithm 3.1 [

Step 1. If ‖ P ( x k ) − g ( x k ) − x k ‖ < ε 1 , stop, input x k .

Step 2. (Backtracking)

Step 2.1 Set λ = α k .

Step 2.2 Set x + = P ( x k − λ g ( x k ) ) .

Step 2.3 If

f ( x + ) ≤ max 0 ≤ j ≤ min { k , M − 1 } f ( x k − j ) + γ 〈 x + − x k , g ( x k ) 〉 , (11)

Then define λ k = λ , x k + 1 = x + , s k = x k + 1 − x k , y k = g ( x k + 1 ) − g ( x k ) , and go to step 3.

If (11) does not hold, define λ n e w ∈ [ σ 1 λ , σ 2 λ ] . Set λ = λ n e w , and go to step 2.2.

Step 3. compute b k = 〈 s k , y k 〉 , If b k ≤ 0 , set α k + 1 = α max ; else compute α k = 〈 s k , s k 〉 , α k + 1 = min { α max , max { α min , a k / b k } } , and go to step 1.

Algorithm 3.2 [

Step 2. (Backtracking)

Step 2.1. Compute d k = P ( x k − α k g ( x k ) ) − x k , Set λ = 1 .

Step 2.2. Set x + = x k + λ d k .

Step 2.3. If

f ( x + ) ≤ max 0 ≤ j ≤ min { k , M − 1 } f ( x k − j ) + γ λ 〈 d k , g ( x k ) 〉 , (12)

Then define λ k = λ , x k + 1 = x + , s k = x k + 1 − x k , y k = g ( x k + 1 ) − g ( x k ) , and go to step 3.

If (12) does not hold, define λ n e w ∈ [ σ 1 λ , σ 2 λ ] . Set λ = λ n e w , and go to step 2.2.

The output point of the first stage is used as the initial point of the next stage.

Algorithm 3.3 [

Step 0. Choose constants ρ , σ , η ∈ ( 0 , 1 ) , p 1 > 0 , p 2 > 2 , Let x 0 = x N ∈ Ω , k : = 0 .

Step 1. Choose V k ∈ ∂ B H ( x k ) , compute ∇ f ( x k ) = V k T H ( x k ) .

Step 2. If x k is a stationary point, stop. Otherwise let

d G k = − γ k ∇ f ( x k ) , (13)

where

γ k = min { 1 , η f ( x k ) / ‖ ∇ f ( x k ) ‖ 2 } , (14)

and go to step 3.

Step 3. If the linear system

H ( x k ) + V k d = 0 (15)

has a solution d N k , and

− ∇ f ( x k ) T d N k ≥ p 1 ‖ d N k ‖ p 2 , (16)

then use the direction d N k . Otherwise, set d N k = d G k .

Step 4. Let m k be the smallest nonnegative integer m satisfying

f ( x k + d ¯ k ( ρ m ) ) ≤ f ( x k ) + σ ∇ f ( x k ) T d ¯ G k ( ρ m ) , (17)

where for any λ ∈ [ 0 , 1 ] ,

d ¯ k ( λ ) = t k ∗ ( λ ) d ¯ G k ( λ ) + [ 1 − t k ∗ ( λ ) ] d ¯ N k ( λ ) , (18)

d ¯ G k ( λ ) = ∏ X [ x + λ d G k ] − x k , d ¯ N k ( λ ) = ∏ X [ x + λ d N k ] − x k (19)

and t k ∗ ( λ ) is an optimal solution to

min t ∈ [ 0 , 1 ] 1 2 ‖ H ( x k ) + V k [ t d ¯ G k ( λ ) + ( 1 − t ) d ¯ N k ( λ ) ] ‖ 2 , (20)

the optimal solution is

t ∗ ( λ ) = max { 0 , min { 1 , t ( λ ) } } , (21)

let λ k = ρ m k , x k + 1 = x k + d ¯ k ( λ k ) .

Step 5. Let k = : k + 1 , and go to step 1.

Theorem 4.1 [

Theorem 4.2 [

Theorem 4.3 [

Many practical problems can be solved by transforming them into constrained semi-smooth equations. For example, mixed complement problem (MCP):

F : R n → R n is a continuous differentiable function, finding a vectors x ∈ X satisfies

F ( x ) T ( y − x ) ≥ 0 , ∀ y ∈ X , (22)

The function ψ α : ℝ 2 → ℝ with α ∈ [ 0 , 1 ] is defined by

ψ α ( a , b ) : = ( [ ϕ α ( a , b ) + ] ) 2 + ( [ − a ] + ) 2 , (23)

where [ a ] + : = max { 0 , a } for any a ∈ ℝ and ϕ α : ℝ 2 → ℝ is the penalized Fischer-Burmeister function introduced by Chen et al. [

ϕ α ( a , b ) : = α ϕ F B ( a , b ) + ( 1 − α ) a + b + , (24)

Here, ϕ α : ℝ 2 → ℝ is an NCP function, which is given by

ϕ F B ( a , b ) : = ( a + b ) − a 2 + b 2 , (25)

The mixed complement problem can be transformed into a semi-smooth system of equations by the above functions.

Let N = { 1 , ⋯ , n }

I f : = { i | l i = − ∞ , u i = ∞ , i ∈ N } , I l : = { i | l i > − ∞ , u i = ∞ , i ∈ N } , I u : = { i | l i = − ∞ , u i < ∞ , i ∈ N } , I l u : = N \ { I l ∪ I u ∪ I f } (26)

MCP can be reformulated as H ( x ) = 0 with

H i ( x ) : = { | F i ( x ) | if i ∈ I f | ϕ α ( x i − l i , F i ( x ) ) | if i ∈ I l | ϕ α ( u i − x i , − F i ( x ) ) | if i ∈ I u ψ α ( x i − l i , F i ( x ) ) + ψ α ( x i − l i , F i ( x ) ) if i ∈ I l u , i = 1 , ⋯ , n (27)

Then we can use the two phase method to solve this problem.

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.

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

Zhang, Y.Z. (2019) A Spectral Projected Gradient-Newton Two Phase Method for Constrained Nonlinear Equations. Journal of Applied Mathematics and Physics, 7, 104-110. https://doi.org/10.4236/jamp.2019.71009