On Solving of Constrained Convex Minimize Problem Using Gradient Projection Method

Let C and Q be closed convex subsets of real Hilbert spacesH1 andH2, respectively, and let g : C 󳨀→ R be a strictly real-valued convex function such that the gradient∇g is an 1/L-ismwith a constant L > 0. In this paper, we introduce an iterative scheme using the gradient projection method, based on Mann’s type approximation scheme for solving the constrained convex minimization problem (CCMP), that is, to find a minimizer q ∈ C of the function g over set C. As an application, it has been shown that the problem (CCMP) reduces to the split feasibility problem (SFP) which is to find q ∈ C such that Aq ∈ Q where A : H1 󳨀→ H2 is a linear bounded operator. We suggest and analyze this iterative scheme under some appropriate conditions imposed on the parameters such that another strong convergence theorems for the CCMP and the SFP are obtained. The results presented in this paper improve and extend the main results of Tian and Zhang (2017) and many others. The data availability for the proposed SFP is shown and the example of this problem is also shown through numerical results.


Introduction
Throughout this paper, we always assume that  be a closed convex subset of a real Hilbert space  with inner product and norm are denoted by ⟨⋅, ⋅⟩ and ‖ ⋅ ‖, respectively.Let  :  → R be a strictly real-valued convex function.
Consider the following constrained convex minimization problem (CCMP): min ∈  () . ( Assume that (1) is consistent (that is, the CCMP has a solution) and we use U to denote its solution set.If  is Fréchet differentiable, then the gradient projection algorithm (GPA) is usually applied to solving the CCMP (1), which generates a sequence {  } through the recursion: +1 =   ( − ∇)   , ∀ = 0, 1, 2, . . ., or more generally,  +1 =   ( −   ∇)   , ∀ = 0, 1, 2, . . ., where the initial guess  0 ∈  is chosen arbitrarily, the parameters  or   are positive real number, and   is the metric projection from  onto .It is well known that the convergence of algorithms ( 2) and (3) depends on the behavior of the gradient ∇.It is known from Levitin and Polyak [1] that if ∇ is -strongly monotone and -Lipschitzian, that is, there exists the constants  > 0 and  > 0 such that ⟨∇ () − ∇ () ,  − ⟩ ≥       − respectively, then, for 0 <  < 2/ 2 , the operator is a contraction; hence, the sequence {  } defined by the GPA (2) converges in norm to the unique minimizer of the CCMP (1).More generally, for 0 <   < 2/ 2 for all  = 0, 1, 2, . .., the operator 2 International Journal of Mathematics and Mathematical Sciences is a contraction; if the sequence {  } is chosen satisfying the property then the sequence {  } defined by the GPA (3) converges in norm to the unique minimizer of the CCMP (1).However, if the gradient ∇ fails to be -strongly monotone (it means that the gradient ∇ only satisfies the -Lipschitzian condition), then the operators  and   defined by ( 6) and (7), respectively, may fail to be contraction; consequently, the sequence {  } generated by algorithms ( 2) and (3) may fail to converge strongly (see also Xu [2]) in the setting of infinitedimensional real Hilbert space, but still converge weakly as the following statement.
We observe from Theorem 1 that if the parameter {  } converges to  ∈ (0, 2/) such that {  } satisfies the condition (9) then  ∈  solves the CCMP (1) which is the unique consistent if and only if  solves the fixed-point equation It is well known that the gradient-projection algorithm is very useful in dealing with the CCMP (1) and has extensively been studied (see [1][2][3][4][5][6][7][8][9] and the references therein).It has recently been applied to solve the split feasibility problems (SFP) (see [10][11][12][13][14][15]) which find applications in image reconstructions and the intensity modulated radiation therapy (see [13][14][15][16][17][18]). We now consider the following regularized minimization (that is, the CCMP (1) has the unique minimizer solution) problem: where   > 0 for all  = 0, 1, 2, . . .and  :  → R is a continuous differentiable function, and we also consider the regularized gradient-projection algorithm which generates a sequence {  } by the following recursive formula: Many researchers studied the strong convergency theorems for solving the CCMP (1) using the sequence {  } which is generated by algorithm (12) for their proposal on the gradient ∇ which is the class of nonexpansive mapping and the class of -Lipschitzian mapping (see [19][20][21][22][23][24][25]) and in case the gradient ∇ is the class of 1/-ism mapping such that  > 0, Xu (2010) introduced the sequence {  } which is generated by algorithm (12), and he proved that this sequence {  } converges weakly to the minimizer of the CCMP (1) in the setting of infinite-dimensional real Hilbert space (see [15]) under some appropriate condition.
Recently, Tian and Zhang (2017) introduced the sequence {  } generated by algorithm (12), and they proved that this sequence {  } converges strongly to the minimizer of the CCMP (1) in the same setting of infinite-dimensional real Hilbert space (see [26]) under the control conditions: In this paper, under the motivated and the inspired by above results, we introduce new iterative scheme, based on Mann's type approximation scheme for solving the CCMP (1) in the case of the gradient ∇ being the class of 1/-ism mapping such that  > 0 as follows: under the mild some appropriate conditions of the parameters {  }, {  }, and , we obtain a strong convergency theorem to solve the CCMP (1), in which condition (iii) ∑ ∞ =0 | +1 −   | < ∞ of Tian and Zhang to be removed.In Section 4 of the applications, it has been shown that the CCMP (1) reduces to the split feasibility problem (SFP) and the data availability for the proposed SFP is shown in Section 5, and the example of this problem is also shown in Section 6 through numerical results.

Preliminaries
Let  be a nonempty closed convex subset of a real Hilbert space .If  :  → R is a differentiable function, then we denote ∇ the gradient of the function .We will also use the notation: → to denote the strong convergency, ⇀ to denote the weak convergency, and Fix() = { :  = } to denote the fixed point set of the mapping .
Recall that the metric projection   :  →  is defined as follows: for each  ∈ ,    is the unique point in  satisfying      −        = inf {      −      :  ∈ } .
Let  :  → R be a function.Recall that the function  is a strictly real-valued convex function if such that  ̸ = .We collect together some known lemmas and definitions which are our main tool in proving our results.

Main Result
Throughout this paper, we let  be a nonempty closed convex subset of a real Hilbert space .First, we will show that   which is defined by has the unique fixed point under the conditions 0 <  < 2/( + 2), 0 ≤   < 1 and 0 <   < 1 where  :  → R be a strictly real-valued convex function such that ∇ is 1/-ism International Journal of Mathematics and Mathematical Sciences with  > 0. Since, ∇ is 1/-ism and the nonexpansiveness of   .Then, for each ,  ∈ , we have So,   is a contraction, therefore, by Banach's contraction principle,   has the unique fixed point.Therefore,   is welldefined.
Let U be the solution set of the CCMP (1).It is clear that U is a closed and convex sets.We now ready to present my main results as follows.
Proof.We divide the proof into 4 steps.
Step 1.We will show that {  } is bounded.Let  ∈ U.By the strictly convexity of , we have that U is a singleton set.Noticing from 1/-ism of ∇ that ∇ is -Lipschitzian.So, by (10), we have  =   ( − ∇).Therefore, by (30) where   =   and It is easy to see that ∑ ∞ =0   = ∞ and limsup →∞   ≤ 0. Therefore, by Lemma 10, we obtain {  } converges strongly to .This completes the proof.

Applications
Let  and  be closed convex subsets of real Hilbert spaces  1 and  2 , respectively, and  :  1 →  2 be a bounded linear operator.We now consider the split feasibility problem (SFP) which introduced in 1994 by Censor and Elfving [13], where this problem is to find an element  ∈  such that  ∈ .Define the convex function  :  → R as follows: It follows by Lemma 7 that the gradient of  as ∇ =  * ( −   ) where  * is the adjoint operator of , and ∇ is 1/‖‖ 2ism.We have the consequence results as follows.

Data Availability
In order of the feasible solution, all algorithms of the iterations have to compute many inner iterations to find the appropriate result, and stack overflow often occurs in which a computer program makes too many subroutine calls and its call stack runs out of space when the parameters of iterations have using many stack arrays to compute the feasible solution.
To avoid the stack overflow, we introduce how to do the mathematical programming without using the stack arrays of its parameters for solving the SFP of the algorithm in Corollary 14.Indeed, the situation of the stack overflow may have occurred from calculating the floating point numbers or the significant decimal digits; to avoid it we ought to be careful of that by always using digit precision command such as the command [ ] in Mathematica, and the command ( ) in Matlab, and also define all matrix in the regular command type without using the matrix palette to avoid it.
Some mathematical software has a command to give the total number of seconds of CPU time used and the total number of seconds since the beginning of computation in the session such as the commands [] and [] in Mathematica, respectively, and the commands  and / in Matlab, respectively.
We now give the formulation of orthogonal projection   where  is a simply closed convex sets as follows, and in the case that  is not a simply closed convex sets, for instance,  is a halfspace, we can found more the formulation in [33].
Proposition 15.For  ∈ R  we have ,  ∉ , ,  ∈ . (50) Proof.Obviously, the results (i) and (ii) hold by the definition of orthogonal projection of   , and the result (iii) also holds by the normal vector of the boundary points set of .
We are ready to introduce how to do the mathematical programming without using the stack arrays of its parameters for solving the SFP of the algorithm in Corollary 14 as follows.Suppose that the SFP has the unique consistent.Taking  1 = (R  , ‖ ⋅ ‖ 2 ) and  2 = (R  , ‖ ⋅ ‖ 2 ) into Corollary 14.Let the sets  and , the operator , the sequence {  }, and the parameters {  },  satisfy the conditions in Corollary 14.We have that {  } is a convergent sequence, and so it is a Cauchy sequence.Hence, we can choose the stopping criteria  > 0 which satisfies ‖ +1 −   ‖ 2 <  for stopping the program, and also the approximate solution refers to the last iteration.
Steps of the mathematical programming of the algorithm in Corollary 14 are shown as follows: Mathematical programming for the split feasibility problem Finding the solution of an augmented matrix equation  ×  ×1 =  ×1 .
Step 1. Declare of all parameters  × ,  ×1 , the starting point Start ×1 and .Step 2. Define the formulations of the orthogonal projections of   and   where If we choose  = R  and  = {} such that  ∈ R  then the orthogonal projections of   and   are easy to calculated, International Journal of Mathematics and Mathematical Sciences and, hence, we do not need to define its formulations in this step, and we can put directly its formulations to process.
The example of the commands in Mathematica is shown as follows.
Step 3. Set the starting index  = 0 and fix parameter  ∈ (0, 2/(2 + ‖‖ 2 )).If the parameter  is not a fix number such that it is a sequence, then we must lie it in the while loop of step 4.
It well known that, in the case of finite-dimensional real space,  * =   where   stands for matrix transposition of , and, hence, the algorithm in Corollary 14 can be reduced to  Step 5. Clear memory of the system.
The example of the command in Mathematica is shown as follows. [];

Conclusion
In this paper, we obtain an iterative scheme using the gradient projection method based on Mann's approximation method for solving the constrained convex minimization problem (CCMP) and also solving the split feasibility problem (SFP) such that another strong convergence theorems for the CCMP and the SFP are obtained.