(Communicated by Xinmin Yang)

Abstract: A large number of problems in social sciences and natural sciences today can be attributed to global optimization problems, which are widely found in economic models, finance, network transportation, database, integrated circuit design image processing, chemical engineering design and control, molecular biology, environmental engineering, etc. At present, the methods for solving global optimization problems can be divided into two categories, namely, deterministic algorithm and stochastic algorithm. Stochastic algorithm, such as genetic algorithm, simulated annealing algorithm and so on, is constructed by using the point column generated by a probabilistic mechanism to describe the iterative process. Deterministic algorithms use the analytic property of the problem to generate deterministic finite (or infinite) point sequences that converge to the global optimal solution, such as interval method, D.C. Planning algorithm, branch and bound method, auxiliary function method and integral level set method, etc. Among them, the filled function method is a deterministic method which can search the optimization problem globally. Ren pu Ge took the lead in proposing the filled function method^[1]. The main idea is to use the filled function to jump out of the current local minimizer to find a better minimizer. The filled function method consists of two parts, one is to local minimization and the other is to fill. In the minimization phase, a local minimization algorithm is used to obtain a local minimizer of the original problem. In the filling phase, the original problem is minimized by means of the filled function, and the two phases are alternated until a global minimizer is found.

     The following global optimization problem will be discussed

min $f(x)$,

s.t.  $x \in \mathbb{R}^n$.                                                                           （1）

Ge states that if a function satisfies the filled function definition and the filling property, then it can be used to solve the problem (1). The first generation filled function formula given in [1] is as follows:

$G(x,x_k^*,r,\rho)=\frac1{r+f(x)}\exp(-\frac{\|x-x_k^*\|^2}{\rho^2})$,                                                   （2）

where $r$ and $\rho$ are given parameters. The selection of parameters in formula (2) is quite difficult. This is a challenge for coming up with new filled function methods. The filled function of equation (2) has some shortcomings. 

It has been discussed in depth. These shortcomings are mainly related to the two parameters $r$ and $\rho$ contained in the function. According to Gao, Ge's filled function contains an exponential term $\exp(-\frac{\|x-x_k^*\|^2}{\rho^2})$, and the picture of the filled function is almost flat if $\|x-x_k^*\|^2$ is large or $\rho$ is small. This makes it difficult for the computer to discern changes in function values.

    In this paper, a new non-parameter filled function is constructed. It satisfies the definition of filled function and has excellent properties. Then, the corresponding algorithm is designed according to the new filled function. Numerical experiments are made and comparisons on several test problems are shown which exhibit the feasibility and effectiveness of the algorithm.

(Communicated by Xinmin Yang)

Abstract:Global optimization problems are common issue in numerous fields, such as molecular biology, economics and finance, environmental engineering, industrial manufacturing. In the global optimization problem, if it is a convex programming, we can use a local minimization algorithm to find its global optimal solution. Unfortunately, in most cases, these problems are non-convex optimization problems with multiple minimum points, which cause a scientific question arise: How to find a better minimizer from the known local minimizer of the objective function.

    The traditional nonlinear programming algorithms, such as Quasi-Newton methods or other algorithms, cannot directly solve the puzzle. Therefore, it is a great challenge to find the global minimizer of non-convex objective function. In recent years, lots of studies have focused on global optimization theory and algorithm. In general, global optimization methods can be divided into two categories: stochastic methods and deterministic methods. The stochastic methods for global optimization include evolutionary algorithm, simulated annealing, and particle swarm optimization, etc. These methods are inspired by physical principles or natural evolution, which jump out of the local minimum by a probability-based approach. Although the stochastic methods have been effectively applied to solve many intricate problems, some problems remain unsolved, such as convergence analysis and parameter selection. The deterministic methods, for example, the filled function method, tunnelling method, covering method, can converge faster and often find a better minimizer than the current minima quickly. Compared with other deterministic methods, the filled function method is a more popular global optimization method because it is relatively easy to implement. The basic idea of filled function is to construct an auxiliary function at the current local minima of the objective function. By minimizing the auxiliary function, the current local minima is quickly jumped out and a better local minima is reached. This process cycles until a better local minimum cannot be found.

    The first filled function introduced by Ge is as follows, i.e.

$P(x,x_{1}^*, r, \rho)=\frac{1}{r+F(x)}\text{exp}(-\frac{\|x-x_{1}^{\ast}\|^{2}}{\rho^{2}}),$

where the parameters $r$ and $\rho$ need to be chosen appropriately. However, Ge's filled function has some shortcomings. Because there are too many restrictions on the parameters $r$ and $\rho$, it is difficult for them to be properly selected so that $P(x,x_{1}^*, r, \rho)$ satisfies the definition of the filled function. Constructing a filled function with good properties is one of the keys to the success of a filled function algorithm. Therefore, researchers have done a lot of beneficial research work in enriching the forms of filled function. Unfortunately, These filled functions indeed possess several merits, yet they also exhibit drawbacks that necessitate further refinement and improvement. The adjustment of parameters for some two-parameter or single-parameter filled functions is still difficult. Non-smooth, non-differentiable or even discontinuous filled functions are not minimized by effective local optimization procedures based on gradient information, which decreases the efficiency of the filled algorithm. To address the difficulties in adjusting parameters, some scholars have proposed parameter-free filled functions. However, some parameter-free filled functions do not have any information about the objective function when $f(x)\geq f(x_{1}^{*})$, which may lead to difficulties in finding the global minimum point. In addition, without the adjustment of parameters, the proposed function in literature may fail to satisfy the definition of the filled function.

    Based on the above considerations, a new one-parameter filled function is proposed in this paper. The main contributions of this work are summarized below:

    (i) The new filled function is continuously differentiable and its gradient information can be used in local minimization procedure. Thus, the realization of the minimization process is simplified, and the algorithm is more efficient and convenient.

      (ii)  In any case, our proposed filled function always contains information about the objective function.

     (iii)  The new filled function has an easily adjustable parameter that can be tuned according to the specific problem, enabling the algorithm to adapt to the problem and avoids any issues similar to those encountered in literature.

    Next, we conducted a theoretical analysis and research on the properties of the new filled function. These good function properties ensure the convergence of the minimization process of the proposed filled function. Therefore, the minimum iteration process of the new filled function algorithm will run successfully in the whole region. Without these function properties, the filled function may behave similarly to that in reference, having no local minima within the entire feasible region. Furthermore, supported by thorough theoretical analysis, we have meticulously conceived a filled function method that is not only easy to implement but also highly efficient. Through a series of rigorous numerical experiments, we have fully verified the effectiveness and reliability of this new algorithm. It is worth mentioning that in the comparative tests with two other filled function methods, our algorithm has demonstrated more significant advantages, and the experimental results convincingly prove its superiority.

(Communicated by Zheng-Hai Huang)

    Overfitting is a common phenomenon in machine learning, wherein models almost can fit the samples on the training set but have poor generalization ability on the test set. Regularization is devoted to tackling this problem by imposing a penalty on the complexity or smoothness of the model. However, the performance of regularization is usually circumscribed by the lack of correlation with data samples, which restricts its potential efficiency for many practical models. In this paper, pursuing the seminal work by Zhu et al. (LDMNet), we develop a coupled tensor norm regularization. It can be customized to the model with small-sized structural samples. The main idea of this regularization, which is built upon empirical manifold observation that input data and output features have a low-dimensional structure, is an alternative representation of low-dimensionality. Concretely, coupled tensor norm regularization is the low-rank approximation of the coupled tensor rank function. Related theoretical properties are presented and we further test this regularization for multinomial logistic regression and deep neural networks by theoretical algorithm analysis and numerical experiments. Numerical simulations on real datasets demonstrate the compelling performance of proposed regularization.

(Communicated by Guihua Lin)

    We consider the optimization problem of minimizing a smooth and convex function. Based on the accelerated coordinate descent method (ACDM) using probabilities $L_i^{1/2}[\sum_{k=1}^n L_k^{1/2}]^{-1}$ for non-uniform sampling (Nesterov Yu. et al., SIAM J. Optim., 110–123, 2017 [3]), we propose an adaptive accelerated coordinate descent method (AACDM) with the same probability distribution determined by $\{L_i\}$ as in ACDM.

    In [1, 3], the step sizes of their algorithms are fixed and determined by the (global) parameters $\{L_i\}$. Note that this may not be preferable for practical applications where the (local) parameter values differ from the global counterparts to some extent. This implies that methods which can be adaptive to the local parameters might improve the performance in practice. Motivated by this, in this paper we study the adaptive ACDM, which still requires (global) Lipschitz constants for non-uniform sampling as a prior, while the (local) coordinate Lipschitz constants are determined by backtracking (not neceessarily monotone) to achieve better performance. Both the strongly and non-strongly cases are discussed in this paper.

    The non-monotone backtracking line search is included in our adaptive scheme, which performs better (compared with the monotone one) for applications whose local coordinate Lipschitz constants oscillate along the trajectory or become smaller when approaching the tail. The adaptive ACDM is indeed not a monotone method, meaning that the sequence of function values it produces is not necessarily nonincreasing. Since the monotone approach can be used to improve numerical stability (see monotone FISTA in [2]), we also propose an adaptive ACDM in monotone version.

    Numerical results on some classic problems show the efficiency of the adaptive scheme.

(Communicated by Xinwei Liu)

    This paper introduces a novel conjugate gradient method that exploits the m-th order Taylor expansion of the objective function and cubic Hermite interpolation conditions. We derive a set of modified secant equations with enhanced accuracy in approximating the Hessian matrix of the objective function. Additionally, we develop a modified Wolfe line search to address the limitations of the conventional constraint imposed on modified secant equations while ensuring the fulfillment of the curvature condition. Consequently, an improved spectral conjugate gradient algorithm is proposed based on the modified secant equation and Wolfe line search. Under standard assumptions, the algorithm is proven to be globally convergent for minimizing general nonconvex functions. Numerical results are provided to demonstrate the effectiveness of this new proposed algorithm.       

(Communicated by Zheng-Hai Huang)

    Recently, many new challenges in Compressed Sensing (CS) arose. In this paper, we present a new algorithm for solving CS with block sparse constraints (BSC) in complex fields. Firstly, based on block sparsity characteristics, we propose a new model to deal with CS with BSC and analyze the properties of the functions involved in this model. We then present a new  -stationary point and analyze corresponding first-order sufficient and necessary conditions. We further develop a block Newton hard-thresholding pursuit (BNHTP) algorithm for efficiently solving CS with BSC. Finally, numerical experiments demonstrate that the BNHTP algorithm has superior performance in terms of recovery accuracy and time.

(Communicated by Nobuo Yamashita)

     In this paper, in the absence of any constraint qualifications, we develop sequential necessary and sufficient optimality conditions for a constrained multiobjective fractional programming problem characterizing a Henig proper efficient solution in terms of the $\epsilon$-subdifferentials and the subdifferentials of the functions. This is achieved by employing a sequential Henig subdifferential calculus rule of the sums of $m\ (m\geq 2)$ proper convex vector valued mappings with a composition of two convex vector valued mappings. In order to present an example illustrating Our results, we establish the classical optimality conditions under Moreau-Rockafellar qualification condition. Our results are presented in the setting of reflexive Banach space in order to avoid the use of nets.

(Communicated by Guanglu Zhou)

    Consider the linear equation

$Kx = y,$                                                                                    (1)

where $x\in\mathbb{R}^n $ is the original signal or image, $K:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is a degradation operator, and $y\in\mathbb{R}^m$ is the observed signal or image. It has some important applications to signal processing, non-destructive testing, etc. The typical $\ell_1$-norm sparse regularization method does not have beneficial stability for ill-posed problems with large condition number. What's more, in many applications, the solution $x$ of the question (1) is not really sparse. To overcome the shortcomings of typical sparsity regularization, some researchers have focused on the elastic-net regularization problem, that is,

$\min_{x\in \mathbb{R}^n} J(x)=\frac{1}{2}\|Kx-y\|^2+\alpha\|x\|_{1}+\frac{\beta}{2}\|x\|^2,$                                                           (2)

where $\alpha>0, \beta>0, $ $\|\cdot\|$ represents $\ell_2$-norm, $\|\cdot\|_1$ represents $\ell_1$-norm. As a species of multi-parameter Tikhonov regularization, elastic-net regularization is more applicable to ill-conditioned issues. Statistically speaking, elastic-net regularization has superior steadiness than the typical regularization. There is an additional $\ell_2$-norm in elastic-net regularization compared to the classical $\ell_1$-norm sparsity regularization. Elastic-net regularization has the advantages of both $\ell_2$-regularization and $\ell_1$-regularization.

    Even though there exist various approaches to solve the elastic-net regularization problem which mainly focus on the convergence of the algorithm, not much work has been carried out in the convergence time of the algorithm. Therefore, it is necessary to devise a method to estimate the convergence time. For (2), we can relax the problem to be the following constrained optimization problem

min $g(x)=\frac{1}{2} \|Kx-y\|^2+\frac{\beta}{2}\|x\|^2$

s.t.  $x\in\Theta,$                                                                                                        (3)

where $\Theta=\{x\in\mathbb{R}^n|\|x\|_{1}\leq r\}, r>0,$ and transform the problem (3) into the fixed point equation form

$x=P_\Theta(x-\lambda K^T(Kx-y)-\lambda\beta x),$                                                          (4)

where the stepsize $\lambda>0$. Obviously, the solution of problem (1) can be obtained by taking the solution of problem (3) when $\beta\rightarrow0$.

    Based on (4), we propose the following  new continuous-time projection algorithm for solving the elastic-net regularization

$\dot{x}(t)=-\varphi(x)(x(t)-z(x(t))),$                                                                       (5)

where

$z(x(t))=P_\Theta(x(t)-\lambda(K^TKx(t)-K^Ty)-\lambda\beta x(t)),$ $k_1>0, k_2>0, k_3>0,\alpha_1\in(0,1),  \alpha_2>1, \lambda>0$ and $\beta>0$.

    The equivalence between the equilibrium point of (5) and the solution of (4) is displayed. Moreover, we prove that for every $\beta>0$ and $\lambda\in(0,\frac{2\beta}{L^2})$, if $x$ is an equilibrium point of (5), then for all $t\in[0,+\infty)$, the solution of (5) exists uniquely for any given initial value. Furthermore, the convergence time of the proposed method can be reckoned by the setting time

$T(x(0))\leq T_{\max}=\frac{1}{d_1(1-p_1)}+\frac{1}{d_2(p_2-1)},$

where $k_1>0, k_2>0, k_3>0, \alpha_1\in(0,1), \alpha_2>1, L=\|K\|^2+\beta, \lambda\in(0,\frac{2\beta}{L^2}), \Phi=\sqrt{\frac{1}{1-\lambda^2 L^2+2\lambda\beta}}\in (0,1),$ $p_1:=\frac{1+\alpha_1}{2},$ $p_2:=\frac{1+\alpha_2}{2},$ $d_1=k_1(1-\Phi)\frac{2^{p_1}}{(\frac{4\beta}{4\beta-\lambda L^2})^{1-\alpha_1}}>0,$ $d_2=2^{p_1}(1-\Phi)^{\alpha_2}k_2>0. $

    To validate the convergence and effectiveness of the proposed algorithm, we apply our algorithm to some numerical examples of signal processing and image restoration. Compared with state-of-the-art methods in the literature such as CAPPA method, THZF method and PNN algorithm,  our algorithm is more effective to calculate the mean squared error (MSE) in signal processing. Furthermore,  our algorithm shows the better performance to achieve the signal-to-noise ratio (SNR), the structural  similarity index (SSIM) and the relative error (RE) in image restoration.

(Communicated by Jie Sun)

    In this paper, we propose an extended CQ algorithm integrated with selection technique to address the multiple-sets split feasibility problem (MSFP). At each iteration, the selection technique is employed to formulate a split feasibility subproblem for MSFP, subsequently it is resolved by means of the CQ algorithm. Under mild conditions, we establish the global convergence results for the extended CQ algorithm. Furthermore, we provide empirical evidence in the form of numerical results, which conclusively affirm the effectiveness and competitiveness of our proposed algorithm.

(Communicated by Chen Ling)

(Communicated by Donghui Li)

     In this paper, we consider a sixth-order paired symmetric Cauchy tensor and its generating vec- tors. Initially, we investigate the conditions for positive definiteness and positive semidefinite- ness of the sixth-order paired symmetric Cauchy tensor. Necessary and sufficient conditions for its positive definiteness are given according to the structural characteristics of a sixth-order paired symmetric Cauchy tensor. Subsequently, we apply the concept of the M-eigenvalue to the sixth-order paired symmetric Cauchy tensor, and further discuss related properties. We give two M-eigenvalue inclusion intervals for a sixth-order paired symmetric Cauchy tensor, which pro- vide two upper bounds for the M-spectral radius. The inclusion relation between them is given. In the end, we provide two numerical examples of eigenvalue inclusion intervals, confirming the inclusion relationship of the intervals.

(Communicated by Robert Csetnek)

      The purpose of this paper is to study the strong convergence of Moudafi's proximal iterative method for finding a common element of the set of solution of finite family of equilibrium problems in a Hilbert space. More precisely, we suppose that, for $i=1,...,m$, $C_i$ is closed convex subset of Hilbert space $H$ and $(F_i)_{i=1,...,m}$ be a sequence of real-valued bifunctions such that $F_i$ is defined on $C_i\times C_i$ for each $i=1,2,\dots,m.$ The corresponding system of equilibrium problems we consider aims at finding a point $x$ in $C=\bigcap_{i=1} ^m C_i$ such that :

(SEP)     $F_{i}{(x,y )}\geq 0$ for all $y\in C_i$ and all $i=1,...,m.$                                       （1）

Under appropriate condition on the parameters and the data, the sequence $(x_k)$ generated by the following algorithm 

$x^{k+1}=\alpha_{k}{x^k}+\beta_{k}{g(x^k)}+\gamma_{k}{J_r^{F_{m}}}{J_r^{F_{m-1}}}\dots{J_r^{F_{1}}}(x)$

is proved to be strongly converging to a point $\bar{x}\in S,$ where $S$ is the set of solutions to the system (1), $J_r^{F_{i}}(x)$ is the resolvent of $F_i$ at $x$ with parameter $r$ for $i=1,2\dots,m$, $g:C_m \longrightarrow C_m$ is a contraction and $\alpha_k ,\beta_k ,\gamma_k$ are positive scalars such that $\alpha_k +\beta_k +\gamma_k=1$, $\beta_k \rightarrow 0$, $ \sum \beta_k=+\infty$ and $(\alpha_k )\subset [c,d]$ for some $c,d\in (0,1)$.

     The second goal of this paper is to use the iterative algorithm proposed for the equilibrium system (1) to solve the split common fixed point problem as defined by Censor and Segal in [The split common fixed-point problem for directed operators J. Convex Anal.16 (2009) 587–600.], i.e., finding a point $\bar{x}$ satisfying

$\overline{x}\in\bigcap_{i=1}^{m}{Fix}{U_{i}}$   and    $A{\overline{x}}\in\bigcap_{j=1}^{m}{Fix}{T_{j}}$                              (3)

where, $H_1$ and $H_2$ are two Hilbert spaces, $A : H_{1} \longrightarrow H_{2}$ is a bounded linear operator, $(U_{i})_{i=1}^{m}: H_{1} \longrightarrow H_{1}$ are F-hemicontinuous operators and $(T_{i})_{i=1}^{m}: H_{2} \longrightarrow H_{2}$ are M-demicontractive mappings. Note that the concept of M-demicontractiveness is new in the literature, we have introduced it in order to develop the relation between problems (1) and (3). Our analysis underlines also that the class of $M$-demicontractive mappings is intermediately situated between nonexpansive operators and demicontractive ones. Finally, to illustrate the convergence of our algorithm, we present a numerical example written in matlab.

(Communicated by Fanwen Meng)

     
In this paper, A new approach is presented for exactly solving the following linear mixed integer programming problem (P) with scenario constraints.

wherecis an n-dimensional column vector,  and for each scenario  .    is an  matrix, d is anm-dimensional column vector,   and  are the lower and upper bounds of x, respectively.

     Consider the following subproblem:

where  and  and .

    We first decompose the subproblem  into two parts: main problem (MP) and the feasibility subproblems   of  scenario constraints.

       The first part is the following problem (MP) by discarding the scenario constraints of :

The optimal objective function value of the continuous relaxation problem of problem (MP) provides a lower bound for problem . Also we can obtain two integer points by rounding the continuous optimal solution of (MP) up or down along some directions.

      Another part is only the scenario constraints. For a given feasible solution of (MP), we determine whether is feasible to the scenario constraints or not by solving the following sub-problems:

     Then we develop the domain partition techniques for several cases according to whether an integer point xˆ being feasible to each constraint or not and thus cut off some integer boxes which do not contain any optimal solution of from .

     Integrating the solution scheme into a branch-and-bound frame, we present the new algorithm for solving problem (P). The proposed solution method reduces the optimality gap gradually in the solution iterations and is of a finite-step convergence.

     Finally, we give the numerical experiments and the comparison numerical results with the traditional branch-and-bound method for different sizes of scenario constraints. The comparison results with the traditional branch-and-bound method also show the proposed algorithm is very efficient.

（Communicated by Gui-hua Lin）

A globally and superlinearly convergent BFGS method is introduced to solve general nonlinear equations without computing exact gradient. Compared with existing Gauss-Newton-based BFGS type methods, the proposed method does not require conditions such as symmetry on the underlying function. Moreover, it can be suitably adjusted to solve nonlinear least squares problems and still guarantee global convergence. Some numerical results are reported to show its eciency.

    We employ the multi way array to generate the clustering costs and use the hypergraph related tensor to construct the constrained optimization model

for the semi-supervised clustering problem. This is a 0-1 integer programming. To make this optimization model easy, we penalize the constraint and add a regularization term  to the objective function. In terms of the discrete 0-1 constraint, it can be deduced that the column vectors of X are unit vectors and orthogonal to each other. Thus, we relax it to the continuous constraint . Then the optimization model (1) is transformed to an orthogonal constrained model

    The model (2) is an optimization constrained on the Steifel manifold . To solve this optimization problem, we first find a descent direction in the tangent space from the current point, and then employ the polar decompostion to map the descent direction to the Stiefel manifold.
    The tangent space at a point  is

The direction  is employed to find a new point in the tangent space, where G is the gradient of function f at the current point X. In the iteration process, we find

in the tangent space from Xk: The next step is to map to the manifold by using a retraction. Consider the univariate function

with its domain , which is the projection of onto the Stiefel manifold.
    Given and the descent direction ; by using the adaptive feasible BB-like method, we find a step size such that

where is a reference objective function value. The new iteration point

    We employ the proposed SSHC method to cluster synthetic and real data, and compare the results of SSHC method with an unsupervised method SHC. Numerical experiments show that our method works well on synthetic and real data.

(Communicated by Chen Ling)

The third one is an infeasible trust-region algorithm, which constructs an infeasible trustregion subproblem, expanding the search scope with the expectation of improving the success rate. The largest Z-eigenvalue of tensors can be obtained by solving the following optimization problem

    To expand the search scope, we construct the following trust-region subproblem with linear constraint at the current iteration point , 

where

    For solving subproblem (2), if the constraints on this subproblem are consistent, the trial steps  is solved by interior-point method. Otherwise, if the constraints are inconsistent, a mechanism is introduced into the algorithm which computes the step in two stages. In the first stage, we try to satisfy the linearized constraints, that is,

here . Then the total trial step dk can be accepted by solving

    Each iteration point of the subproblem (2) may not be feasible (). To expand the search scope, jump out of the local optimal region and find better points, the algorithm can accept the reduction in the objective function f. To prevent moving away from the sphere constraint while the objective function descends, the constraints must remain feasible when accepting a decrease in the objective function. Hence, we employ a trial step and project it onto the unit sphere.
     We demonstrate the global convergence of the infeasible trust region algorithm, which can converge to a KKT point, a strong stationary point, or a better point that jumps out of the local region of the original problem (1). The above three algorithms are considered as the preprocessing strategies based on the feasible trust-region method. We combine the three preprocessing strategies with the feasible trust-region method and conduct comparisons. Numerical experimental results show that the success rate of calculating the extreme Z-eigenvalue is greatly increased with the help of the global optimization strategy.

    For linear imaging problem

the EM algorithm can be derived by minimizing the Kullback-Leibler (KL) divergence between forward problem data and the measurements b. However, the Kullback-Leibler divergence is sensitive to noise due to part of small boundary measurements. Thus in this manuscript, a parameter is introduced into Kullback-Leibler divergence and the difference between  and b is measured by the -KL divergence

Then the linear imaging problem (1) is reduced to a constrained minimization problem

and solved by a novel parameterized iteration formula

which is derived from the Karush-Kuhn-Tucker condition of (3).
    To get the convergent property of the iterative method, the original optimization problem is spit into two optimization subproblems
    1. Find minimizer of  with respect to r,

    2. Find minimizer of   with respect to ,

where and are two mn-dimensional hidden vectors with entries , and . Then the iteration sequence generated by (4) has the following properties

In particular, if , then .

     In the first subproblem, the minimizer of the with respect to r with the constraint  is

 In the second subproblem, fix and then there are the following results

Combining the property that a bounded point sequence must have a convergent subsequence, the convergence of the iterative sequence can be obtained.
    To test the performance of the new iterative formula, a fan-beam X-ray CT reconstruction problem of low-contract Shepp-Logan phantom is used. Compared with the traditional EM algorithm,
the parameterized algorithm can achieve a relatively better reconstruction effect and faster convergent speed by adjusting the parameter .

(Communicated by Michel Théra)

Abstract: The aim of our paper is to establish optimality conditions on dual spaces for constrained and unconstrained set-valued optimization problems with respect to some new notions of directional Pareto efficiency. The methods we employ are different for each of the two cases. For the constrained problem we use a fuzzy extremal principle, and for the unconstrained one we use the incompatibility between openness and minimality to prove that the limit of a sequence of directional minima for a perturbation of the objective set-valued map F is a critical point for F.