We study the regularity criterion for the 3D nematic liquid crystal flows in the framework of anisotropic Lebesgue space. More precisely, we proved some sufficient conditions in terms of velocity or the fractional derivative of velocity in one direction.

We prove a strong convergence theorem for a common fixed point of a finite family of right Bregman strongly nonexpansive mappings in the framework of real reflexive Banach spaces. Furthermore, we apply our method to approximate a common zero of a finite family of maximal monotone mappings and a solution of a finite family of convex feasibility problems in reflexive real Banach spaces. Our theorems complement some recent results that have been proved for this important class of nonlinear mappings.

We introduce a new construction—$F{S}_{+}$-domain—and prove that the category with $F{S}_{+}$-domains as objects and Scott continuous functions as morphisms is a Cartesian closed category. We obtain that the Plotkin powerdomain $P^P(L)$ over an $FS$-domain $L$ is an $F{S}_{+}$-domain.

The aim of this paper is to show the strong convergence theorems of the $CQ$ algorithm for $H$-monotone operators in Hilbert spaces by hybrid method in the mathematical programming. The main results extend and improve the corresponding results. Moreover, the assumption conditions of our results are weaker than those of the corresponding results.

This paper presents a novel optimization method for effectively solving nonconvex quadratically constrained quadratic programs (NQCQP) problem. By applying a novel parametric linearizing approach, the initial NQCQP problem and its subproblems can be transformed into a sequence of parametric linear programs relaxation problems. To enhance the computational efficiency of the presented algorithm, a cutting down approach is combined in the branch and bound algorithm. By computing a series of parametric linear programs problems, the presented algorithm converges to the global optimum point of the NQCQP problem. At last, numerical experiments demonstrate the performance and computational superiority of the presented algorithm.

A generalized Newton method for the solution of a kind of complementarity problem is given. The method is based on a nonsmooth equations reformulation of the problem by F-B function and on a generalized Newton method. The merit function used is a differentiable function. The global convergence and superlinear local convergence results are also given under suitable assumptions. Finally, some numerical results and discussions are presented.

This paper presents a global optimization algorithm for solving globally the generalized nonlinear multiplicative programming (MP) with a nonconvex constraint set. The algorithm uses a branch and bound scheme based on an equivalently reverse convex programming problem. As a result, in the computation procedure the main work is solving a series of linear programs that do not grow in size from iterations to iterations. Further several key strategies are proposed to enhance solution production, and some of them can be used to solve a general reverse convex programming problem. Numerical results show that the computational efficiency is improved obviously by using these strategies.

Under the condition that the values of the objective function and its subgradient are computed approximately, we introduce a cutting plane and level bundle method for minimizing nonsmooth nonconvex functions by combining cutting plane method with the ideas of proximity control and level constraint. The proposed algorithm is based on the construction of both a lower and an upper polyhedral approximation model to the objective function and calculates new iteration points by solving a subproblem in which the model is employed not only in the objective function but also in the constraints. Compared with other proximal bundle methods, the new variant updates the lower bound of the optimal value, providing an additional useful stopping test based on the optimality gap. Another merit is that our algorithm makes a distinction between affine pieces that exhibit a convex or a concave behavior relative to the current iterate. Convergence to some kind of stationarity point is proved under some looser conditions.

This work is to consider Furuta type inequalities and their applications. Firstly, some Furuta type inequalities under $A\ge B\ge 0$ are obtained via Loewner-Heinz inequality; as an application, a proof of Furuta inequality is given without using the invertibility of operators. Secondly, we show a unified satellite theorem of grand Furuta inequality which is an extension of the results by Fujii et al. At the end, a kind of Riccati type operator equation is discussed via Furuta type inequalities.

A practical algorithm for solving large-scale box-constrained optimization problems is developed, analyzed, and tested. In the proposed algorithm, an identification strategy is involved to estimate the active set at per-iteration. The components of inactive variables are determined by the steepest descent method at first finite number of steps and then by conjugate gradient method subsequently. Under some appropriate conditions, we show that the algorithm converges globally. Numerical experiments and comparisons by using some box-constrained problems from CUTEr library are reported. Numerical comparisons illustrate that the proposed method is promising and competitive with the well-known method—L-BFGS-B.

Nonlinear Lagrangian algorithm plays an important role in solving constrained optimization problems. It is known that, under appropriate conditions, the sequence generated by the first-order multiplier iteration converges superlinearly. This paper aims at analyzing the second-order multiplier iteration based on a class of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints. It is suggested that the sequence generated by the second-order multiplier iteration converges superlinearly with order at least two if in addition the Hessians of functions involved in problem are Lipschitz continuous.

We study a nonmonotone adaptive Barzilai-Borwein gradient algorithm for $l_1$-norm minimization problems arising from compressed sensing. At each iteration, the generated search direction enjoys descent property and can be easily derived by minimizing a local approximal quadratic model and simultaneously taking the favorable structure of the $l_1$-norm. Under some suitable conditions, its global convergence result could be established. Numerical results illustrate that the proposed method is promising and competitive with the existing algorithms NBBL1 and TwIST.

We present a new iterative scheme with errors to solve the problems of finding common zeros of finite $m$-accretive mappings in a real Banach space. Strong convergence theorems are established, which extend the corresponding works given by some authors. Moreover, the relationship between zeros of $m$-accretive mappings and one kind of nonlinear elliptic systems is investigated, from which we can see that some restrictions imposed on the iterative scheme are valid and the solution of one kind of nonlinear elliptic systems can be approximated by a suitably defined iterative sequence.

Let , let , and let $\phi$ be an analytic self-map of $𝔻$ and $g\in H(𝔻)$. The boundedness and compactness of generalized composition operators $(C_{\phi }^{g}f)(z)={\int }_0^z\mathrm{‍}{f}^{\mathrm{\text{'}}}(\phi (\xi ))g(\xi )d\xi ,\hspace{0.17em}z\in 𝔻,\hspace{0.17em}f\in H(𝔻)$, from ${ℬ}_{\mu }$ (${ℬ}_{\mu ,0}$) spaces to $Q_{K,\omega }(p,q)$ spaces are investigated.

For the multiclass classification problem of microarray data, a new optimization model named multinomial regression with the elastic net penalty was proposed in this paper. By combining the multinomial likeliyhood loss and the multiclass elastic net penalty, the optimization model was constructed, which was proved to encourage a grouping effect in gene selection for multiclass classification.

This paper presents a global optimization algorithm for solving the signomial geometric programming (SGP) problem. In the algorithm, by the straight forward algebraic manipulation of terms and by utilizing a transformation of variables, the initial nonconvex programming problem (SGP) is first converted into an equivalent monotonic optimization problem and then is reduced to a sequence of linear programming problems, based on the linearizing technique. To improve the computational efficiency of the algorithm, two range reduction operations are combined in the branch and bound procedure. The proposed algorithm is convergent to the global minimum of the (SGP) by means of the subsequent solutions of a series of relaxation linear programming problems. And finally, the numerical results are reported to vindicate the feasibility and effectiveness of the proposed method.

This paper is devoted to studying the Cauchy problem for a fifth-order equation. We prove that it is locally well-posed for the initial data in the Sobolev space $H^s(\mathbf{R})$ with $s\ge 1/4$. We also establish the ill-posedness for the initial data in $H^s(\mathbf{R})$ with . Thus, the regularity requirement for the fifth-order dispersive equations $s\ge 1/4$ is sharp.

A new modified mixed Ishikawa iterative sequence with error for common fixed points of two asymptotically quasi pseudocontractive type non-self-mappings is introduced. By the flexible use of the iterative scheme and a new lemma, some strong convergence theorems are proved under suitable conditions. The results in this paper improve and generalize some existing results.

The multiple-sets split equality problem (MSSEP) requires finding a point $x\in {\cap }_{i=1}^N{C}_i$, $y\in {\cap }_{j=1}^M{Q}_j$ such that $Ax=By$, where $N$ and $M$ are positive integers, $\{C_1,C_2,\dots ,C_N\}$ and $\{Q_1,Q_2,\dots ,Q_M\}$ are closed convex subsets of Hilbert spaces $H_1$, $H_2$, respectively, and $A:H_1\to H_3$, $B:H_2\to H_3$ are two bounded linear operators. When $N=M=1$, the MSSEP is called the split equality problem (SEP). If $B=I$, then the MSSEP and SEP reduce to the well-known multiple-sets split feasibility problem (MSSFP) and split feasibility problem (SFP), respectively. One of the purposes of this paper is to introduce an iterative algorithm to solve the SEP and MSSEP in the framework of infinite-dimensional Hilbert spaces under some more mild conditions for the iterative coefficient.

Let $n$ be a positive integer, and an operator $T\in B(\mathscr{H})$ is called a class $A(n)$ operator if ${\left|T^{1+n}\right|}^{2/\left(1+n\right)}\ge {|T|}^2$ and $n$-paranormal operator if ${∥T^{1+n}x∥}^{1/\left(1+n\right)}\ge ||Tx||$ for every unit vector $x\in \mathscr{H}$, which are common generalizations of class $A$ and paranormal, respectively. In this paper, firstly we consider the tensor products for class $A(n)$ operators, giving a necessary and sufficient condition for $T\otimes S$ to be a class $A(n)$ operator when $T$ and $S$ are both non-zero operators; secondly we consider the properties for $n$-paranormal operators, showing that a $n$-paranormal contraction is the direct sum of a unitary and a $C_{.0}$ completely non-unitary contraction.

It is one of the important issues in the field of today's sewage treatment of researching the MBR membrane flux prediction for membrane fouling. Firstly this paper used the principal component analysis method to achieve dimensionality and correlation of input variables and obtained the three major factors affecting membrane fouling most obvious: MLSS, total resistance, and operating pressure. Then it used the BP neural network to establish the system model of the MBR intelligent simulation, the relationship between three parameters, and membrane flux characterization of the degree of membrane fouling, because the BP neural network has slow training speed, is sensitive to the initial weights and the threshold, is easy to fall into local minimum points, and so on. So this paper used genetic algorithm to optimize the initial weights and the threshold of BP neural network and established the membrane fouling prediction model based on GA-BP network. As this research had shown, under the same conditions, the BP network model optimized by GA of MBR membrane fouling is better than that not optimized for prediction effect of membrane flux. It demonstrates that the GA-BP network model of MBR membrane fouling is more suitable for simulation of MBR membrane fouling process, comparing with the BP network.

We provide some conditions for $2\times 2$ operator matrices whose diagonal entries are $M$-hyponormal operators to be subscalar. As a consequence, we obtain that Weyl type theorem holds for such operator matrices.