A Successive Projection Algorithm for Solving the Multiple-Sets Split Feasibility Problem

The multiple-sets split feasibility problem (MSFP) arises in many areas and it can be unified as a model for many inverse problems where the constraints are required on the solutions in the domain of a linear operator as well as in the operator's range. Some existing algorithms, in order to get the suitable step size, need to compute the largest eigenvalue of the related matrix, estimate the Lipschitz constant, or use some step-size search scheme, which usually requires many inner iterations. In this article, we introduce a successive projection algorithm for solving the multiple-sets split feasibility problem. In each iteration of this algorithm, the step size is directly computed, which is not needed to compute the largest eigenvalue of the matrix or estimate the Lipschitz constant. It also does not need any step-size search scheme. Its theoretical convergence results are also given.     

激光破岩技术的研究现状及进展

激光破岩技术是应用光学领域的重要研究方向,它是集合了多相态、多耦合、多尺度的复杂高温、高压物理与化学过程。为了厘清激光破岩研究中的核心难点,给研究者提供有效的理论参考和趋向信息,本文对激光破岩技术的研究概况进行了分析总结。首先明确了激光破岩机理;之后分别从石油井下作业用激光器、激光破岩影响因素、激光破岩的温度场相变传热、物理力学性质以及激光破岩应用可行性等方面对国内外激光破岩技术研究进行了总结分析,指出了现阶段所取得的创新成果和不足;最后阐明了激光破岩技术的优点和发展趋势。研究结果表明,激光破岩技术可在现场配套设施研发、多影响因素评价、多场耦合作用机制和井下适用性理论体系研究等方向着力突破。     

A projection-fixed point method for a class of bilevel variational inequalities with split fixed point constraints

We propose a method for solving bilevel split variational inequalities involving strongly monotone operators in the leader problems and nonexpansive mappings in the follower ones. The proposed method is a combination between the projection method for variational inequality and the Krasnoselskii–Mann scheme for fixed points of nonexpansive mappings. Strong convergence of the iterative process is proved. Special cases are considered.     

Extragradient and linesearch methods for solving split feasibility problems in Hilbert spaces

Utilizing the Tikhonov regularization method and extragradient and linesearch methods, some new extragradient and linesearch algorithms have been introduced in the framework of Hilbert spaces. In the presented algorithms, the convexity of optimization subproblems is assumed, which is weaker than the strong convexity assumption that is usually supposed in the literature, and also, the auxiliary equilibrium problem is not used. Some strong convergence theorems for the sequences generated by these algorithms have been proven. It has been shown that the limit point of the generated sequences is a common element of the solution set of an equilibrium problem and the solution set of a split feasibility problem in Hilbert spaces. To illustrate the usability of our results, some numerical examples are given. Optimization subproblems in these examples have been solved by FMINCON toolbox in MATLAB.     

凸可行问题的平行近似次梯度投影算法

对凸可行问题提出了包括上松弛的平行近似次梯度投影算法和加速平行近似次梯度投影算法．与序列近似次梯度投影算法相比, 平行近似次梯度投影算法(每次迭代同时运用多个凸集的近似次梯度超平面上的投影)能够保证迭代序列收敛到离各个凸集最近的点． 上松弛的迭代技术和含有外推因子的加速技术的应用, 减少了数据存储量, 提高了收 敛速度. 最后在较弱的条件下证明了算法的收敛性, 数值实验结果验证了算法的有效性和优越性.     

Extragradient methods for split feasibility problems and generalized equilibrium problems in Banach spaces

The purpose of this paper is the presentation of a new extragradient algorithm in 2‐uniformly convex real Banach spaces. We prove that the sequences generated by this algorithm converge strongly to a point in the solution set of split feasibility problem, which is also a common element of the solution set of a generalized equilibrium problem and fixed points of of two relatively nonexpansive mappings. We give a numerical example to investigate the behavior of the sequences generated by our algorithm.     

Catalytic Cracking of Naphtha to Light Olefins

A catalytic process that produces light olefins from naphtha was developed to improve the yield of the conventional steam cracker. In laboratory-scale tests, a newly developed zeolite-based catalyst at a reaction temperature of 650°C produced an ethylene-plus-propylene yield of about 60%, which is about 10% higher than the conventional process operated at around 820°C. A feasibility study carried out for a catalytic cracking process using the developed catalyst, that cracks 3000 tons-naphtha/day, indicated an energy saving of about 20% compared with the conventional process.     

Reflection-Projection Method for Convex Feasibility Problems with an Obtuse Cone

The convex feasibility problem asks to find a point in the intersection of finitely many closed convex sets in Euclidean space. This problem is of fundamental importance in the mathematical and physical sciences, and it can be solved algorithmically by the classical method of cyclic projections.In this paper, the case where one of the constraints is an obtuse cone is considered. Because the nonnegative orthant as well as the set of positive-semidefinite symmetric matrices form obtuse cones, we cover a large and substantial class of feasibility problems. Motivated by numerical experiments, the method of reflection-projection is proposed: it modifies the method of cyclic projections in that it replaces the projection onto the obtuse cone by the corresponding reflection.This new method is not covered by the standard frameworks of projection algorithms because of the reflection. The main result states that the method does converge to a solution whenever the underlying convex feasibility problem is consistent. As prototypical applications, we discuss in detail the implementation of two-set feasibility problems aiming to find a nonnegative [resp. positive semidefinite] solution to linear constraints in ⁿ [resp. in , the space of symmetric n×n matrices] and we report on numerical experiments. The behavior of the method for two inconsistent constraints is analyzed as well.     

An Iterative Approach to Quadratic Optimization

Assume that C ₁, . . . , C _N are N closed convex subsets of a real Hilbert space H having a nonempty intersection C. Assume also that each C _i is the fixed point set of a nonexpansive mapping T _i of H. We devise an iterative algorithm which generates a sequence (x _n ) from an arbitrary initial x ₀H. The sequence (x_n) is shown to converge in norm to the unique solution of the quadratic minimization problem min _xC (1/2)Ax, x–x, u, where A is a bounded linear strongly positive operator on H and u is a given point in H. Quadratic–quadratic minimization problems are also discussed.     

A Simple Primal-Dual Feasible Interior-Point Method for Nonlinear Programming with Monotone Descent

We propose and analyze a primal-dual interior point method of the feasible type, with the additional property that the objective function decreases at each iteration. A distinctive feature of the method is the use of different barrier parameter values for each constraint, with the purpose of better steering the constructed sequence away from non-KKT stationary points. Assets of the proposed scheme include relative simplicity of the algorithm and of the convergence analysis, strong global and local convergence properties, and good performance in preliminary tests. In addition, the initial point is allowed to lie on the boundary of the feasible set.