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Abstract: In abstract of the paper [A. Rafiq, A note on “A family of methods for solving nonlinear equations”, Appl. Math. Comput. 195 (2008) 819–821] we can find the following sentences. We cite: Ujević et al. introduced a family of methods for solving nonlinear equations. However the main Algorithm 1 put forward by Ujević et al. (p. 7) is wrong. This is the main aim of this note. We also point out some major bugs in the results of Ujević et al. – the end of the citation. Here it is shown that all of the mentioned assertions are not true. In other words, the Algorithm 1 is correct (up to an obvious misprint, which is not mentioned in the above paper) and there are no major bugs in the paper by Ujević et al. In fact, these observations, which will be given in this note, show that the main aim of the paper by Rafiq is wrong. [Copyright &y& Elsevier]

The fundamental purpose of the present paper is to apply an effective numerical algorithm based on the mixture of homotopy analysis technique, Sumudu transform approach and homotopy polynomials to obtain the approximate solution of a nonlinear fractional Drinfeld–Sokolov–Wilson equation. The nonlinear Drinfeld–Sokolov–Wilson equation naturally occurs in dispersive water waves. The uniqueness and convergence analysis are shown for the suggested technique. The convergence of the solution is fixed and managed by auxiliary parameter ℏ. The numerical results are shown graphically. Results obtained by the application of the technique disclose that the suggested scheme is very accurate, flexible, effective and simple to use. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the Stancu type generalisation of Lupaş–Schurer operators based on ( p, q )-integers. We are concerned with the basic convergence of the constructed operators based on Korovkin’s type approximation theorem. Further, we obtain the rate of convergence for the new operators in terms of the modulus of continuity, with the help of functions of Lipschitz class and Peetre’s K-functionals. Then, we present three significant numerical mathematical algorithms. Finally, in order to confirm our theoretical results we obtain error estimation and illustrate the convergence of the ( p, q )-Lupaş–Schurer–Stancu operators to certain functions by using MATLAB. [ABSTRACT FROM AUTHOR]

In this paper, convergence capability of the conventional fractional order gradient methods (FOGMs) is analyzed and a new FOGM with guaranteed and faster convergence ability is proposed. In particular, we further consider the case that the derivative order varies between 1 and 2, and give some discussion on how to determine the derivative order in the algorithm settings. Simulation results from a number of numerical examples are provided to illustrate the approaches. [ABSTRACT FROM AUTHOR]

Abstract: Multipoint iterative methods belong to the class of the most efficient methods for solving nonlinear equations. Recent interest in the research and development of this type of methods has arisen from their capability to overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. This survey paper is a mixture of theoretical results and algorithmic aspects and it is intended as a review of the most efficient root-finding algorithms and developing techniques in a general sense. Many existing methods of great efficiency appear as special cases of presented general iterative schemes. Special attention is devoted to multipoint methods with memory that use already computed information to considerably increase convergence rate without additional computational costs. Some classical results of the 1970s which have had a great influence to the topic, often neglected or unknown to many readers, are also included not only as historical notes but also as genuine sources of many recent ideas. To a certain degree, the presented study follows in parallel main themes shown in the recently published book (Petković et al., 2013) [53], written by the authors of this paper. [Copyright &y& Elsevier]

Abstract: Very recently, Yao et al. (2010) [4] proposed a hybrid iterative algorithm. Under the parameter sequences satisfying some quite restrictive conditions, they derived a strong convergence theorem in a Hilbert space. In this paper, under the weaker conditions, we prove the strong convergence of the sequence generated by their iterative algorithm to a common fixed point of an infinite family of nonexpansive mappings, which solves a variational inequality. An appropriate example, such that all conditions of this result that are satisfied and that other conditions are not satisfied, is provided. Moreover, the method of our paper is different from the method in Yao et al. (2010) [4]. [Copyright &y& Elsevier]

The aim of this paper is to present the convergence analysis of a very general class of gradient projection methods for smooth, constrained, possibly nonconvex, optimization. The key features of these methods are the Armijo linesearch along a suitable descent direction and the non Euclidean metric employed to compute the gradient projection. We develop a very general framework from the point of view of block-coordinate descent methods, which are useful when the constraints are separable. In our numerical experiments we consider a large scale image restoration problem to illustrate the impact of the metric choice on the practical performances of the corresponding algorithm. [ABSTRACT FROM AUTHOR]

Abstract: The hybrid algorithm that combined particle swarm optimization with simulated annealing behavior (SA-PSO) is proposed in this paper. The SA-PSO algorithm takes both of the advantages of good solution quality in simulated annealing and fast searching ability in particle swarm optimization. As stochastic optimization algorithms are sensitive to their parameters, proper procedure for parameters selection is introduced in this paper to improve solution quality. To verify the usability and effectiveness of the proposed algorithm, simulations are performed using 20 different mathematical optimization functions with different dimensions. The comparative works have also been conducted among different algorithms under the criteria of quality of the solution, the efficiency of searching for the solution and the convergence characteristics. According to the results, the SA-PSO could have higher efficiency, better quality and faster convergence speed than compared algorithms. [Copyright &y& Elsevier]

Abstract: Asian options are popular path-dependent financial derivatives. This paper uses lattices to price fixed-strike European-style Asian options that are discretely monitored. The algorithm proposed can also be applied to floating-strike Asian options as well because fixed-strike and floating-strike Asian options are related through an equation. The discretely monitored version is usually found in practice instead of the continuously monitored version usually encountered in the literature. This paper presents the first provably quadratic-time convergent lattice algorithm for pricing fixed-strike European-style discretely monitored Asian options. It is the most efficient lattice algorithm with convergence guarantees. The algorithm relies on the Lagrange multipliers to choose the number of states for each node of the lattice. Extensive numerical experiments and comparisons with many existing numerical methods confirm the performance claims and the competitiveness of our algorithm. This result places fixed-strike European-style discretely monitored Asian options in the same complexity class as vanilla options. [Copyright &y& Elsevier]

Abstract: In this paper a new continuous formulation for the zero-one programming problem is presented, followed by an investigation of the algorithm for it. This paper first reformulates the zero-one programming problem as an equivalent mathematical programs with complementarity constraints, then as a smooth ordinary nonlinear programming problem with the help of the Fischer–Burmeister function. After that the augmented Lagrangian method is introduced to solve the resulting continuous problem, with optimality conditions for the non-smooth augmented Lagrangian problem derived on the basis of approximate smooth variational principle, and with convergence properties established. To our benefit, the sequence of solutions generated converges to feasible solutions of the original problem, which provides a necessary basis for the convergence results. [ABSTRACT FROM AUTHOR]

Abstract: Let X be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. First purpose of this paper is to introduce a modified viscosity iterative process with perturbation for a continuous pseudocontractive self-mapping T and prove that this iterative process converges strongly to x ∗ ∈ F(T)≔{x ∈ X∣x = T(x)}, where x ∗ is the unique solution in F(T) to the following variational inequality: Second aim of the paper is to propose two modified implicit iterative schemes with perturbation for a continuous pseudocontractive self-mapping T and prove that these iterative schemes strongly converge to the same point x ∗ ∈ F(T). Basically, we show that if the perturbation mapping is nonexpansive, then the convergence property of the iterative process holds. In this respect, the results presented here extend, improve and unify some very recent theorems in the literature, see [L.C. Zeng, J.C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear Anal. 64 (2006) 2507–2515; H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291; Y.S. Song, R.D. Chen, Convergence theorems of iterative algorithms for continuous pseudocontractive mappings, Nonlinear Anal. (2006)]. [Copyright &y& Elsevier]

Abstract: This paper proposes a nonlinear Lagrangian Based on Fischer-Burmeister NCP function for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. Moreover, the paper develops the dual approach associated with the proposed nonlinear Lagrangian, in which the related duality theorem is demonstrated. Furthermore, it is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Numerical results for solving several nonlinear programming problems are reported, showing that the new nonlinear Lagrangian is superior over other known nonlinear Lagrangians for solving some nonlinear programming problems. [Copyright &y& Elsevier]

Abstract: This paper presents the new convergence analysis for the Newton-like method proposed by [Xinyuan Wu, A new continuation Newton-like method and its deformation, Appl. Math. Comput. 112 (2000) 75–78.] Compared with the original version of the convergence analysis, the restriction imposed on f″(x) is removed thoroughly. In order to guarantee the quadratic convergence of the Newton-like method, it is suffices to suppose that f′(x ∗)≠0 and f′(x) is local Lipschitz near x ∗, where x ∗ is a solution of nonlinear equation f(x)=0. Moreover, some comments are given with examples for the Newton-like method, in comparison with Newton’s method. It can be concluded that the new algorithm is more feasible, effective than Newton’s method. In particularly, when it happens that, the derivative of the function f(x) at an iterate is singular or almost singular, the Newton-like method is vast superior to the classical Newton method. The numerical results of the paper strongly support the conclusion. [Copyright &y& Elsevier]

Abstract: The particle swarm optimization algorithm was showed to converge rapidly during the initial stages of a global search, but around global optimum, the search process will become very slow. On the contrary, the gradient descending method can achieve faster convergent speed around global optimum, and at the same time, the convergent accuracy can be higher. So in this paper, a hybrid algorithm combining particle swarm optimization (PSO) algorithm with back-propagation (BP) algorithm, also referred to as PSO–BP algorithm, is proposed to train the weights of feedforward neural network (FNN), the hybrid algorithm can make use of not only strong global searching ability of the PSOA, but also strong local searching ability of the BP algorithm. In this paper, a novel selection strategy of the inertial weight is introduced to the PSO algorithm. In the proposed PSO–BP algorithm, we adopt a heuristic way to give a transition from particle swarm search to gradient descending search. In this paper, we also give three kind of encoding strategy of particles, and give the different problem area in which every encoding strategy is used. The experimental results show that the proposed hybrid PSO–BP algorithm is better than the Adaptive Particle swarm optimization algorithm (APSOA) and BP algorithm in convergent speed and convergent accuracy. [Copyright &y& Elsevier]

Abstract: In this paper, a heuristic algorithm, different from the Frank–Wolfe and its modified methods, is introduced for network equilibration. By using the column generation technique and the network equilibrium conditions, the new method need not enumerate initially all feasible paths for all origin/destination (O/D) pairs, but can give all paths used between each O/D pair and the path flows accordingly while the new algorithm obtains an optimal traffic assignment. Some convergence issues of the new method is discussed in this paper. Numerical experiments show that the new method is efficient and robust. [Copyright &y& Elsevier]

Abstract: This paper presents an approach to the well-known traveling salesman problem (TSP) using Kohonen self-organizing maps (SOM). We investigate these questions, ‘Does it converge to a feasible solution?’ ‘Is there a best adaptation law of parameters for the algorithm?’ ‘What is the effect of different sequencing of the nodes along the initial path?’ ‘How do we organize the index of winner neurons and the initial order of the cities?’ Despite many different types of SOM algorithm having already been approached to solve the TSP in the literature, the purpose of this paper is to look for the incorporation of an efficient initialization method and the definition of a parameter adaptation law to achieve better quality results and a faster convergence. Complexity of the modified SOM algorithm is analyzed. The results show an average deviation of 2.4372% from the optimal tour length for a set of 12 TSP instances. This result is better than an average deviation of 3.69% by using the same 12 TSP instances in the literature [C.V. Frederico, D.D.N. Adiao, A.F. Jose, An efficient approach to the traveling salesman problem using self-organizing maps, International Journal of Neural Systems 13(2) (2003) 59–66]. [Copyright &y& Elsevier]

The slow-convergence characteristics of the regular convex combination algorithm (such as Frank–Wolfe method) are well known particularly when the optimal solution is being reached. In this paper, a new convex combination algorithm with non-monotone line search technique proposed for solving the equilibrium assignment problem, together with the proof of its global convergence. Moreover, it should be pointed out that the conditions which ensure the global convergence of the algorithm proposed in this paper are much milder than those suggested by Bonnans et al.; Grippo et al.; Panier and Tits; Xu et al. [SIAM J. Number. Anal. 29 (4) (1992) 1187; SIAM J. Number. Anal. 23 (4) (1986) 707; SIAM J. Number. Anal. 28 (4) (1991) 1183; Comput. Optimiz. Appl. 18 (2001) 221]. The new algorithm can be viewed as a generalization of the regular convex combination algorithm. Numerical results indicate that the proposed algorithm would lead to a considerable saving both in the number of line searches and in the number of function evaluations. [Copyright &y& Elsevier]

In this paper, we propose a new iterative algorithm to approximate fixed point of Suzuki’s generalized nonexpansive mappings. We establish some weak and strong convergence theorems in a uniformly convex Banach space. We also provide examples to illustrate the convergence behavior of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The sequential block-iterative scheme based on Landweber’s method is a general iterative one for image reconstruction. In this paper, we give the finite termination convergence for this scheme, which provides an approach to choose relaxation coefficients. Furthermore, sufficient and necessary convergence conditions are also established for the sequential block-iterative scheme case. [ABSTRACT FROM AUTHOR]

A trust-region algorithm in association with line search filter technique for solving nonlinear equality constrained programming is proposed in this paper. In the current iteration, the trial step providing sufficient descent is generated by solving a corresponding trust-region subproblem. Then, the step size is decided by backtracking line search together with filter technique to obtain the next iteration point. The advantage of this method is that resolving trust-region subproblem many times to determine a new iteration point in traditional trust-region method can be avoided and hence the expensive computation can be lessened. And the difficult decisions in regard to the choice of penalty parameters in the merit functions can be avoided by using filter technique. Second order correction steps are introduced in the proposed algorithm to overcome Maratos effect. Convergence analysis shows that fast local convergence can be achieved under some mild assumptions. The preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

A new statistical analysis model is used to analyze the pseudo affine projection (PAP) algorithm with the optimal step-size in this paper. By setting the weight error to be zero in the direction of the adaptive weight update, the optimal step-size is obtained. Two assumptions for the direction vector are proposed, which simplify the analysis of the convergence behavior for the optimal step-size PAP algorithm. Under this condition, deterministic recursive equations for the weight error and for the mean-square error are derived in the direction of the adaptive weight update. Simulation results are provided to corroborate the analytical results. [ABSTRACT FROM AUTHOR]

Class of algorithms referring to the affine projection algorithms (APA) applies updates to the weights in a direction that is orthogonal to the most recent input vectors. This speeds up the convergence of the algorithm over that of the normalized least mean square (NLMS) algorithm, especially for highly colored input processes. In this paper a new statistical analysis model is used to analyze the APA class of algorithms with unity step size. Four assumptions are made, which are based on the direction vector for the APA class. Under these assumptions, deterministic recursive equations for the weight error and for the mean-square error are derived. We also analyze the steady-state behavior of the APA class. The new model is applicable to input processes that are autoregressive as well as autoregressive-moving average, and therefore is useful under more general conditions than previous models for prediction of the mean square error of the APA class. Simulation results are provided to corroborate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, a fast proximal point algorithm (PPA) is proposed for solving ℓ 1 -minimization problem arising from compressed sensing. The proposed algorithm can be regarded as a new adaptive version of customized proximal point algorithm, which is based on a novel decomposition for the given nonsymmetric proximal matrix M . Since the proposed method is also a special case of the PPA-based contraction method, its global convergence can be established using the framework of a contraction method. Numerical results illustrate that the proposed algorithm outperforms some existing proximal point algorithms for sparse signal reconstruction. [ABSTRACT FROM AUTHOR]

In this paper we consider relations between the principal iterations from the Padé family of Kenney and Laub for computing the matrix sign function and the principal iterations from the reciprocal Padé family of Greco, Iannazzo and Poloni, and the dual Padé family of Ziȩtak. We show global convergence of the principal reciprocal Padé iterations and the principal dual Padé iterations. We adopt the dual Padé family of iterations, to which the Newton method belongs, for computing the unitary polar factor of a nonsingular matrix. We present numerical experiments with the scaled Newton method of Higham for computing the unitary polar factor, which show how the quality of the computed inverses of matrices in the scaled Newton method affects the accuracy of the computed polar factorization of a nonsingular matrix. It indicates that assumptions, under which the backward stability of the scaled Newton method has been proved in the literature, cannot be weaker. [ABSTRACT FROM AUTHOR]

A class of differential quadratic programming problems in finite dimensional spaces is introduced in this paper. First, the existence of solutions for the differential quadratic programming problem is established under some suitable assumptions. Second, an algorithm for solving the differential quadratic programming problem is given and the convergence analysis for the algorithm is shown. Finally, some numerical experiments are reported to verify the validity of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a D–N alternating algorithm based on the natural boundary reduction (NBR) is discussed to solve exterior three dimensional (3-D) Poisson problem with prolate spheroid artificial boundary. By the principle of the natural boundary reduction, we obtain the natural integral equation on prolate spheroid artificial boundary, suggest a D–N alternating algorithm, and analysis its convergence of the algorithm. Finally, some numerical examples are presented to show the effectiveness of this method. [ABSTRACT FROM AUTHOR]

The sparse representation of a multiple measurement vector (MMV) is an important problem in compressed sensing theory, the old alternating direction method (ADM) is an optimization algorithm that has recently become very popular due to its capabilities to solve large-scale or distributed problems. The MMV–ADM algorithm to solve the MMV problem by ADM has been proposed by H. Lu, et al. (2011)[24], but the theoretical result about the convergence of matrix iteration sequence generated by the algorithm is left as a future research topic. In this paper, based on the subdifferential property of the two-norm for vector, a shrink operator associated with matrix is established. By using the operator, a convergence theorem is proved, which shows the MMV–ADM algorithm can recover the jointly sparse vectors. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of absolute value equations associated with circular cone (CCAVE for short), which is a generalization of the absolute value equations discussed recently in the literature, analogously to the fact that circular cone is the very generalization of the second-order cone. We show that the CCAVE is equivalent to a class of circular cone linear complementarity problems, and hence generalize the well-known equivalence between absolute value equations and linear complementarity problems. Useful properties of the generalized differential of the absolute value function over the circular cone are investigated, which helps to propose a generalized Newton method for solving the CCAVE. The convergence of this method is established under mild conditions, as well as the efficiency of which is illustrated by some preliminary numerical results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two finite families of demicontractive mappings. The strong convergence theorem of the proposed algorithm is established under some suitable control conditions in a real Hilbert space. Our result generalizes several recent results in the current literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new class of conjugate gradient algorithms for training neural networks which is based on a new modified nonmonotone scheme proposed by Shi and Wang (2011). The utilization of a nonmonotone strategy enables the training algorithm to overcome the case where the sequence of iterates runs into the bottom of a curved narrow valley, a common occurrence in neural network training process. Our proposed class of methods ensures sufficient descent, avoiding thereby the usual inefficient restarts and it is globally convergent under mild conditions. Our experimental results provide evidence that the proposed nonmonotone conjugate gradient training methods are efficient, outperforming classical methods, proving more stable, efficient and reliable learning. [ABSTRACT FROM AUTHOR]

In this paper, we combine the conjugate gradient method with the Barzilai and Borwein gradient method, and propose a Barzilai and Borwein scaling conjugate gradient method for nonlinear unconstrained optimization problems. The new method does not require to compute and store matrices associated with Hessian of the objective functions, and has an advantage of less computational efforts. Moreover, the descent direction property and the global convergence are established when the line search fulfills the Wolfe conditions. The limited numerical experiments and comparisons show that the proposed algorithm is potentially efficient. [ABSTRACT FROM AUTHOR]

In the CQ-like algorithms for the split feasibility problem, in order to get the step-size, one has to compute the largest eigenvalue of the related matrix or use some line search scheme. Our contribution in this short note is to give a simple CQ-like algorithm in which the step-size is directly computed. The algorithm presented in this paper not only need not to compute the largest eigenvalue of the related matrix but also need not to use any line search scheme. The theoretical convergence and numerical results are also given. [ABSTRACT FROM AUTHOR]

In this paper we present the convergence and stochastic stability analysis of a set of PSO variants: those that differ with the classical PSO in the statistical distribution of the three PSO tuning parameters: inertia weight, local and global acceleration. We provide an analytical expression for the upper limit of the second order stability regions (the so called USL curves) of the particle trajectories that can be applied to most of these PSO algorithms. Thus, this work generalizes to this set the result found in the literature for the classical PSO. We apply this analysis to some of these variants. Finally, numerical experiments have been performed that confirm the known fact that the best algorithm performance is provided tuning the PSO parameters close to the USL curve. [ABSTRACT FROM AUTHOR]

The paper deals with an effective implementation of some algorithms for the solution of convex QPQC problems with elliptic and other separable constraints with strong curvature. Here we discuss robust quantitative refinement of the Karush–Kuhn–Tucker conditions, extend existing results on the decrease of the cost function along the projected gradient path to separable constraints with elliptic components, and plug them into the existing algorithms for the solution of the QPQC problems with R-linear rate of convergence in the bounds on the spectrum. The results are then extended to the problems with separable inequality and linear equality constraints. The performance of the algorithms is demonstrated on the solution of a problem of two cantilever beams in mutual contact with orthotropic Tresca and Coulomb friction discretized by up to one and half million nodal variables. [ABSTRACT FROM AUTHOR]

This paper presents an improved multiple objective particle swarm optimization (MOPSO) algorithm to solve bilevel linear programming problems with multiple objective functions at the upper level. The algorithm aims to produce a good approximation of the entire Pareto front of the problem. We have previously designed a MOPSO algorithm for the same class of problems, in which several techniques for the global best selection were tested, including a new one. The algorithm revealed a good convergence towards the Pareto front but the diversity of the solutions was a drawback. The algorithm we propose herein uses a hybrid strategy for the global best selection and an adaptive mutation operator. The incorporation of these mechanisms led to an improved algorithm, which also showed better overall performance than considering alternative options usually employed in MOPSO algorithms. The algorithm and computational results are presented. [ABSTRACT FROM AUTHOR]

In this paper, we establish a class of accelerated Uzawa (AU) algorithms for solving the large sparse nonsingular saddle point problems by making use of the extrapolation technique. This extrapolation technique is based on the eigenvalues of the iterative matrix. These AU algorithms involve two iteration parameters whose special choices can cover the known classical Uzawa method, as well as yield new ones. Firstly, the accelerated model for the Uzawa algorithm is established and the detail algorithm description of AU method is presented. Then the convergence analyse of the AU method is given. Moreover, theoretical analyses show that the AU algorithm converges faster than some Uzawa-type methods (the Uzawa method is also included in) when the eigenvalues of the iterative matrix and the parameter τ satisfy some conditions. Numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the AU method. [ABSTRACT FROM AUTHOR]

In this paper we propose a unification approach for a class of Han-type algorithms for approximating least squares solutions of inconsistent systems of linear inequalities. Regarding this aspect we establish two basic assumptions and we prove that all the Han-type algorithms fulfill them, which give us the possibility to obtain certain convergence results. Numerical experiments on two sets of linear separability problems are also presented. [ABSTRACT FROM AUTHOR]

In this paper, the ill-posedness of computing the two dimensional Fourier transform is discussed. A regularized algorithm for computing the two dimensional Fourier transform of band-limited signals is presented. The convergence of the regularized Fourier series is studied and compared with the Fourier series by some examples. [ABSTRACT FROM AUTHOR]

Abstract: A new method used to prove the global convergence of the nonlinear conjugate gradient methods, the spectral method, is presented in this paper, and it is applied to a new conjugate gradient algorithm with sufficiently descent property. By analyzing the descent property, several concrete forms of this algorithm are suggested. Under standard Wolfe line searches, the global convergence of the new algorithm is proven for nonconvex functions. Preliminary numerical results for a set of 720 unconstrained optimization test problems verify the performance of the algorithm and show that the new algorithm is competitive with CG_DESCENT algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, we establish convergence theorems for a general algorithm of an asymptotically nonexpansive map in a uniformly convex hyperbolic space. Our results generalize simultaneously the approximation results of Rhoades (1994) [18], Suantai (2005) [20] and Xu and Noor (2002) [26] on a nonlinear domain. Our results are refinements and generalizations of the corresponding ones in uniformly convex Banach spaces and spaces. [Copyright &y& Elsevier]

Abstract: In this paper, we study geometric properties of surfaces of the generalized Fischer–Burmeister function and its induced merit function. Then, a visualization is proposed to explain how the convergent behaviors are influenced by two descent directions in merit function approach. Based on the geometric properties and visualization, we have more intuitive ideas about how the convergent behavior is affected by changing parameter. Furthermore, geometric view indicates how to improve the algorithm to achieve our goal by setting proper value of the parameter in merit function approach. [Copyright &y& Elsevier]

Abstract: In this paper, based on a symmetrically perturbed smoothing Fischer–Burmeister function, a non-interior continuation method is proposed for solving the second-order cone complementarity problem (SOCCP). The proposed algorithm solves only one linear system of equations and performs only one line search at each iteration. Under monotonicity, it is shown that our algorithm is globally and locally superlinearly convergent without requiring strict complementarity assumption at the SOCCP solution. Furthermore, the proposed algorithm has local quadratic convergence under mild conditions. Some numerical results are reported which indicate the effectiveness of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, a type of smooth nonlinear optimization problems with inequality constraints is considered, and a new sequential linear equation algorithm is proposed by introducing a new constructing technique for the system of linear equations (SLE), which depends on the perturbation of the constraints’ gradients. At each iteration of the proposed algorithm, one or more SLEs need to be solved. Specially, only one SLE is required to be solved when the iterates are sufficiently close to the solution (i.e., after a finite number of iterations), which decreases the amount of computations. The search technique is a combination of the line Armijo-type and Newton step size. Under mild assumptions without the strict complementarity, it is shown that the proposed algorithm enjoys the properties of global and superlinear convergence. Finally, some preliminary numerical tests are reported, and the numerical results show that the proposed algorithm is promising. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a subgradient extragradient method for solving multi-valued variational inequality. It is showed that the method converges globally to a solution of multi-valued variational inequality, provided the multi-valued mapping is pseudomonotone with nonempty compact convex values. Convergence rate is analyzed. Preliminary computational experience is also reported. [Copyright &y& Elsevier]

Abstract: In this paper, nonlinear minimax problems with inequality constraints are discussed. Combined the norm-relaxed SQP method with the idea of strongly sub-feasible directions method, a new method of quasi-strongly sub-feasible directions (MQSSFD) with arbitrary initial point for the discussed problems is presented. At each iteration of the proposed algorithm, an improved search direction is obtained by solving a quadratic program (QP) which always has a solution, and a high-order correction direction is yielded via a system of linear equations (SLE) to avoid the Maratos effect. After finite iterations, the iteration point always get into the feasible set by introducing a new non-monotone curve search. Under some mild conditions including the weak Mangasarian–Fromovitz constraint qualification (MFCQ), the proposed algorithm possesses global convergence, and the superlinear convergence is obtained without the strict complementarity. Finally, some elementary numerical experiments are implemented and reported. [Copyright &y& Elsevier]

Abstract: The alternating direction method of multipliers (ADMM) is known to be a classic and efficient method for constrained optimization problem with two blocks of variables, and its empirical efficiency has been well illustrated in various fields. Specially, for improving its speed performance, Ye and Yuan suggested to do an additional extension with an optimal step size on the variables after each iteration of the primary ADMM. Indeed, the numerical experiments indicate that this modified ADMM improves the speed performance of the ADMM by around 40% without changing the algorithmic framework much. Recently, the convergence rate of the primary ADMM is established. Inspired by its idea, in this paper, we show that this improved ADMM also has convergence rate. The reason that larger γ yields better speed performance is also investigated and explained. [Copyright &y& Elsevier]

Abstract: The purpose of this article is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive semigroups which contains many kinds of semi-groups as its special cases, and then to modify the Halpern–Mann-type iteration algorithm for this kind of semi-groups to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding recent results announced by many authors. [Copyright &y& Elsevier]

Abstract: In this paper, the steepest descent algorithm without line search is proposed for p-Laplacian. Its search direction is the weighted preconditioned steepest descent one, and step length is estimated by a formula except the first iteration. Continuation method is applied for solving the p-Laplacian with very large p. Lots of numerical experiments are carried out on these algorithms. All numerical results show the algorithm without line search can cut down some computational time. Fast convergence of these new algorithms is displayed by their step length figures. These figures show that if search direction is the steepest descent one, exact step lengths can be substituted properly with step lengths obtained by the formula. [Copyright &y& Elsevier]

Abstract: This paper deals with the null controllability of an initial-boundary value problem for a parabolic coupled system with nonlinear terms of local and nonlocal kinds. The control is distributed in space and time and is exerted through one scalar function whose support can be arbitrarily small. We first prove that, if the initial data are sufficiently small and the linearized system at zero satisfies an appropriate coupling condition, the equations can be driven exactly to zero. We also present an iterative algorithm of the quasi-Newton kind for the computation of the control and we prove a convergence result. The behavior of this algorithm is illustrated with some numerical experiments. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the convergence of the double-shift and multi-shift QR algorithms for symmetric tridiagonal matrices. We analyze how to choose multi-shifts by comparing the relationships between the number of iterations, CPU time and the number of multi-shifts. Numerical tests and figures are performed. [Copyright &y& Elsevier]