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Abstract: The purpose of this article is to propose a new hybrid algorithm with variable coefficients and prove convergence theorems for asymptotically pseudocontractive mappings in the intermediate sense on unbounded domains. The results of the paper improve and extend the recent results of Qin et al. [X.L. Qin, S.Y. Cho, J.K. Kim, Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense, Fixed Point Theory Appl. 2010, doi:10.1155/2010/186874, Article ID 186874, 14 pages] and several others. The algorithm with variable coefficients introduced in this paper is of independent interest. [Copyright &y& Elsevier]

Abstract: This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness and Lipschitzian assumptions, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and -differentiable versions of Newton’s method for nonsmooth Lipschitzian equations. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the projection algorithm studied by Iiduka and Takahashi (2008) for finding a solution of the variational inequality problem for an inverse strongly monotone operator in a Banach space. We first remark that, under the assumptions imposed on the operator in their paper, the iterative sequence converges weakly to a zero of the operator, not just a solution of the variational inequality problem. In our proof, slightly modified from the original, we do not assume the uniform smoothness of a space as was the case there. Finally, using Halpern’s type method, we modify this algorithm to obtain the strong convergence to a zero of an inverse strongly monotone operator which is nearest to the initial element of the algorithm in the sense of the Bergman distance associated with the function . [Copyright &y& Elsevier]

Abstract: In this paper, the problem of identifying the active constraints for constrained nonlinear programming and minimax problems at an isolated local solution is discussed. The correct identification of active constraints can improve the local convergence behavior of algorithms and considerably simplify algorithms for inequality constrained problems, so it is a useful adjunct to nonlinear optimization algorithms. Facchinei et al. [F. Facchinei, A. Fischer, C. Kanzow, On the accurate identification of active constraints, SIAM J. Optim. 9 (1998) 14–32] introduced an effective technique which can identify the active set in a neighborhood of a solution for nonlinear programming. In this paper, we first improve this conclusion to be more suitable for infeasible algorithms such as the strongly sub-feasible direction method and the penalty function method. Then, we present the identification technique of active constraints for constrained minimax problems without strict complementarity and linear independence. Some numerical results illustrating the identification technique are reported. [Copyright &y& Elsevier]

Abstract: In this paper, the penalty function method is used to study duality in nonconvex mathematical programming problems. In particular, we prove the zero duality gap between optimization problems involving invex functions with respect to the same function η and their Lagrangian dual problems. The results proved in the paper are illustrated by suitable examples of nonconvex optimization problems. [Copyright &y& Elsevier]

Abstract: In this paper we use the penalty approach in order to study minimization problems with mixed constraints in Banach spaces. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property holds and is stable under perturbations of objective functions, constraint functions and the right-hand side of constraints. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to propose several modified hybrid projection algorithms and prove several strong convergence theorems for common fixed points of a family of Lipschitz quasi-pseudo-contractions in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding ones announced by many others. [Copyright &y& Elsevier]

Abstract: In the present paper, we propose two kinds of new algorithms for a family of quasi-asymptotic pseudo-contractions in real Hilbert spaces. By using the proposed algorithms, we prove several strong convergence theorems for a family of quasi-asymptotic pseudo-contractions. The results of this paper are interesting extensions of those known results. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and consider a new system of general variational inequalities involving four different operators. Using the projection operator technique, we suggest and analyze some new explicit iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of variational inequalities involving three operators, variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with the observability of piecewise-affine hybrid systems. Our aim is to give sufficient conditions to observe the discrete and continuous states, in terms of algebraic and geometrical conditions. Firstly, we will give the algebraic conditions to observe the discrete state based on the switch function reconstruction for linear hybrid systems. Secondly, we will give a geometrical condition based on the transversality concept for nonlinear hybrid systems. Throughout this paper, we illustrate our propositions with examples and simulations. [Copyright &y& Elsevier]

Abstract: Let be real Hilbert spaces, let , be two nonempty closed convex level sets, let , be two bounded linear operators. Our interest is in solving the following new convex feasibility problem which allows asymmetric and partial relations between the variables and . In this paper, we present and study the convergence of a relaxed alternating CQ-algorithm (RACQA) and show that the sequences generated by such an algorithm weakly converge to a solution of (1.1). The interest of RACQA is that we just need projections onto half-spaces, thus making the relaxed CQ-algorithm implementable. Note that, by taking , in (1.1), we recover the split convex feasibility problem originally introduced in Censor and Elfving (1994) [13] and used later in intensity-modulated radiation therapy (Censor et al. (2006) [11]). We also recover the relaxed CQ-algorithm introduced by Yang (2004) [8] by particularizing both and a given parameter. [Copyright &y& Elsevier]

Abstract: The notion of quasi-Fejér monotonicity has proven to be an efficient tool to simplify and unify the convergence analysis of various algorithms arising in applied nonlinear analysis. In this paper, we extend this notion in the context of variable metric algorithms, whereby the underlying norm is allowed to vary at each iteration. Applications to convex optimization and inverse problems are demonstrated. [Copyright &y& Elsevier]

Abstract: In a Hilbert space, we study the convergence of a proximal point method to a common zero of a finite family of maximal monotone operators under the presence of computational errors. Most results known in the literature establish the convergence of proximal point methods, when computational errors are summable. In the present paper, the convergence of the method is established for nonsummable computational errors. We show that the proximal point method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. [Copyright &y& Elsevier]

Abstract: It is a known fact that the method of alternating projections introduced long ago by von Neumann fails to converge strongly for two arbitrary nonempty, closed and convex subsets of a real Hilbert space. In this paper, a new iterative process for finding common zeros of two maximal monotone operators is introduced and strong convergence results associated with it are proved. If the two operators are subdifferentials of indicator functions, this new algorithm coincides with the old method of alternating projections. Several other important algorithms, such as the contraction proximal point algorithm, occur as special cases of our algorithm. Hence our main results generalize and unify many results that occur in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we study the convergence of a Halpern type proximal point algorithm for accretive operators in Banach spaces. Our results fill the gap in the work of Zhang and Song (2012) and, consequently, all the results there can be corrected accordingly. [Copyright &y& Elsevier]

Abstract: Isac and Németh [G. Isac and A. B. Németh, Projection method, isotone projection cones and the complementarity problem, J. Math. Anal. App., 153, 258-275(1990)] proved that solving a coincidence point equation (fixed point problem) in turn solves the corresponding implicit complementarity problem (nonlinear complementarity problem) and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems (nonlinear complementarity problems) defined by these cones. In this paper, the notion of *-isotone projection cones is employed and an iterative algorithm is presented in connection with an implicit complementarity problem on *-isotone projection cones. It is proved that if the sequence generated through the defined algorithm is convergent, then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem. Sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by *-isotone projection cones. The question of finding nonzero solutions of these problems is also studied. [Copyright &y& Elsevier]

Abstract: The main objectives of this paper are to employ a new proof technique to prove the strong convergence of and , defined respectively by to some zero of accretive operator in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm (or with a weakly continuous duality mapping ) whenever and are sequences in and satisfying the conditions: , , ,. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new multi-step iterative method. We prove the strong convergence of the method to a common fixed point of a finite number of nonexpansive mappings that also solves a suitable equilibrium problem. [Copyright &y& Elsevier]

Abstract: In this paper, it is proved that the sequence defined by Mann’s algorithm -converges to a fixed point in complete CAT() spaces. [Copyright &y& Elsevier]

Abstract: In this paper, a general system of nonlinear variational inequality problem in Banach spaces was considered, which includes some existing problems as special cases. For solving this nonlinear variational inequality problem, we construct two methods which were inspired and motivated by Korpelevich’s extragradient method. Furthermore, we prove that the suggested algorithms converge strongly to some solutions of the studied variational inequality. [Copyright &y& Elsevier]

Abstract: Let be a nonempty closed convex subset of a real Hilbert space . Let be a non-expansive mapping and be an infinite family of non-expansive mappings. The purpose of this paper is to find the minimum norm solution of the following general hierarchical fixed point problem We introduce an explicit regularized algorithm with strong convergence for finding the minimum norm solution of the above hierarchical fixed point problem. [Copyright &y& Elsevier]

Abstract: The work of Hundal [H. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal. 57 (1) (2004) 35–61] has revealed that the sequence generated by the method of alternating projections converges weakly, but not strongly in general. In this paper, we present several algorithms based on alternating resolvents of two maximal monotone operators, and , that can be used to approximate common zeros of and . In particular, we prove that the sequences generated by our algorithms converge strongly. A particular case of such algorithms enables one to approximate minimum values of certain convex functionals. [Copyright &y& Elsevier]

Abstract: In this paper, we use Conley index theory to develop necessary conditions for stability of equilibrium and periodic solutions of nonlinear continuous-time systems. The Conley index is a topological generalization of the Morse theory which has been developed to analyze dynamical systems using topological methods. In particular, the Conley index of an invariant set with respect to a dynamical system is defined as the relative homology of an index pair for the invariant set. The Conley index can then be used to examine the structure of the system invariant set as well as the system dynamics within the invariant set, including system stability properties. Efficient numerical algorithms using homology theory have been developed in the literature to compute the Conley index and can be used to deduce the stability properties of nonlinear dynamical systems. [Copyright &y& Elsevier]

Abstract: In this paper, we propose, analyze, and test a new method for solving -norm regularization problems arising from the spare solution recovery in compressed sensing, basis-pursuit problems, and linear inverse problems. The method aims to minimize a non-smooth minimization problem consisting of a least-squares data fitting term and a -norm regularization term. The problem is firstly formulated for a convex quadratic program problem, and then for an equivalent non-smooth equation. At each iteration, a spectral gradient method is applied to the resulting problem without requiring Jacobian matrix information. Convergence of the proposed method follows directly from the results which already exist. The algorithm is easily performed, where only the matrix–vector inner product is required at each and every step. Numerical experiments to decode a sparse signal arising in compressed sensing and image deconvolution are performed. The numerical results illustrate that the proposed method is practical and promising. [Copyright &y& Elsevier]

Abstract: Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm — one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008) while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996) and later developed by Xu (2006) . The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to construct an Ishikawa type of hybrid algorithm for pseudo-contractive mappings in Hilbert spaces. Our results extend the recent ones announced by Yao et al. [Y.H. Yao, Y.C. Liou, G. Marino, A hybrid algorithm for pseudo-contractive mappings, Nonlinear Anal. 71 (2009) 4997–5002] and many others. [Copyright &y& Elsevier]

Abstract: Some new classes of extended general nonconvex set-valued variational inequalities and the extended general Wiener–Hopf inclusions are introduced. By the projection technique, equivalence between the extended general nonconvex set-valued variational inequalities and the fixed point problems as well as the extended general nonconvex Wiener–Hopf inclusions is proved. Then by using this equivalent formulation, we discuss the existence of solutions of the extended general nonconvex set-valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set-valued variational inequalities. We also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set-valued variational inequalities. The results presented in this paper extend and improve some known results from the literature. [ABSTRACT FROM AUTHOR]

Abstract: In this paper a class of polynomial interior-point algorithms for the Cartesian second-order cone linear complementarity problem based on a parametric kernel function, with parameters and , are presented. The proposed parametric kernel function is used both for determining the search directions and for measuring the distance between the given iterate and the -center for the algorithms. Moreover, the currently best known iteration bounds for the large- and small-update methods, namely, and , are obtained, respectively, which reduce the gap between the practical behavior of the algorithms and its theoretical performance results. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to study hybrid iterative schemes of Halpern types for a semigroup of relatively nonexpansive mappings on a closed and convex subset of a Banach space with respect to a sequence of asymptotically left invariant means defined on an appropriate invariant subspace of . We prove that given a certain sequence in , , we can generate an iterative sequence which converges strongly to where is the generalized projection from onto the fixed point set . Our main result is even new for the case of a Hilbert space. [Copyright &y& Elsevier]

Abstract: In this paper, by introducing mappings and using the idea of the aggregate function method, a new aggregate constraint homotopy method is proposed to solve the Karush–Kuhn–Tucker (KKT) point of nonconvex nonlinear programming problems. Compared with the previous results, the choice scope of initial points is greatly enlarged, so use of the new aggregate constraint homotopy method may improve the computational efficiency of reduced predictor–corrector algorithms. [ABSTRACT FROM AUTHOR]

Abstract: This paper uses tools in quasi-homogeneous normal form theory to discuss certain aspects of reversible vector fields around an equilibrium point. Our main result provides an algorithm, via Lie Triangle, that detects the non-reversibility of vector fields. That is, it is possible to decide whether a planar center is not reversible. Some of the theory developed is also applied to get further results on nilpotent and degenerate polynomial vector fields. We find several families of nilpotent centers which are non-reversible. [Copyright &y& Elsevier]

Abstract: A smoothing method for solving the linear complementarity systems (LCS) has been proposed by Zheng in his thesis. In Zheng’s algorithm, the LCS is reformulated as a parameterized differential-algebraic equation (DAE). However, the existence of the solution of the parameterized DAE is not proved. The aim of this paper is to establish the existence of the solution of the parameterized DAE and to demonstrate the continuous dependence of the solution on the parameter and the initial value of the parameterized DAE. [Copyright &y& Elsevier]

Abstract: In this paper, we present an algorithm for calculating an element of Clarke generalized Jacobian for a vector-valued max-type function. The algorithm reduces the computational cost of an existing algorithm. [Copyright &y& Elsevier]

Abstract: This paper deals with a general fixed point method which unifies relaxation factors and a two step inertial type extrapolation. These strategies are intended to improve the convergence of many existing algorithms. A convergence theorem, which improves the known ones, is established in this new setting. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to introduce the notion of general -monotone operators in Banach spaces and the new proximal mapping associated with the general -monotone operators. By using Alber’s inequalities and the new proximal mapping technique, the variational convergence of a new proximal algorithm for nonlinear general -monotone operator equation systems with relaxed cocoercive operators in Banach spaces is introduced and studied. Our results improve and generalize the corresponding results of recent works. [Copyright &y& Elsevier]

Abstract: In this paper we study iterative algorithms for finding a common element of the set of fixed points of -strict pseudocontractions or finding a solution of a variational inequality problem for a monotone, Lipschitz continuous mapping. The last problem being related to finding fixed points of pseudocontractions. These algorithms were already studied in [G.L. Acedo, H.-K. Xu, Iterative methods for strict pseudo-contractions in hilbert spaces, Nonlinear Analysis 67 (2007) 2258–2271] and [N. Nadezhkina, W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and lipschitz-continuous monotone mappings, SIAM Journal on Optimization 16 (4) (2006) 1230–1241] but our aim here is to provide the links between these known algorithms and the general framework of -class algorithms studied in [H.H. Bauschke, P.L. Combettes, A weak-to-strong convergence principle for fejér-monotone methods in hilbert spaces, Mathematics of Operations Research 26 (2) (2001) 248–264]. [Copyright &y& Elsevier]

Abstract: In this paper, we suggest and analyze a hybrid algorithm for pseudo-contractive mappings in Hilbert spaces. Further, we prove the strong convergence of the proposed iterative algorithm for Lipschitz pseudo-contractive mappings in Hilbert spaces. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new primal-dual interior-point algorithm for solving a special case of convex quadratic semi-definite optimization based on a parametric kernel function. The proposed parametric kernel function is used both for determining the search directions and for measuring the distance between the given iterate and the -center for the algorithm. These properties enable us to derive the currently best known iteration bounds for the algorithm with large- and small-update methods, namely, and , respectively, which reduce the gap between the practical behavior of the algorithm and its theoretical performance results. [Copyright &y& Elsevier]

Abstract: In this paper, we use the auxiliary principle technique in conjunction with the Bregman function to suggest and analyze a predictor–corrector method for solving invex equilibrium problems. We also study the convergence criteria of this new method under some mild conditions. We obtain, as special cases, new and known methods for solving variational-like inequalities and equilibrium problems. [Copyright &y& Elsevier]

Abstract: We present primal–dual interior-point algorithms for second-order cone optimization based on a wide variety of kernel functions. This class of kernel functions has been investigated earlier for the case of linear optimization. In this paper we derive the iteration bounds for large- and for small-update methods, respectively. Here denotes the number of second-order cones in the problem formulation and the desired accuracy. These iteration bounds are currently the best known bounds for such methods. Numerical results show that the algorithms are efficient. [Copyright &y& Elsevier]

Abstract: In this paper we develop the necessary conditions of optimality for a class of distributed parameter systems (partial differential equations) determined by operator valued measures and controlled by vector measures. Based on some recent results on existence of optimal controls from the space of vector measures, we develop necessary conditions of optimality for a class of control problems. The main results are the necessary conditions of optimality for problems without state constraints and those with state constraints. Also, a conceptual algorithm along with a brief discussion of its convergence is presented. [Copyright &y& Elsevier]

Abstract: The purpose in this paper is to prove a theorem of strong convergence to a common solution for a finite family of accretive operators in a strictly convex Banach space by means of a new iterative algorithm, which is a generalization and extension of the results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60], and Zegeye and Shahzad [H. Zegeye, N. Shahzad, Strong convergence theorems for a common zero of a finite family of m-accretive mappings, Nonlinear Anal. 66 (2007) 1161–1169]. Further using the result, the theorem of strong convergence to a common fixed point is discussed for a finite family of pseudocontractive mappings under certain conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we modify the normal Mann’s iterative process to have strong convergence for a -strictly pseudo-contractive non-self mapping in the framework of Hilbert spaces. Our results improve and extend the corresponding results announced by many others. [Copyright &y& Elsevier]

Abstract: In this paper, Hopf bifurcation for two-species Lotka–Volterra competition systems with delay dependence is investigated. By choosing the delay as a bifurcation parameter, we prove that the system is stable over a range of the delay and beyond that it is unstable in the limit cycle form, i.e., there are periodic solutions bifurcating out from the positive equilibrium. Our results show that a stable competition system can be destabilized by the introduction of a maturation delay parameter. Further, an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the theory of normal forms and center manifolds, and numerical simulations supporting the theoretical analysis are also given. [Copyright &y& Elsevier]

Abstract: In [A. Ouorou, A primal-dual algorithm for monotropic programming and its application to network optimization, Computational Optimization and Application 15 (2002) 125–143], a block-wise Gauss–Seidel method has been developed for monotropic programming problems, using two different quadratic augmented Lagrangian functions defined for the primal and the dual problems. In this paper, we extend the concept by introducing a nonlinear re-scaling principle obtained recently by Polyak [R. Polyak, Nonlinear rescaling vs smoothing technique in constrained optimization, Mathematical Programming 92 (2002) 197–235]. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to investigate the asymptotic behavior of algorithms for finding solutions for a certain class of variational inequalities involving nonexpansive type mappings in smooth Banach spaces. We study the existence of solutions of variational inequalities when is the set of solutions of zeros of accretive operators or the set of fixed points of nonexpansive mappings or the set of fixed points of pseudocontractive mappings. Our convergence analysis covers proximal point algorithm for finding zeros of accretive operators as well as functional Halpern algorithm for finding fixed points of nonexpansive mappings in Banach spaces. Our results improve a number of results concerned with viscosity approximation methods in the context of weakly contraction mappings. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and investigate a new class of mixed quasi-variational-like inequalities in reflexive Banach spaces. By applying a minimax inequality due to Ding–Tan and a lemma due to Chang, we establish some existence and uniqueness results of solution for the mixed quasi-variational-like inequality. Next, by using a KKM theorem due to Fan and an auxiliary principle technique due to Cohen, we suggest two iterative algorithms and study the convergence criteria of iterative sequences generated by the iterative algorithms. Our results extend, improve and unify several known results in the literature. [Copyright &y& Elsevier]

Abstract: Stackelberg games, which was originally introduced by Stackelberg, are widely applied in such fields as economics, management, politics and behavioral sciences. Stackelberg games can be modelled as a bi-level optimization problem. There exists an extensive literature about static bi-level optimization problems. However, the studies on dynamic bi-level optimization problems are fairly scarce in spite of the importance in explaining and predicting some phenomena rationally. In this paper, we consider discrete time dynamic Stackelberg games with feedback information. In general, the lower-level strategies are non-unique in practice. For a unique solution, dynamic programming algorithms have been presented with multiple players. We revisit dynamic programming for feedback information dynamic Stackelberg games with non-unique lower-level solution. First, we define some kind of solutions related to the decisions styles. Then, we analyze them, respectively. Moreover, dynamic programming algorithm is successful in solving solve feedback information dynamic Stackelberg games with non-unique lower-level solutions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with obtaining the conditions under which the origin is an isochronous centre for certain planar polynomial systems. The necessity of these conditions is proved using a computer implementation of an algorithm first proposed by I.I. Pleshkan. Their sufficiency is then proved using various methods. [Copyright &y& Elsevier]

Abstract: Let be a bounded smooth domain in . This paper deals with the Moser–Trudinger inequality for functions with mean value zero. Using blowing up analysis, the author proves that is attained for any , and the supremum is infinity for any , where , and is the area of the unit sphere in . [Copyright &y& Elsevier]