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Abstract: In this paper, a trust-region sequential quadratic programming algorithm with a modified filter acceptance mechanism is proposed for nonlinear equality constrained optimization. The most important advantage of the proposed algorithm is its avoidance of any feasibility restoration phase, a necessity in traditional filter methods. We solve quadratic programming subproblems based on the well-known Byrd–Omojokun trust-region method. Inexact solutions to these subproblems are allowed. Under some standard assumptions, global convergence of the proposed algorithm is established. Numerical results show our approach is potentially useful. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new partial splitting augmented Lagrangian method for solving the separable constrained convex programming problem where the objective function is the sum of three separable convex functions and the constraint set is also separable into three parts. The proposed algorithm combines the alternating direction method (ADM) and parallel splitting augmented Lagrangian method (PSALM), where two operators are handled by a parallel method, while the third operator and the former two are dealt with by an alternating manner. Under mild conditions, we prove the global convergence of the new method. We also report some preliminary numerical results on constrained matrix optimization problem, illustrating the advantage of the new algorithm over the most recently PADALM of Peng and Wu (2010) [12]. [Copyright &y& Elsevier]

Abstract: An adaptive parameter tuning of particle swarm optimization based on velocity information (APSO-VI) algorithm is proposed. In this paper the velocity convergence of particles is first analyzed and the relationship between the velocity of particle and the search failures is pointed out, which reveals the reasons why PSO has relative poor global searching ability. Then this algorithm introduces the velocity information which is defined as the average absolute value of velocity of all the particles. A new strategy is presented that the inertia weight is dynamically adjusted according to average absolute value of velocity which follows a given nonlinear ideal velocity by feedback control, which can avoid the velocity closed to zero at the early stage. Under the guide of the nonlinear ideal velocity, APSO-VI can maintain appropriate swarm diversity and alleviate the premature convergence validly. Numerical experiments are conducted to compare the proposed algorithm with different variants of PSO on some benchmark functions. Experimental results show that the proposed algorithm remarkably improves the ability of PSO to jump out of the local optima and significantly enhance the convergence speed and precision. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and analyze the new generalized mixed equilibrium problems (NGMEP) and the bilevel new generalized mixed equilibrium problems (BNGMEP) in Banach spaces. First, by using a minimax inequality, some new existence theorems of the solution and the behavior of solution set for the NGMEP and the BNGMEP are obtained in both arbitrary Banach spaces with the coercive conditions and a reflexive Banach space without the coercive conditions. Next, by using auxiliary principle technique, some new iterative algorithms for solving the NGMEP and the BNGMEP are suggested and analyzed. The strong convergence of the iterative sequences generated by the algorithms are also proved in both arbitrary Banach spaces with the coercive conditions and a reflexive Banach space without the coercive conditions. These results are new and generalize some recent results in this field. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a compact alternating direction implicit (ADI) scheme for solving the two-dimensional parabolic equation. The maximum norm convergence is proven, and the convergence rate is second-order in time and fourth-order in space. We develop a Richardson extrapolation algorithm to increase the accuracy to sixth-order both in time and space. Numerical experiments show the effectiveness of the method. [Copyright &y& Elsevier]

Abstract: The split feasibility problem is to find with , if such points exist, where A is a given real matrix, C and Q are nonempty closed convex sets in and , respectively. Byrne (2002) [2] proposed a well-known CQ algorithm for solving this problem. In this paper, we propose a modification for the CQ algorithm, which computes the stepsize adaptively, and performs an additional projection step onto some simple closed convex set in each iteration. We further give a relaxation scheme for this modification to make it more easily implemented. Convergence results of both algorithms are analyzed, and preliminary numerical results are reported. We also extend these modified algorithms to solve the multiple-sets split feasibility problem, which is to find a point closest to the intersection of a family of closed convex sets in one space such that its image under a linear transformation will be closest to the intersection of another family of closed convex sets in the image space. [Copyright &y& Elsevier]

Abstract: In this paper, we consider numerical simulation to a class of constrained dynamic problems where the overall dynamics are determined by the interactions between two sub-systems. We present an iterative algorithm that naturally decouples the computation of the two sub-systems and that ensures an accurate and efficient solution procedure. We conduct rigorous error analysis for the convergence of the iterative algorithm, and verify the analytical results through careful numerical tests. [Copyright &y& Elsevier]

Abstract: In this paper, a new technique of homotopy analysis method (nHAM) is proposed for solving second order nonlinear differential equations. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM). The proposed provides an approximate solution by rewriting the second order nonlinear differential equation in the form of two first order differential equations. The solution of these two differential equations is obtained as a power series solution. This scheme is tested on four non-linear exactly solvable differential equations. Three of the examples are initial value problems and the fourth is boundary value problem. The results demonstrate reliability and efficiency of the algorithm developed. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new penalty-free-type method for nonlinear equality constrained problems. The new algorithm uses trust region framework and feasibility safeguarding technique. Moreover, it has no choice of penalty parameter and penalty function as a merit function, and it does not use the filter technique to avoid the penalty function either. We analyze the global convergence of the main algorithm under the standard assumptions. The preliminary numerical tests are reported. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a filter algorithm for nonlinear systems of equalities and inequalities. In the algorithm, the systems of nonlinear equalities and inequalities are transformed into a nonlinear programming problem. The new algorithm uses the non-monotone technique in conjunction with filter idea and a global line search strategy that ensures convergence from any starting point. Under mild conditions, global convergence and local superlinear convergence rate are obtained. Numerical results demonstrate that the new algorithm is effective. [Copyright &y& Elsevier]

Abstract: This paper investigates the overall convergence analysis and proposes a novel Random C/D Switchings PSO algorithm with random switchings between convergence operator and divergence operator. With respect to the standard PSO algorithm, its convergence analysis provides a fundamental theory of selecting convergence operator and divergence operator. During the process of finding the suboptimal or global solution, the random switchings between two typical operators, namely Operator C and Operator D, which are two different ways to update velocities of all particles, are conducted by a so-called convergence ratio parameter, which can determine the tradeoff between exploration ability and exploitation ability from the quantitative perspective. Numerical results on several benchmark functions demonstrate the following observations: (1) The proper convergence ratio is closely related to the landscape of objective function, the dimension of solution space and the number of local optimums. (2) Small convergence ratio, setting to 0.60 or 0.65, may benefit the optimization problem which has many local optimums in the high dimensional space; while large convergence ratio, setting to 0.85 or 0.9, is probably helpful for the optimization problem with few local optimums or flat landscape. [Copyright &y& Elsevier]

Abstract: In this paper, we give the notion of M-η-proximal mapping for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space, which is an extension of proximal mappings studied in [X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383; K.R. Kazmi, M.I. Bhat, Convergence and stability of iterative algorithms of generalized set-valued variational-like inclusions in Banach spaces, Appl. Math. Comput. 113 (2005) 153–165; K.R. Kazmi, M.I. Bhat, N. Ahmad, An iterative algorithm based on M-proximal mappings for a system of generalized implicit variational inclusions in Banach spaces, J. Comput. Appl. Math. 233 (2009) 361–371]. We prove its existence and Lipschitz continuity in reflexive Banach space. Further, we consider a system of generalized implicit variational-like inclusions in Banach spaces and show its equivalence with a system of implicit equations using the concept of M-η-proximal mappings. Using this equivalence, we propose a new iterative algorithm for the system of generalized implicit variational-like inclusions. Furthermore, we prove the existence of solution of the system of generalized implicit variational-like inclusions and discuss the convergence and stability analysis of the iterative algorithm in the setting of uniformly smooth Banach spaces. [Copyright &y& Elsevier]

Abstract: This paper considers Nyström type methods for solving a generalized aerofoil equation with weakly singular kernel. Based on sigmoidal transformations and Nyström method, we obtain new algorithms for solving this equation. Convergence analysis for this methods are given. Finally, numerical examples complete this work. [Copyright &y& Elsevier]

Abstract: In this paper, a new numerical algorithm is provided to solve nonlinear three-point boundary value problems in a very favorable reproducing kernel space which satisfies all boundary conditions. Its reproducing kernel function is discussed in detail. We also prove that the approximate solution and its first and second order derivatives all converge uniformly. The numerical experiments show that the algorithm is quite accurate and efficient for solving nonlinear second order three-point boundary value problems. [Copyright &y& Elsevier]

Abstract: Much research on Artificial Intelligence (AI) has been focusing on exploring various potential applications of intelligent systems. In most cases, the researches attempt to model human intelligence by mimicking the brain structure and function, but they ignore an important aspect in human learning and decision making: the artificial emotion. In this paper, we present a new unconstrained global optimization method, hybrid chaos optimization algorithm with artificial emotion (HCOAAE), which avoids trapping to local minima, and improves convergence in large space and high-dimension optimization problems. The main purpose of artificial emotion is to mimic decision making behavior process of humans, to choose most suitable parameters of HCOAAE and decide whether to change current search strategy or not in the next iteration. Numerical simulations of 13 benchmark functions with different dimensions are used to test the performance of HCOAAE. Experimental results show that the proposed method significantly outperforms the existing methods in terms of convergence speed, computational effectiveness, and numerical stability. [Copyright &y& Elsevier]

Abstract: The purpose of this article is to modify the Halpern-type iteration algorithm for total quasi-ϕ-asymptotically nonexpansive mapping 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 results of [X.L. Qin, Y.J. Cho, S.M. Kang, H. Y. Zhou, Convergence of a modified Halpern-type iterative algorithm for quasi-ϕ-nonexpansive mappings, Appl. Math. Lett. 22 (2009) 1051–1055], [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364–2371], [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890–3906], [C. Martinez-Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400–2411] and others. [Copyright &y& Elsevier]

Abstract: Parabolic equations with nonlocal boundary conditions have been given considerable attention in recent years. In this paper new high-order algorithms for the linear diffusion–reaction problem are derived. The convergence of the new schemes is studied and numerical examples are given to show the efficiency of the new methods to solve linear and nonlinear diffusion–reaction equations with these non classical conditions. [Copyright &y& Elsevier]

Abstract: This paper presents extended artificial physics optimization (EAPO), a population-based, stochastic, evolutionary algorithm (EA) for multidimensional search and optimization. EAPO extends the physicomimetics-based Artificial Physics Optimization (APO) algorithm by including each individual’s best fitness history. Including the history improves EAPO’s search capability compared to APO. EAPO and APO invoke a gravitational metaphor in which the force of gravity may be attractive or repulsive, the aggregate effect of which is to move individuals toward local and global optima. A proof of convergence is presented that reveals the conditions under which EAPO is guaranteed to converge. Discrete-time linear system theory is used to develop a second-order difference equation for an individual’s stochastic position vector as a function of time step. Stable solutions require eigenvalues inside the unit circle, leading to explicit convergence criteria relating the run parameters {m i , w, G}. EAPO is tested against several benchmark functions with excellent results. The algorithm converges more quickly than APO and with better diversity. [Copyright &y& Elsevier]

Abstract: In this paper, we study a new system of generalized mixed equilibrium problems involving skew-symmetric bifunctions (SGMEP) in reflexive Banach spaces. A system of auxiliary mixed equilibrium problems (SAMEP) for solving the SGMEP is introduced and the existence and uniqueness of the solutions of the SAMEP is first proved. Next, by using the auxiliary principle technique, a new iterative algorithm to compute the approximate solutions of the SGMEP is suggested and analyzed. Finally, the strong convergence of the iterative sequences generated by the algorithm is also proved under quite mild conditions. These results improve, unify and generalize some known results in recent literature. [Copyright &y& Elsevier]

Abstract: In the present paper, we discuss three point difference method based on nonpolynomial spline basis for the second order ordinary differential equation. Difference schemes are derived for linear and nonlinear case and are used to solve via two parameter alternating group explicit iterative algorithm. The schemes have a fourth and second order of uniform convergence for the choice of the parameters involved in the method. Computational results are presented comparing the two methods in terms of accuracy and execution times. The results indicate the advantage of using parallel implementation of the new method. [Copyright &y& Elsevier]

Abstract: This paper describes a numerical approximation to the solution of Volterra integral equations of the second kind. This algorithm combines trapezoidal and Simpson rules. We prove the convergence of the method. Numerical examples are provided to illustrate the accuracy of the method. [Copyright &y& Elsevier]

Abstract: This paper proposes a novel hybrid immune algorithm (HIA) that can overcome the typical drawback of the artificial immune algorithm (AIA), which runs slowly and experiences slow convergence. The HIA combines the adaptive AIA based on the steepest descent algorithm. The HIA fully displays global search ability and the global convergence of the immune algorithm. At the same time, it inserts a quasi-descent operator to strengthen its local search ability. A good convergence of the HIA with the quasi-descent idea is shown as well. Numerical experiment results show that the HIA successfully improves running speed and convergence performance. [Copyright &y& Elsevier]

Abstract: This paper deals with fixed points methods related to the general class of demicontractive mappings (including the well-known classes of nonexpansive and quasi-nonexpansive mappings) in Hilbert spaces. Specifically, we point out some historical aspects concerning the concept of demicontactivity and we investigate a regularized variant of the Krasnoselski-Mann iteration that can be alternatively regarded as a simplified form of the inertial iteration (P-E. Maingé, J. Math. Anal. Appl. 344 (2008) 876–887) with non-constant relaxation factors. These two methods ensure the strong convergence of the generated sequence towards the least norm element of the set of fixed-points of demicontractive mappings. However, for convergence, our method does not require anymore the knowledge of some constant related to the involved demicontractive operator. A new and simpler proof is also proposed for its convergence even when involving non-constant relaxation factors. We point out the simplicity of this algorithm (at least from computational point of view) in comparison with other existing methods. We also present some numerical experiments concerning a convex feasibility problem, experiments that emphasize the characteristics of the considered algorithm comparing with a classical cyclic projection-type iteration. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a double projection algorithm for a generalized variational inequality with a multi-valued mapping. Under standard conditions, our method is proved to be globally convergent to a solution of the variational inequality problem. Moreover, we present a unified framework of projection-type methods for multi-valued variational inequalities. Preliminary computational experience is also reported. [Copyright &y& Elsevier]

Abstract: This paper is concerned with iterative solutions to a class of complex matrix equations, which include some previously investigated matrix equations as special cases. By applying the hierarchical identification principle, an iterative algorithm is constructed to solve this class of matrix equations. A sufficient condition is presented to guarantee that the proposed algorithm is convergent for an arbitrary initial matrix with a real representation of a complex matrix as tools. By using some properties of the real representation, a convergence condition that is easier to compute is also given in terms of original coefficient matrices. A numerical example is employed to illustrate the effectiveness of the proposed methods. [Copyright &y& Elsevier]

Abstract: In the paper, an incomplete active set algorithm is given for mathematical programs with linear complementarity constraints (MPLCC). At each iteration, a finite number of inner-iterations are contained for approximately solving the relaxed nonlinear optimization problem. If the feasible region of the MPLCC is bounded, under the uniform linear independence constraint qualification (LICQ), any cluster point of the sequence generated from the algorithm is a B-stationary point of the MPLCC. Preliminary numerical tests show that the algorithm is promising. [Copyright &y& Elsevier]

Abstract: In this paper, the issue of multi-degree reduction of Bézier curves with C 1 and G 2-continuity at the end points of the curve is considered. An iterative method, which is the first of this type, is derived. It is shown that this algorithm converges and can be applied iteratively to get the required accuracy. Some examples and figures are given to demonstrate the efficiency of this method. [Copyright &y& Elsevier]

Abstract: This paper proposes a line search filter reduced Hessian method for nonlinear equality constrained optimization. The feature of the presented algorithm is that the reduced Hessian method is used to produce a search direction, a backtracking line search procedure to generate step size, some filtered rules to determine step acceptance, second order correction technique to reduce infeasibility and overcome the Maratos effects. It is shown that this algorithm does not suffer from the Maratos effects by using second order correction step, and under mild assumptions fast convergence to second order sufficient local solutions is achieved. The numerical experiment is reported to show the effectiveness of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, differential transform method (DTM), which is one of the approximate methods is implemented for solving the nonlinear Hirota–Satsuma coupled KdV partial differential equation. A variety of initial value system is considered, and the convergence of the method as applied to the Hirota–Satsuma coupled KdV equation is illustrated numerically. The obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient. Numerical examples are illustrated the pertinent features of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: A nonparametric adaptive filtering approach is proposed in this paper. The algorithm is obtained by exploiting a time-varying step size in the traditional NLMS weight update equation. The step size is adjusted according to the square of a time-averaging estimate of the autocorrelation of a priori and a posteriori error. Therefore, the new algorithm has more effective sense proximity to the optimum solution independent of uncorrelated measurement noise. Moreover, this algorithm has fast convergence at the early stages of adaptation and small final misadjustment at steady-state process. It works reliably and is easy to implement since the update function is nonparametric. Furthermore, the experimental results in system identification applications are presented to illustrate the principle and efficiency of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, we first investigate a two-parametric class of smoothing functions which contains the penalized smoothing Fischer–Burmeister function and the penalized smoothing CHKS function as special cases. Then we present a smoothing Newton method for the nonlinear complementarity problem based on the class of smoothing functions. Issues such as line search rule, boundedness of the level set, global and quadratic convergence are studied. In particular, we give a line search rule containing the common used Armijo-type line search rule as a special case. Also without requiring strict complementarity assumption at the P0-NCP solution or the nonemptyness and boundedness of the solution set, the proposed algorithm is proved to be globally convergent. Preliminary numerical results show the efficiency of the algorithm and provide efficient domains of the two parameters for the complementarity problems. [Copyright &y& Elsevier]

Abstract: Many real life problems can be stated as a minimax optimization problem, such as the problems in economics, finance, management, engineering and other fields. In this paper, we present an algorithm with nonmonotone strategy and second-order correction technique for minimax optimization problems. Using this scheme, the new algorithm can overcome the difficulties of the Maratos effect occurred in the nonsmooth optimization, and the global and superlinear convergence of the algorithm can be achieved accordingly. Numerical experiments indicate some advantages of this scheme. [Copyright &y& Elsevier]

Abstract: The purpose of this article is to prove the strong convergence theorems for hemi-relatively nonexpansive mappings in Banach spaces. In order to get the strong convergence theorems for hemi-relatively nonexpansive mappings, a new monotone hybrid iteration algorithm is presented and is used to approximate the fixed point of hemi-relatively nonexpansive mappings. Noting that, the general hybrid iteration algorithm can be used for relatively nonexpansive mappings but it can not be used for hemi-relatively nonexpansive mappings. However, this new monotone hybrid algorithm can be used for hemi-relatively nonexpansive mappings. In addition, a new method of proof has been used in this article. That is, by using this new monotone hybrid algorithm, we firstly claim that, the iterative sequence is a Cauchy sequence. The results of this paper modify and improve the results of Matsushita and Takahashi, and some others. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to incorporate the preconditioned gradient path in a nonmonotone stabilization algorithm for unconstrained optimization. The global convergence and locally superlinear convergence are established for this class of algorithms. Finally, we report in details the numerical results which show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a nonmonotone trust region algorithm for unconstrained optimization problems is presented. In the algorithm, a kind of nonmonotone technique, which is evidently different from Grippo, Lampariello and Lucidi’s approach, is used. Under mild conditions, global and local convergence results of the algorithm are established. Preliminary numerical results show that the new algorithm is efficient. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a predictor–corrector smoothing Newton method for solving nonlinear symmetric cone complementarity problems (SCCP) based on the symmetrically perturbed smoothing function. Under a mild assumption, the solution set of the problem concerned is just nonempty, we show that the proposed algorithm is globally and locally quadratic convergent. Also, the algorithm finds a maximally complementary solution to the SCCP. Numerical results for second order cone complementarity problems (SOCCP), a special case of SCCP, show that the proposed algorithm is effective. [Copyright &y& Elsevier]

Abstract: In this paper, we will illustrate and visualize the chaotic behavior for solving the sign-changing solutions of concave–convex nonlinearities. We propose contractor iteration algorithm and bifurcation method to improve the computation speed and avoid local convergence of Newton–Raphson method. By this method, sign-changing numerical solutions will be computed and visualized in various domains. [Copyright &y& Elsevier]

Abstract: This paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation AXB + CX T D = F . The basic idea is to decompose the matrix equation (system) under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving Ax = b and AXB = F . The analysis shows that when the matrix equation has a unique solution (under the sense of least squares), the iterative solution converges to the exact solution for any initial values. A numerical example verifies the proposed theorems. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with iterative solutions to a class of complex matrix equations. By applying the hierarchical identification principle, an iterative algorithm is constructed to solve this class of complex matrix equations. The range of the convergence factor is given to guarantee that the proposed algorithm is convergent for arbitrary initial matrix by applying a real representation of a complex matrix as a tool. By using some properties of the real representation, a sufficient convergence condition that is easier to compute is also given by original coefficient matrices. Two numerical examples are given to illustrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an efficient branch and bound algorithm for globally solving sum of geometric fractional functions under geometric constraints, which arise in various practical problems. By using an equivalent transformation and a new linear relaxation technique, a linear relaxation programming problem of the equivalent problem is obtained. The proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems. Numerical results are reported to show the feasibility of our algorithm. [Copyright &y& Elsevier]

Abstract: The smoothing-type algorithm has been successfully applied to solve various optimization problems. In general, the smoothing-type algorithm is designed based on some monotone line search. However, in order to achieve better numerical results, the non-monotone line search technique has been used in the numerical computations of some smoothing-type algorithms. In this paper, we propose a smoothing-type algorithm for solving the nonlinear complementarity problem with a non-monotone line search. We show that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. The preliminary numerical results are also reported. [Copyright &y& Elsevier]

Abstract: In the paper, a new alternating-direction iterative method is proposed based on matrix splittings for solving saddle point problems. The convergence analysis for the new method is given. When the better values of parameters are employed, the proposed method has faster convergence rate and less time cost than the Uzawa algorithm with the optimal parameter and the Hermitian and skew-Hermitian splitting iterative method. Numerical examples further show the effectiveness of the method. [Copyright &y& Elsevier]

Abstract: This paper focuses on the computation issue of portfolio optimization with scenario-based CVaR. According to the semismoothness of the studied models, a smoothing technology is considered, and a smoothing SQP algorithm then is presented. The global convergence of the algorithm is established. Numerical examples arising from the allocation of generation assets in power markets are done. The computation efficiency between the proposed method and the linear programming (LP) method is compared. Numerical results show that the performance of the new approach is very good. The remarkable characteristic of the new method is threefold. First, the dimension of smoothing models for portfolio optimization with scenario-based CVaR is low and is independent of the number of samples. Second, the smoothing models retain the convexity of original portfolio optimization problems. Third, the complicated smoothing model that maximizes the profit under the CVaR constraint can be reduced to an ordinary optimization model equivalently. All of these show the advantage of the new method to improve the computation efficiency for solving portfolio optimization problems with CVaR measure. [Copyright &y& Elsevier]

Abstract: This paper addresses the convergence of two nonmonotone Levenberg–Marquardt algorithms for nonlinear complementarity problem. Under some mild assumptions, and requiring only the solution of a linear system at each iteration, the nonmonotone Levenberg–Marquardt algorithms are shown to be globally convergent. [Copyright &y& Elsevier]

Abstract: In last decades, there has been much effort on the solution and the analysis of the mixed complementarity problem (MCP) by reformulating MCP as an unconstrained minimization involving an MCP function. In this paper, we propose a new modified one-step smoothing Newton method for solving general (not necessarily ) mixed complementarity problems based on well-known Chen–Harker–Kanzow–Smale smooth function. Under suitable assumptions, global convergence and locally superlinear convergence of the algorithm are established. [Copyright &y& Elsevier]

Abstract: In this paper, a new SQP algorithm is presented to solve the general nonlinear programs with mixed equality and inequality constraints. Quoted from P. Spellucci (see ), this method maybe be named sequential equality constrained quadratic programming (SECQP) algorithm. Per single iteration, based on an active set strategy ( see ), this SECQP algorithm requires only to solve equality constrained quadratic programming subproblems or system of linear equations. The theoretical analysis shows that global and superlinear convergence can be induced under some suitable conditions. [Copyright &y& Elsevier]

Abstract: In his seminal paper published in 2002, Passino pointed out how individual and groups of bacteria forage for nutrients and how to model it as a distributed optimization process, which he called the Bacterial Foraging Optimization Algorithm (BFOA). One of the major driving forces of BFOA is the reproduction phenomenon of virtual bacteria each of which models a trial solution of the optimization problem. During reproduction, the least healthier bacteria (with a lower accumulated value of the objective function in one chemotactic lifetime) die and the other healthier bacteria each split into two, which then starts exploring the search place from the same location. This keeps the population size constant in BFOA. The phenomenon has a direct analogy with the selection mechanism of classical evolutionary algorithms. In this letter we provide a simple mathematical analysis of the effect of reproduction on bacterial dynamics. Our analysis reveals that the reproduction event contributes to the quick convergence of the bacterial population near optima. [Copyright &y& Elsevier]

Abstract: This paper investigates an enhanced proximal algorithm with interesting practical features and convergence properties for solving non-smooth convex minimization problems, or approximating zeroes of maximal monotone operators, in Hilbert spaces. The considered algorithm involves a recent inertial-type extrapolation technique, the use of enlargement of operators and also a recently proposed hybrid strategy, which combines inexact computation of the proximal iteration with a projection. Compared to other existing related methods, the resulting algorithm inherits the good convergence properties of the inertial-type extrapolation and the relaxed projection strategy. It also inherits the relative error tolerance of the hybrid proximal-projection method. As a special result, an update of inexact Newton-proximal method is derived and global convergence results are established. [Copyright &y& Elsevier]

Abstract: In this paper, a new descent algorithm for solving unconstrained optimization problem is presented. Its search direction is descent and line search procedure can be avoided except for the first iteration. It is globally convergent under mild conditions. The search direction of the new algorithm is generalized and convergence of corresponding algorithm is also proved. Numerical results show that the algorithm is efficient for given test problems. [Copyright &y& Elsevier]

Abstract: In this paper, we revisit the chaotic number of iterations needed by Newton’s method to converge to a root. Here, we consider a simple modified Newton method depending on a parameter. It is demonstrated using polynomiography that even in the simple algorithm the presence and the position of the convergent regions, i.e. regions where the method converges nicely to a root, can be complicatedly a function of the parameter. [Copyright &y& Elsevier]