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Abstract: A counterexample is provided in this short note to show that some of local convergence theorems established in [D.K.R. Babajee, M.Z. Dauhoo, An analysis of the properties of the variants of Newton’s method with third order convergence, Appl. Math. Comput. 183 (2006) 659–684] are not always true. Some mistakes in the proofs of these theorems are pointed out. [Copyright &y& Elsevier]

Abstract: The particle swarm optimization (PSO) computational method has recently become popular. However, it has limitations. It may trap into local optima and cause the premature convergence phenomenon, especially for multimodal and high-dimensional problems. In this paper, we focus on investigating the fitness evaluation in terms of a particle’s position. Particularly, we find that the fitness evaluation strategy in the standard PSO has two drawbacks, i.e., “two steps forward and one step back” and “two steps back and one step forward”. In addition, we propose a general fitness evaluation strategy (GFES), by which a particle is evaluated in multiple subspaces and different contexts in order to take diverse paces towards the destination position. As demonstrations of GFES, a series of PSOs with GFES are presented. Experiments are conducted on several benchmark optimization problems. The results show that GFES is effective at handling multimodal and high-dimensional problems. [Copyright &y& Elsevier]

Abstract: In this paper, a self-adaptive projection method with a new search direction for solving pseudomonotone variational inequality (VI) problems is proposed, which can be viewed as an extension of the methods in [B.S. He, X.M. Yuan, J.Z. Zhang, Comparison of two kinds of prediction-correction methods for monotone variational inequalities, Computational Optimization and Applications 27 (2004) 247–267] and [X.H. Yan, D.R. Han, W.Y. Sun, A self-adaptive projection method with improved step-size for solving variational inequalities, Computers & Mathematics with Applications 55 (2008) 819–832]. The descent property of the new search direction is proved, which is useful to guarantee the convergence. Under the relatively relaxed condition that F is continuous and pseudomonotone, the global convergence of the proposed method is proved. Numerical experiments are provided to illustrate the efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper, a kind of optimization problems with nonlinear inequality constraints is discussed. Combined the ideas of norm-relaxed SQP method and strongly sub-feasible direction method as well as a pivoting operation, a new fast algorithm with arbitrary initial point for the discussed problem is presented. At each iteration of the algorithm, an improved direction is obtained by solving only one direction finding subproblem which possesses small scale and always has an optimal solution, and to avoid the Maratos effect, another correction direction is yielded by a simple explicit formula. Since the line search technique can automatically combine the initialization and optimization processes, after finite iterations, the iteration points always get into the feasible set. The proposed algorithm is proved to be globally convergent and superlinearly convergent under mild conditions without the strict complementarity. Finally, some numerical tests are reported. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a vector-valued Tikhonov-type regularization algorithm for an extended-valued multiobjective optimization problem. Under some mild conditions, we prove that any sequence generated by this algorithm converges to a weak Pareto optimal solution of the multiobjective optimization problem. Our results improve and generalize some known results. [Copyright &y& Elsevier]

Abstract: This paper presents an affine scaling optimal path approach in association with nonmonotonic interior backtracking line search technique for nonlinear optimization subject to linear constraints. We shall employ the eigensystem decomposition and affine scaling mapping to form affine scaling optimal curvilinear path very easily. By using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The nonmonotone criterion is used to speed up the convergence progress in the contours of objective function with large curvature. Theoretical analysis is given which prove that the proposed algorithm is globally convergent and has a local superlinear convergence rate under some reasonable conditions. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: The least -norm solution for a possibly inconsistent system of nonlinear inequalities is studied in this paper. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a smoothing Levenberg–Marquardt method is applied to solve the parameterized optimization problems. The global convergence of the proposed method is established under suitable assumptions. [Copyright &y& Elsevier]

Abstract: This paper studies a successive partitioning group correction algorithm and its some modified algorithms for solving large scale sparse unconstrained optimization problems. The methods depend on a symmetric consistent partition of the columns of the Hessian matrix. A q-superlinear convergence result and an r-convergence rate estimate show that the methods have good local convergence properties. The numerical results show that the methods, especially the modified algorithms, may be competitive with some current used algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, a class of optimization problems with nonlinear equality and inequality constraints is discussed. Firstly, the original problem is transformed to an associated simpler auxiliary optimization problem with only inequality constraints and a penalty parameter, and the later problem is showed to be equivalent to the original problem if the parameter is large enough (but finite). Then, combining the norm-relaxed Method of Feasible Direction (MFD) with the idea of Method of Strongly Sub-Feasible Direction (MSSFD), we present an algorithm with arbitrary initial point for the original problem. At each iteration of the auxiliary problem, an improved search direction is obtained by solving one Direction Finding Subproblem (DFS), i.e., a quadratic program, which always possesses a solution. In the process of iteration, the feasibility of the iteration points is monotone increasing. Furthermore, whenever an iteration point enters the feasible set, the proposed algorithm reduces to a feasible and decent method for the auxiliary problem. Under some suitable assumptions, the global and strong convergence of the proposed algorithm can be obtained. Finally, some elementary numerical experiments are reported. [Copyright &y& Elsevier]

Abstract: For solving nonlinear, univariate and unconstrained optimization problems, Newton method is an important and basic method which convergences quadratically. In this paper, we suggest a family of three new modifications of the classical Secant method where the iteration formula including an approximation of f″(x k ) is satisfied by a recursive scheme. The efficiencies of the new methods are analyzed in terms of the most popular and widely used criterion; the number of iterations, in comparison with the Newton and Secant methods using six test functions. [Copyright &y& Elsevier]

Abstract: This paper concerns the step-size estimation in line search method for unconstrained optimization problems. Some new approaches for estimating the step size are proposed and the global convergence of related line search methods are analyzed. Finally, we discuss the choice of the initial step size in Armijo line search rule. [Copyright &y& Elsevier]

Line search methods are traditional and successful methods for solving unconstrained optimization problems. Its convergence has attracted more attention in recent years. In this paper we analyze the general results on convergence of line search methods with seven line search rules. It is clarified that the search direction plays a main role in these methods and that step-size guarantees the global convergence in some cases. It is also proved that many line search methods have same convergence property. These convergence results can enable us to design powerful, effective, and stable algorithms in practice. Finally, a class of special line search methods is investigated. [Copyright &y& Elsevier]