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Abstract: In this paper, we propose a method to smooth exact penalty function for inequality constrained optimization. It is shown that an approximate global solution of the original problem can be obtained by searching a global solution of the smoothed penalty problem. Under some mild conditions, the method based on our smoothing function is shown to be globally convergent. Some numerical examples are given to illustrate the applicability of the present smoothing method. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the Polak–Ribière (or Polak–Ribière plus) conjugate gradient method for solving optimality condition of an unconstrained minimization problem. We give two new steplength rules only using gradient, and under gradient-Lipschitz assumption prove this method’s global convergence correspondingly. Then, we develop a practical Polak–Ribière plus method whose steplength is located by one inequality only using gradient, and report promising numerical results on high accuracy solution for some standard test problems when compared to the state-of-art methods in this research direction. Importantly, our work provides a new idea of devising a practical version of the celebrated Polak–Ribière (or Polak–Ribière plus) method. [Copyright &y& Elsevier]

Abstract: Nonlinear conjugate gradient (CG) methods have played an important role in solving large-scale unconstrained optimization. Their wide application in many fields is due to their low memory requirements and global convergence properties. Numerous studies and modifications have been conducted recently to improve this method. In this paper, a new class of conjugate gradient coefficients (βk ) that possess global convergence properties is presented. The global convergence result is established using exact line searches. Numerical result shows that the proposed formula is superior and more efficient when compared to other CG coefficients. [Copyright &y& Elsevier]

Abstract: A modified Artificial Bee Colony algorithm to solve constrained numerical optimization problems is presented in this paper. Four modifications related with the selection mechanism, the scout bee operator, and the equality and boundary constraints are made to the algorithm with the aim to modify its behavior in a constrained search space. Six performance measures found in the specialized literature are employed to analyze different capabilities in the proposed algorithm such as the ability and cost to generate feasible solutions, the capacity and cost to locate the feasible global optimum solution and the competency to improve feasible solutions. Three experiments, including a comparison with state-of-the-art algorithms, are considered in the test design where twenty four well-known benchmark problems with different features are utilized. The overall results show that the proposed algorithm differs in its behavior with respect to the original Artificial Bee Colony algorithm but its performance is improved, mostly in problems with small feasible regions due to the presence of equality constraints. [Copyright &y& Elsevier]

Abstract: In this paper, we present a global convergence theory for a class of nonmonotone filter trust region methods. At each iteration, the trial step is decomposed into a quasi-normal step and a tangential step. Comparable to the traditional filter and monotone methods, the new approach is more flexible and less computational scale. Under some reasonable conditions, we show that there exists at least one accumulate point of the sequence of iterates that is a KKT point. [Copyright &y& Elsevier]

Abstract: We establish in this paper a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the quadratic Hermitian matrix function with respect to the variable matrix by using a linearization method and some known formulas for extremum ranks and inertias of linear Hermitian matrix functions, where both and are complex Hermitian matrices and is the conjugate transpose of . We then derive the global maximum and minimum ranks and inertias of the two quadratic Hermitian matrix functions and subject to a consistent matrix equation , respectively, by using some pure algebraic operations of matrices and their generalized inverses. As consequences, we establish necessary and sufficient conditions for the solutions of the matrix equation to satisfy the quadratic Hermitian matrix equalities and , respectively, and for the quadratic matrix inequalities and in the Löwner partial ordering to hold, respectively. In addition, we give complete solutions to four Löwner partial ordering optimization problems on the matrix functions and subject to . Examples are also presented to illustrative applications of the equality-constrained quadratic optimizations in some matrix completion problems. [Copyright &y& Elsevier]

Abstract: In this paper, we give new results on the Alienor method of dimension reduction. This technique is used to solve multidimensional global optimization problems of type min x∈X f(x) where f is a non convex Lipschitz function and X a compact set of defined by Lipschitz constraints. The idea is to construct an α-dense curve h in the feasible set X. The global minimum of f on X is then approximated by the global minimum of f on the curve h. That is, our problem has become a one-dimensional problem which can be solved by the Piyavskii–Shubert method. Examples of these curves and numerical implementations on several test functions are given. [Copyright &y& Elsevier]

Abstract: The filled function method is considered as an efficient approach to solve the global optimization problems. In this paper, a new filled function method is proposed. Its main idea is as follows: a new continuously differentiable filled function with only one parameter is constructed for unconstrained global optimization when a minimizer of the objective function is found, then a minimizer of the filled function will be found in a lower basin of the objective function, thereafter, a better minimizer of the objective function will be found. The above process is repeated until the global optimal solution is found. The numerical experiments show the efficiency of the proposed filled function method. [Copyright &y& Elsevier]

Abstract: Global optimization problem is known to be challenging, for which it is difficult to have an algorithm that performs uniformly efficient for all problems. Stochastic optimization algorithms are suitable for these problems, which are inspired by natural phenomena, such as metal annealing, social behavior of animals, etc. In this paper, subset simulation, which is originally a reliability analysis method, is modified to solve unconstrained global optimization problems by introducing artificial probabilistic assumptions on design variables. The basic idea is to deal with the global optimization problems in the context of reliability analysis. By randomizing the design variables, the objective function maps the multi-dimensional design variable space into a one-dimensional random variable. Although the objective function itself may have many local optima, its cumulative distribution function has only one maximum at its tail, as it is a monotonic, non-decreasing, right-continuous function. It turns out that the searching process of optimal solution(s) of a global optimization problem is equivalent to exploring the process of the tail distribution in a reliability problem. The proposed algorithm is illustrated by two groups of benchmark test problems. The first group is carried out for parametric study and the second group focuses on the statistical performance. [Copyright &y& Elsevier]

Abstract: The filled function method is an efficient approach for finding global minimizers of multi-dimensional and nonlinear functions in the absence of any restrictions. In this paper, we give a new definition of filled function and the idea of constructing a new filled function, and then a new class of filled functions with one parameter on the basis of the new definition, which possesses better quality, is presented. Theoretical properties of the new class of filled functions are investigated. A new algorithm is developed from the new filled function method. The implementation of the algorithm on seven test problems with dimensions up to 30 is reported, and comparisons with other filled function methods demonstrate that the new algorithm is more efficient. [Copyright &y& Elsevier]

Abstract: In this paper, a new spectral PRP conjugate gradient algorithm has been developed for solving unconstrained optimization problems, where the search direction was a kind of combination of the gradient and the obtained direction, and the steplength was obtained by the Wolfe-type inexact line search. It was proved that the search direction at each iteration is a descent direction of objective function. Under mild conditions, we have established the global convergence theorem of the proposed method. Numerical results showed that the algorithm is promising, particularly, compared with the existing several main methods. [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: 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 new local optimization method for mixed integer quadratic programming problems with box constraints is presented by using its necessary global optimality conditions. Then a new global optimization method by combining its sufficient global optimality conditions and an auxiliary function is proposed. Some numerical examples are also presented to show that the proposed optimization methods for mixed integer quadratic programming problems with box constraints are very efficient and stable. [Copyright &y& Elsevier]

Abstract: The global optimal control problem is proposed for a special class of hybrid dynamical systems, i.e. impulsive switching systems. Then the necessary condition of the above problem, the minimum principle, is given. Ekeland’s variational principle and the matrix cost functional structure expression are utilized in the process of the proof. Based on the main result, a special linear hybrid impulsive and switching system (HISS) is illustrated and the optimal control algorithm is presented. Moreover, the cases of pure impulsive systems and pure switched systems are included in this paper. [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: The filled function method is considered as an efficient method to find the global minimum of multidimensional functions. A number of filled functions were proposed recently, most of which have one or two adjustable parameters. However, there is no efficient criterion to choose the parameter appropriately. In this paper, we propose a filled function without parameter. And this function includes neither exponential terms nor logarithmic terms so it is superior to the traditional ones. Theories of the filled function are investigated. And an algorithm which does not compute gradients during minimizing the filled function is presented. Moreover, the numerical experiments demonstrate the efficiency of the proposed filled function. [Copyright &y& Elsevier]

Abstract: One class of the lately developed methods for solving optimization problems are filter methods. In this paper we attached a multidimensional filter to the Gauss–Newton-based BFGS method given by Li and Fukushima [D. Li, M. Fukushima, A globally and superlinearly convergent Gauss–Newton-based BFGS method for symmetric nonlinear equations, SIAM Journal of Numerical Analysis 37(1) (1999) 152–172] in order to reduce the number of backtracking steps. The proposed filter method for unconstrained minimization problems converges globally under the standard assumptions. It can also be successfully used in solving systems of symmetric nonlinear equations. Numerical results show reasonably good performance of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: Differential evolution is a novel evolutionary approach capable of handling non-differentiable, nonlinear and multimodal objective functions. It has been consistently ranked as one of the best search algorithm for solving global optimization problems in several case studies. In the present study we propose five new mutation schemes for the basic DE algorithm. The corresponding versions are termed as MDE1, MDE2, MDE3, MDE4 and MDE5. These new schemes make use of the absolute weighted difference between the two points and instead of using a fixed scaling factor F, use a scaling factor following the Laplace distribution. The performance of the proposed schemes is validated empirically on a suit of ten benchmark problems having box constraints. Numerical analysis of results shows that the proposed schemes improves the convergence rate of the DE algorithm and also maintains the quality of solution. Efficiency of the proposed schemes is further validated by applying it to a real life electrical engineering problem dealing with the optimization of directional over-current relay settings. It is a highly constrained nonlinear optimization problem. A constraint handling mechanism based on repair methods is used for handling the constraints. Once again the simulation results show the compatibility of the proposed schemes for solving the real life problem. [Copyright &y& Elsevier]