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Abstract: In this paper, we first weaken some assumptions in Jiao et al. (2005) [9] and discuss the continuity and the directional differentiability of the objective function. Furthermore, we investigate the optimality conditions for unconstrained and constrained generalized minimax programming. These optimality conditions are mixed ones that include different conditions at different directions, which is suitable for the non-differentiable objective function. For constrained problem we exploit two kinds of analysis methods to obtain some different optimality conditions. Moreover, we show these optimality conditions by some examples. [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: Many practical optimization problems involve nonsmooth (that is, not necessarily differentiable) functions of hundreds or thousands of variables with various constraints. In this paper, we describe a new efficient adaptive limited memory interior point bundle method for large, possible nonconvex, nonsmooth inequality constrained optimization. The method is a hybrid of the nonsmooth variable metric bundle method and the smooth limited memory variable metric method, and the constraint handling is based on the primal–dual feasible direction interior point approach. The preliminary numerical experiments to be presented confirm the effectiveness of the method. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to propose and compare different one-dimensional maps as chaotic search patterns in the constraint nonlinear optimization problems. For this purpose, about 10 one-dimensional maps are introduced that can be used as search pattern in chaos optimization algorithms. We apply these maps in specific optimization algorithm (weighted gradient direction based chaos optimization algorithm) and compare them based on numerical simulation results. [Copyright &y& Elsevier]

Abstract: Many practical problems can be formulated as constrained optimization problems. Due to the simple concept and easy implementation, the penalty function method has been one of the most common techniques to handle constraints. However, the performance of this technique greatly relies on the setting of penalty factors, which are usually determined by manual trial and error, and the suitable penalty factors are often problem-dependent and difficult to set. In this paper, a differential evolution approach based on a co-evolution mechanism, named CDE, is proposed to solve the constrained problems. First, a special penalty function is designed to handle the constraints. Second, a co-evolution model is presented and differential evolution (DE) is employed to perform evolutionary search in spaces of both solutions and penalty factors. Thus, the solutions and penalty factors evolve interactively and self-adaptively, and both the satisfactory solutions and suitable penalty factors can be obtained simultaneously. Simulation results based on several benchmark functions and three well-known constrained design problems as well as comparisons with some existed methods demonstrate the effectiveness, efficiency and robustness of the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper a class of augmented Lagrangians is considered, for solving equality constrained nonlinear optimization problems via unconstrained minimization techniques. This class of augmented Lagrangians is obtained by multiplying the penalty term on the first order necessary optimality condition in a class of augmented Lagrangians of Di Pillo and Grippo by a penalty parameter. Under suitable assumptions, the exactly corresponding relationship is established between the solution of the original constrained problem and the unconstrained minimization of this class of augmented Lagrangians on the product space of problem variables and multipliers for sufficiently large but finite values of penalty parameters. Therefore, a solution of the original constrained problem and the corresponding values of the Lagrange multipliers can be found by performing a single unconstrained minimization of an augmented Lagrangian on the product space of problem variables and multipliers. In particular, for quadratic programming problems with equality constraints, the optimizer is obtained by minimizing a quadratic function on the expanded space. [Copyright &y& Elsevier]