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Abstract: This paper proposes a nonlinear Lagrangian Based on Fischer-Burmeister NCP function for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. Moreover, the paper develops the dual approach associated with the proposed nonlinear Lagrangian, in which the related duality theorem is demonstrated. Furthermore, it is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Numerical results for solving several nonlinear programming problems are reported, showing that the new nonlinear Lagrangian is superior over other known nonlinear Lagrangians for solving some nonlinear programming problems. [Copyright &y& Elsevier]

A trust-region algorithm in association with line search filter technique for solving nonlinear equality constrained programming is proposed in this paper. In the current iteration, the trial step providing sufficient descent is generated by solving a corresponding trust-region subproblem. Then, the step size is decided by backtracking line search together with filter technique to obtain the next iteration point. The advantage of this method is that resolving trust-region subproblem many times to determine a new iteration point in traditional trust-region method can be avoided and hence the expensive computation can be lessened. And the difficult decisions in regard to the choice of penalty parameters in the merit functions can be avoided by using filter technique. Second order correction steps are introduced in the proposed algorithm to overcome Maratos effect. Convergence analysis shows that fast local convergence can be achieved under some mild assumptions. The preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

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 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, 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 this paper, a quadratically approximate algorithm framework for solving general constrained minimax problems is presented. The framework contains the idea of the sequential quadratic programming method, the sequential quadratically constrained quadratic programming method, norm-relaxed method and strong sub-feasible method. The global convergence of the algorithm framework is obtained under the Mangasarian–Fromovitz constraint qualification (MFCQ), and the conditions for superlinear convergence of the algorithm framework are presented under the MFCQ, the constant rank constraint qualification (CRCQ) as well as the strong second-order sufficiency conditions (SSOSC). And quadratic convergence rate is obtained under the MFCQ and SSOSC. [Copyright &y& Elsevier]

Abstract: In this paper, based on combined homotopy interior point method we propose an interior point algorithm for convex nonlinear programming. The algorithm ensures that the obtained iterative points are interior points of the feasible set in terms of the technique of β-cone neighborhood. We establish the global convergence of the algorithm. Furthermore, it is shown that the algorithm has iteration complexity. The preliminary numerical experiments indicate that the algorithm is efficient. [Copyright &y& Elsevier]

Abstract: Method of feasible directions (MFD) is an important method for solving nonlinearly constrained optimization. However, various types of MFD all need an initial feasible point, which can not be found easily in generally. In addition, the computational cost of some MFD with superlinearly convergent property is rather high. On the other hand, the strongly sub-feasible direction method does not need an initial feasible point, but most of the proposed algorithm do not have the superlinearly convergent property, and can not guarantee that the iteration point is feasible after finite iterations. In this paper, we present a new superlinearly convergent algorithm with arbitrary initial point. At each iteration, a master direction is obtained by solving one direction finding subproblem (DFS), and an auxiliary direction is yielded by an explicit formula. After finite iterations, the iteration point goes into the feasible set and the master direction is a feasible direction of descent. Since a new generalized projection technique is contained in the auxiliary direction formula, under some mild assumptions without the strict complementarity, the global convergence and superlinear convergence of the algorithm can be obtained. [Copyright &y& Elsevier]

Abstract: The paper presents a new rectangle branch-and-reduce approach for solving nonconvex quadratic programming problems, in which a new lower approximate linear functions of the quadratic function over an n-rectangle is given to determine a lower bound of the global optimal value of the original problem over each rectangle, and a simple two-partition technique on rectangle is used, as well as the tactics on reducing and deleting subrectangles is used to accelerate the convergence of the proposed algorithm. The proposed algorithm is proved to be convergent and shown to be effective with numerical results. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a QP-free type algorithm which solves the problem of minimizing a smooth function subject to smooth inequality constraints. In contrast with the SQP methods, each iteration this algorithm only needs to solve systems of linear equations which are derived from the equality part in the KKT first order optimality conditions. It is observed that, if the quasi-Newton direction is zero, we can obtain a direction of descent by dropping a constraint from the active set at the current iterate. A high order modified direction is introduced in order to prevent Maratos effect. Global and superlinear convergence are proven under some suitable conditions. [Copyright &y& Elsevier]

In this paper, a new method is presented to deal with nonconvex nonlinear inequality constrained problem by using the idea of #x03B5;-effective set. It is only required to solve two systems of linear equations per single iteration. The theoretical analysis shows that the algorithm is global convergence under some suitable conditions. [Copyright &y& Elsevier]