Abstract In this paper,a quasidifferentiable programming problem with inequality constraints is considered.First,a general form of optimality conditions for this problem is given,which contains the results of Luderer,...Abstract In this paper,a quasidifferentiable programming problem with inequality constraints is considered.First,a general form of optimality conditions for this problem is given,which contains the results of Luderer,Kuntz and Scholtes.Next,a new generalized K T condition is derived.The new optimality condition doesnt use Luderers regularity assumption and its Lagrangian multipliers dont depend on the particular elements in the superdifferentials of the object function and constraint functions.Finally,a penalty function for the problem is studied.Sufficient conditions of the penalty function attaining a global minimum are obtained.展开更多
Yang et al gave some criteria of prequasi-invex functions, semistrictly prequasi-invex functions and strictly prequasi-invex functions in 2001, under a certain set of conditions. In this note, some of these conditions...Yang et al gave some criteria of prequasi-invex functions, semistrictly prequasi-invex functions and strictly prequasi-invex functions in 2001, under a certain set of conditions. In this note, some of these conditions can be weakened to get the same results, and another simplified proof for a criterion of prequasi-invex functions established under the condition of lower semicontinuity is given. MR Subject Classification 90C30 Keywords prequasi-invex function - semistrictly prequasi-invex function - strictly prequasi-invex function - criteria Supported by the Zhejiang Province Natural Science Foundation (602095).展开更多
A interior point scaling projected reduced Hessian method with combination of nonmonotonic backtracking technique and trust region strategy for nonlinear equality constrained optimization with nonegative constraint on...A interior point scaling projected reduced Hessian method with combination of nonmonotonic backtracking technique and trust region strategy for nonlinear equality constrained optimization with nonegative constraint on variables is proposed. In order to deal with large problems, a pair of trust region subproblems in horizontal and vertical subspaces is used to replace the general full trust region subproblem. The horizontal trust region subproblem in the algorithm is only a general trust region subproblem while the vertical trust region subproblem is defined by a parameter size of the vertical direction subject only to an ellipsoidal constraint. Both trust region strategy and line search technique at each iteration switch to obtaining a backtracking step generated by the two trust region subproblems. By adopting the l 1 penalty function as the merit function, the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion and the second order correction step are used to overcome Maratos effect and speed up the convergence progress in some ill-conditioned cases. MR Subject Classification 90C30 - 65K05 - 49M40 Keywords trust region method - backtracking step - reduced Hessian - nonmonotonic technique - interior point Supported partially by the National Natural Science Foundation of China (10071050), Science Foundation (02ZA14070) of Shanghai Technical Sciences Committee and Science Foundation (02DK06) of Shanghai Education Committee.展开更多
In this paper we present a filter-successive linearization method with trust region for solutions of nonlinear semidefinite programming. Such a method is based on the concept of filter for nonlinear programming introd...In this paper we present a filter-successive linearization method with trust region for solutions of nonlinear semidefinite programming. Such a method is based on the concept of filter for nonlinear programming introduced by Fletcher and Leyffer in 2002. We describe the new algorithm and prove its global convergence under weaker assumptions. Some numerical results are reported and show that the new method is potentially efficient.展开更多
In general normed spaces, we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior. We establish that the weak Pareto optimal solution set o...In general normed spaces, we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior. We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function. Moreover, we provide necessary and sufficient conditions about the existence of weak (sharp) Pareto solutions.展开更多
The new trust region subproblem with the conic model was proposed in 2005, and was divided into three different cases. The first two cases can be converted into a quadratic model or a convex problem with quadratic con...The new trust region subproblem with the conic model was proposed in 2005, and was divided into three different cases. The first two cases can be converted into a quadratic model or a convex problem with quadratic constraints, while the third one is a nonconvex problem. In this paper, first we analyze the nonconvex problem, and reduce it to two convex problems. Then we discuss some dual properties of these problems and give an algorithm for solving them. At last, we present an algorithm for solving the new trust region subproblem with the conic model and report some numerical examples to illustrate the efficiency of the algorithm.展开更多
文摘Abstract In this paper,a quasidifferentiable programming problem with inequality constraints is considered.First,a general form of optimality conditions for this problem is given,which contains the results of Luderer,Kuntz and Scholtes.Next,a new generalized K T condition is derived.The new optimality condition doesnt use Luderers regularity assumption and its Lagrangian multipliers dont depend on the particular elements in the superdifferentials of the object function and constraint functions.Finally,a penalty function for the problem is studied.Sufficient conditions of the penalty function attaining a global minimum are obtained.
基金Supported by the Zhejiang Province Natural Science Foundation( 60 2 0 95 )
文摘Yang et al gave some criteria of prequasi-invex functions, semistrictly prequasi-invex functions and strictly prequasi-invex functions in 2001, under a certain set of conditions. In this note, some of these conditions can be weakened to get the same results, and another simplified proof for a criterion of prequasi-invex functions established under the condition of lower semicontinuity is given. MR Subject Classification 90C30 Keywords prequasi-invex function - semistrictly prequasi-invex function - strictly prequasi-invex function - criteria Supported by the Zhejiang Province Natural Science Foundation (602095).
基金Supported partially by the National Natural Science Foundation of China( 1 0 0 71 0 5 0 ) ScienceFoundation ( 0 2 ZA1 4 0 70 ) of Shanghai Technical Sciences Committee and Science Foundation ( 0 2 DK0 6) ofShanghai Education Committee.
文摘A interior point scaling projected reduced Hessian method with combination of nonmonotonic backtracking technique and trust region strategy for nonlinear equality constrained optimization with nonegative constraint on variables is proposed. In order to deal with large problems, a pair of trust region subproblems in horizontal and vertical subspaces is used to replace the general full trust region subproblem. The horizontal trust region subproblem in the algorithm is only a general trust region subproblem while the vertical trust region subproblem is defined by a parameter size of the vertical direction subject only to an ellipsoidal constraint. Both trust region strategy and line search technique at each iteration switch to obtaining a backtracking step generated by the two trust region subproblems. By adopting the l 1 penalty function as the merit function, the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion and the second order correction step are used to overcome Maratos effect and speed up the convergence progress in some ill-conditioned cases. MR Subject Classification 90C30 - 65K05 - 49M40 Keywords trust region method - backtracking step - reduced Hessian - nonmonotonic technique - interior point Supported partially by the National Natural Science Foundation of China (10071050), Science Foundation (02ZA14070) of Shanghai Technical Sciences Committee and Science Foundation (02DK06) of Shanghai Education Committee.
基金supported by National Natural Science Foundation of China (Grant No. 10871098)Science Foundation of Jiangsu Province (Grant No. BK2006214)
文摘In this paper we present a filter-successive linearization method with trust region for solutions of nonlinear semidefinite programming. Such a method is based on the concept of filter for nonlinear programming introduced by Fletcher and Leyffer in 2002. We describe the new algorithm and prove its global convergence under weaker assumptions. Some numerical results are reported and show that the new method is potentially efficient.
基金supported by the National Natural Science Foundation of China (Grant No. 10761012)theNatural Science Foundation of Yunnan Province,China (Grant No. 2003A002M) the Research GrantsCouncil of Hong Kong (Grant No. B-Q771)
文摘In general normed spaces, we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior. We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function. Moreover, we provide necessary and sufficient conditions about the existence of weak (sharp) Pareto solutions.
基金the National Natural Science Foundation of China (Grant No.10471062)the Natural Science Foundation of Jiangsu Province (Grant No. BK2006184)
文摘The new trust region subproblem with the conic model was proposed in 2005, and was divided into three different cases. The first two cases can be converted into a quadratic model or a convex problem with quadratic constraints, while the third one is a nonconvex problem. In this paper, first we analyze the nonconvex problem, and reduce it to two convex problems. Then we discuss some dual properties of these problems and give an algorithm for solving them. At last, we present an algorithm for solving the new trust region subproblem with the conic model and report some numerical examples to illustrate the efficiency of the algorithm.
