In classical nonlinear programming, it is a general method of developing optimality conditions that a nonlinear programming problem is linearized as a linear programming problem by using first order approximations of ...In classical nonlinear programming, it is a general method of developing optimality conditions that a nonlinear programming problem is linearized as a linear programming problem by using first order approximations of the functions at a given feasible point. The linearized procedure for differentiable nonlinear programming problems can be naturally generalized to the quasi differential case. As in classical case so called constraint qualifications have to be imposed on the constraint functions to guarantee that for a given local minimizer of the original problem the nullvector is an optimal solution of the corresponding 'quasilinearized' problem. In this paper, constraint qualifications for inequality constrained quasi differentiable programming problems of type min {f(x)|g(x)≤0} are considered, where f and g are qusidifferentiable functions in the sense of Demyanov. Various constraint qualifications for this problem are presented and a new one is proposed. The relations among these conditions are investigated. Moreover, a Wolf dual problem for this problem is introduced, and the corresponding dual theorems are given.展开更多
The main purpose of this article is to establish an effective version of the Grunwald-Wang theorem, which asserts that given a family of local characters Xv of Kv of exponent m, where v C S for a finite set S of prime...The main purpose of this article is to establish an effective version of the Grunwald-Wang theorem, which asserts that given a family of local characters Xv of Kv of exponent m, where v C S for a finite set S of primes of K, there exists a global character X of the idele class group CK of exponent m （unless some special case occurs, when it is 2m） whose local component at v is Xv. The effectiveness problem for this theorem is to bound the norm N（X） of the conductor of X in terms of K, m, S and N（Xv） （v C S）. The Kummer case （when K contains pro） is easy since it is almost an application of the Chinese remainder theorem. In this paper, we solve this problem completely in general case, and give three versions of bound, one is with GRH, and the other two are unconditional bounds. These effective results have some interesting applications in concrete situations. To give a simple example, if we fix p and l, one gets a good least upper bound for N such that p is not an /-th power rood N. One also gets the least upper bound for N such that lr ｜φ｜（N） and p is not an/-th power mod N. Some part of this article is adopted （with some revision） from the unpublished thesis by Wang （2001）.展开更多
文摘In classical nonlinear programming, it is a general method of developing optimality conditions that a nonlinear programming problem is linearized as a linear programming problem by using first order approximations of the functions at a given feasible point. The linearized procedure for differentiable nonlinear programming problems can be naturally generalized to the quasi differential case. As in classical case so called constraint qualifications have to be imposed on the constraint functions to guarantee that for a given local minimizer of the original problem the nullvector is an optimal solution of the corresponding 'quasilinearized' problem. In this paper, constraint qualifications for inequality constrained quasi differentiable programming problems of type min {f(x)|g(x)≤0} are considered, where f and g are qusidifferentiable functions in the sense of Demyanov. Various constraint qualifications for this problem are presented and a new one is proposed. The relations among these conditions are investigated. Moreover, a Wolf dual problem for this problem is introduced, and the corresponding dual theorems are given.
基金supported by National Basic Research Program of China(973 Program)(Grant No.2013CB834202)National Natural Science Foundation of China(Grant No.11321101)the One Hundred Talent’s Program from Chinese Academy of Sciences
文摘The main purpose of this article is to establish an effective version of the Grunwald-Wang theorem, which asserts that given a family of local characters Xv of Kv of exponent m, where v C S for a finite set S of primes of K, there exists a global character X of the idele class group CK of exponent m （unless some special case occurs, when it is 2m） whose local component at v is Xv. The effectiveness problem for this theorem is to bound the norm N（X） of the conductor of X in terms of K, m, S and N（Xv） （v C S）. The Kummer case （when K contains pro） is easy since it is almost an application of the Chinese remainder theorem. In this paper, we solve this problem completely in general case, and give three versions of bound, one is with GRH, and the other two are unconditional bounds. These effective results have some interesting applications in concrete situations. To give a simple example, if we fix p and l, one gets a good least upper bound for N such that p is not an /-th power rood N. One also gets the least upper bound for N such that lr ｜φ｜（N） and p is not an/-th power mod N. Some part of this article is adopted （with some revision） from the unpublished thesis by Wang （2001）.
