| 1. | Duality theorems of multiobjective programming for - arcwise convex function
弧式凸函数多目标规划的对偶定理 |
| 2. | Optimality and duality for fuzzy weak efficient solutions in nonsmooth multiobjective programming
弱有效解的最优性条件及其对偶定理 |
| 3. | In this method , we get the optimal value of semidefinite programming by finding the optimal descent direction of its dual problem
该方法根据次微分理论与半定规划的强对偶定理，通过求解对偶问题得到原始问题的最优值。 |
| 4. | Zhu had deduced generalized fenchel ' s duality theorem in [ 8 ] , and further applied it to minimum discrimination information ( mdi ) problem
朱德通在文[ 8 ]中导出广义的fenchel对偶定理，并将此定理成功地应用于带约束的最小区别信息量问题(简称mdi问题) 。 |
| 5. | With the help of the express theorem of weakly major efficient solution , the duality relation between weakly major efficient solution and sub - weakly major efficient solution is established
借助弱较多有效解的表示定理，讨论了弱较多有效解和次弱较多有效解之间的对偶关系，建立了相应的对偶定理。 |
| 6. | The author discusses the duality theorem of infinite dimensional hopf algebras in braided tensor categories in the first chapter , and shows the theorem by the use of braiding diagram
本文第一章讨论了辫子张量范畴中无限维hopf代数的对偶定理，应用辫子图对此定理给出证明，得到如下结果：命题2 |
| 7. | Under the different conditions of the constrained functions , dividing constrained sets properly , and under generalized ( f , ) convexity , the theorems of the weak duality , strong duality and strictly reverse duality are testified
依所给约束函数的不同条件来恰当划分约束集，在广义( f ， )凸性下，证明了弱对偶定理、强对偶定理以及严格逆对偶定理。 |
| 8. | In [ 6 ] , rockafellar had deduced the fenchel theory , but some nonlinear programming could not get the dual by fenchel theory directly and dual theory could not be applied in solving some problem
在文[ 6 ]中， rockafellar导出了fenchel对偶定理，然而有许多凸规划问题直接使用fenchel对偶定理很难用显式表示其对偶，有的问题甚至不能直接使用fenchel对偶定理。 |
| 9. | Furthermore , a mixed type dual for the primal problem is established and duality results are obtained . after that , the optimality sufficiency conditions and duality results for the nonlinear fractional programming problem are presented under generalized ( f , or , p , d ) - convexity assumptions
接着建立了含有等式约束和不等式约束的非线性多目标规划问题的wolfe型对偶和mond - weir型对偶及原问题的混合类型对偶，并获得了相应类型下的弱对偶定理和强对偶定理。 |
| 10. | In view of these , the second part of this paper presents two sufficient conditions and two mixed type duals for the generalized fractional programming only under ( f , ) - convexity assumptions . these sufficient conditions apply to a broader class of mathematical programming problems . the results about weak duality , strong duality and strictly reverse duality are also presented under more suitable conditions
鉴于此，本文的第二部分，我们仅在函数( f ， ) ?凸性假设下，给出了广义分式规划的二个最优性充分条件，这些充分条件较文献中的相关的条件有更广泛的适用性；同时还给出了混合型对偶，并且在适当的条件下，给出了相应的弱对偶定理、强对偶定理，以及严格逆对偶定理。 |
