凸规划

This course offers an introduction to primary convex analysis, linear programming, non-linear programming, dynamic programming, multi-objective programming, multi-level programming, calculus of variations, and optimal control, which are the optimization methods commonly used in economics and finance.

课程内容简介：本课程介绍凸分析基础、线性规划、非线性规划、动态规划、多目标规划、多层规划、变分法、最优控制，这些都是西方经济学、金融学中常用的最优化方法。

In anti-fuzzy number space, the conjugate mapping of convex fuzzy mapping is concerned, and convexities of conjugate set and conjugate mapping of convex fuzzy mapping are proved.

在研究模糊向量空间、模糊凸集和模糊凸映射的基础上，我们研究了模糊规划的Lagrange对偶和凸模糊规划的KKT条件，并将有关结果应用到模糊线性规划和模糊二次规划的研究中。

To increase the running speed of the collision detection algorithm, a new approach is proposed, which uses convex Bounding Volume Hierarchies to express convex polyhedron. Thus the distance problem of two convex polyhedrons is come down to a nonlinear programming problem with constraints. The nonlinear programming problem can be solved by simulated annealing genetic algorithm.

为了提高碰撞检测算法的速度，提出用顶点的凸包表示凸多面体，将两个凸多面体间距离的问题归结为一个带约束条件的非线性规划问题，利用模拟退火遗传算法对该问题进行求解。

Along with the development of convex fuzzy set, people investigate the convexity of fuzzy mapping on convex sets and its application in fuzzy programming .

随着凸模糊集研究的深入与发展，人们讨论了模糊映射在凸集上的凸性及其在模糊规划中的应用。

Furthermor, we consider their nondiferentiable situation, we define nonsmooth univex functions for Lipschitz functions by using Clarke generalized directional derivative and study nonsmooth multiobjective fractional programming with the new convexity. We establish generalized Karush-Kuhn-Tucker necessary and sufficient optimality condition and prove weak, strong and strict converse duality theorems for nonsmooth multiobjective fractional programming problems containing univex functions.

而且，本文利用Clarke广义方向导数针对Lipschitz函数在原来一致凸函数概念的基础上定义了不可微的一致凸函数，并利用这类新凸性，我们研究了非光滑多目标分式规划，获得了广义Karush-Kuhn-Tucher最优性条件；弱对偶定理、强对偶定理和严格逆对偶定理。

Firstly,the mixed-integer bi-level programming is transformed into a single level continuous optimization problem by virtue of penalty function concept and optimal value function tool.

利用罚函数思想和最优值函数的概念将混合整数双层规划转化为连续变量的单层非线性规划，然后用事先确定步长的凸组合算法迭代求解此单层非线性规划，进而得到原双层规划的局部最优解

We define I-quasi-invex vector function ., I-strictly quasi-invex vector function and KT-I-strictly quasi invex vector function, and derive the above equivalent condition for unconstrained or constrained multiobjective programming.

于是，在本文的第三部分，我们定义了Ⅰ类不变拟凸、Ⅰ类严格不变拟凸、KT-Ⅰ类严格不变拟凸的向量值函数，并且在无约束或约束多目标规划中，获得了每个驻点是有效解的等价条件。

Robust programming is, as stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem.

半定规划是一个子字段的凸优化的基本变量半矩阵。这是泛化的线性和凸二次规划。

Optimization; parametric quadratic convex programming; set-valued map; directional derivative; linear stability; solution-set map; parametric linear programming; error bound; subdifferential map; lower locally directionally Lipschitzian; upper locally di-rectionally Lipschitzian; locally directionally Lipschitzian; convex function; quasidiferential; kernelled quasidiferential; quasi-kernel; star-kernel; star-diferential; Penot diferential; subderivative; superderivative; epiderivative; set-valued optimization; set-valued analysis; subdifferential; optimization condition;ε-dual; scalization; generalized subconvexlike-cone;ε-Lagrange multiplier

基础科学，数学，运筹学最优化；集值映射；方向导数；线性稳定；最优解集映射；参数线性规划；参数凸二次规划；误差界；次微分映射；下局部方向Lipschitzian；上局部方向Lipschitzian；局部方向Lipschitzian；凸函数；拟微分；核拟微分；拟核；星核；星微分； Penot-微分；上导数；下导数； Epi-导数；集值优化；集值分析；集值映射的次微分；最优性条件；广义锥次类凸；ε-对偶；数乘；ε-Lagrange乘子

This dissertation is devoted to discussing the optimality conditions and analyzing the augmented Lagrangian method for nonconvex semidefinite programming.

本论文主要讨论非凸半定规划问题的最优性条件，分析非凸半定规划问题的增广Lagrange方法。

Cynanchum Lingtai apricot production in the average weight 65 grams, the brightly-colored fruit, juicy rich, sweet-sour taste, sweet from the nucleolus, when the late Qing Dynasty famous Shaanxi, Gansu provinces, the Qing imperial court Tongzhi tribute for years.

灵台生产的牛心杏平均单果重65克，果实色泽鲜艳，汁多味浓，甜酸适口，离核仁甜，清末时就驰名陕、甘两省，清同治年间曾为朝廷贡品。

Chenopodium album,Solanum nigrum, and Amaranthus retroflexus were very susceptible to the herbicides. Polygonum persicaria and Abutilon theophrasti were relatively less susceptible to the herbicides, and Lycopersicon esculentum was not susceptible to it. The relationship between reduction rates of weed biomass and PPM values of weed leaves 2,4, and 6 days after treatment was established.

供试的6种杂草对该混剂的敏感性存在显著差异：红心藜Chenopodium album、龙葵Solanum nigrum和反枝苋Amaranthus retroflexus对该混剂最敏感，ED90值分别为47.65、71.67和29.17g/hm2；春蓼Polygonum persicaria和苘麻Abutilon theophrasti敏感，ED90值分别为96.91、114.20g/hm2；而番茄不敏感。