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## 最优化

(2009-04-04 16:30:03)

### 杂谈

The first optimization technique, which is known as steepest descent, goes back to Gauss. Historically, the first term to be introduced was linear programming, which was invented by George Dantzig in the 1940s.

The term programming in this context does not refer to
computer programming (although computers are nowadays used extensively to solve mathematical problems). Instead, the term comes from the use of program by the United States military to refer to proposed training and logistics schedules, which were the problems that Dantzig was studying at the time. (Additionally, later on, the use of the term "programming" was apparently important for receiving government funding, as it was associated with high-technology research areas that were considered important.)

Other important mathematicians in the optimization field include:

## Major subfields

• Linear programming studies the case in which the objective function f is linear and the set A is specified using only linear equalities and inequalities. Such a set is called a polyhedron or a polytope if it is bounded.
• 线性规划研究的情况下，目标函数F是线性和设置指定只用线性平等和不平等现象。这样一套被称为多面体或多面体如果是有界的。

• 线性整数规划研究项目，其中部分或全部变量约束采取的整数值。
二次规划的目标函数可以有二次条件，而设置必须指定与线性等式和不等式。
非线性规划研究的一般情况下，目标函数或限制或都包含非线性部分。
• Integer programming studies linear programs in which some or all variables are constrained to take on integer values.
• Quadratic programming allows the objective function to have quadratic terms, while the set A must be specified with linear equalities and inequalities.
• Nonlinear programming studies the general case in which the objective function or the constraints or both contain nonlinear parts.
• Convex programming studies the case when the objective function is convex and the constraints, if any, form a convex set. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.
• 凸规划研究的情况下，目标函数是凸的限制，如果有的话，形成一个凸集。这可以被看作是一个特殊情况非线性规划或泛化的线性或凸二次规划。
• Semidefinite programming (SDP) is a subfield of convex optimization where the underlying variables are semidefinite matrices. It is generalization of linear and convex quadratic programming.
• Stochastic programming studies the case in which some of the constraints or parameters depends on random variables.
• 二阶锥规划（ SOCP ） 。
半定规划（社民党）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。
随机规划研究的情况，其中的一些限制或参数取决于随机变量。
强大的节目，如随机规划，试图捕捉数据的不确定性所依据的优化问题。不这样做通过使用随机变量，而是解决问题是要考虑到不正确的输入数据。
组合优化问题是有关的一套可行的解决办法是离散或可归结为一个独立的一个。
• Robust programming is, as stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem. This is not done through the use of random variables, but instead, the problem is solved taking into account inaccuracies in the input data.
• Combinatorial optimization is concerned with problems where the set of feasible solutions is discrete or can be reduced to a discrete one.
• Infinite-dimensional optimization studies the case when the set of feasible solutions is a subset of an infinite-dimensional space, such as a space of functions.
• 无限维优化的情况下，研究一套可行的解决办法是一个子集，一个无限的三维空间，如空间的功能。
启发式算法
• Heuristic algorithms
• Constraint satisfaction studies the case in which the objective function f is constant (this is used in artificial intelligence, particularly in automated reasoning).
• Disjunctive programming used where at least one constraint must be satisfied but not all. Of particular use in scheduling.
• Trajectory optimization is the speciality of optimizing trajectories for air and space vehicles.
• Metaheuristics
约束满足研究的情况下，目标函数F是常数（这是用在人工智能，特别是在自动推理） 。
约束规划。
析取编程用于指至少有一个限制，必须符合而不是所有。尤其在安排使用。
轨迹优化是专业的优化轨迹的空中和空间飞行器。
在一些分支，该技术的目的主要是为了优化的动态范围内（即决策的时间） ：

In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):

• Calculus of variations seeks to optimize an objective defined over many points in time, by considering how the objective function changes if there is a small change in the choice path.
• 变分法寻求优化的目标界定了许多时间点，考虑如何目标函数的变化，如果有一个小变化，选择的道路。

• 最优控制理论是一个泛化的变分。
动态规划研究的情况下，优化战略的基础是分裂成较小的子问题。方程涉及这些子被称为贝尔曼方程。
数学规划与平衡约束的制约因素包括变分不等式和互补性。
• Optimal control theory is a generalization of the calculus of variations.
• Dynamic programming studies the case in which the optimization strategy is based on splitting the problem into smaller subproblems. The equation that relates these subproblems is called the Bellman equation.
• Mathematical programming with equilibrium constraints is where the constraints include variational inequalities or complementarities

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