☆ Mathematical Programming (Optimization Problem)
• Linear Programming(线性规划) • Nonlinear Programming(非线性规划)
– Unconstrained Problems(无约束最优化方法) – Constrained Problems(约束最优化方法) ☆ Variational Inequality (VI)(变分不等式)
• Monotone VI • Non-monotone VI
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Optimization Problem
(Mathematical) Optimization Problem
minimize f x subject to gi x bi , i 1,L ,m
g x x1,L , xn : optimization variables (优化变量)
2
Optimization Process
real world problem
analysis
validation, sensitivity analysis
algorithm, model, solution technique
numerical methods
verification 验证
computer implementation
the roads, the travel times between intersections can be considered as
constant, but if the traffic is heavy the travel times can increase
体运行时间) for handling a (long) sequence of tasks(处理一个任务序
列), by appropriately assigning a fraction of each type of task to each
worker.
Worker
Information
Policy
Suppose that a manufacturer of kitchen cabinets is trying to maximize the weekly revenue of a factory. Various orders have come in that the company could accept. They include bookcases(书橱) with open shelves, cabinets with doors, cabinets with drawers, and custom-designed(定制的) cabinets. The following Table indicates the quantities of materials and labor required to assemble the four types of cabinets, as well as the revenue earned. Suppose that 5000 units of wood and 1500 units of labor are available.
Chapter 1
Introduction & Preliminaries
Optimization
• Optimization models attempt to express, in mathematical terms(数 学术语), the goal of solving a problem in the “best” way
g f : n : objective function (目标函数) g gi : n , i 1,L , m : constraint functions (约束函数)
optimal solution(最优解) x* has smallest value(最小值) of f among all vectors(矢量) that satisfy the constraints(满足约束条件).
• Optimization models have been used for centuries, as their purpose is so appealing, and in recent times, they have come to be essential
• The concept of optimization is well rooted as a principle underlying the analysis of many complex decision or allocation problems
Data fitting(数据拟合) • variables: model parameters • constraints: prior information, parameter limits • objective: measure of misfit or prediction error
• A particular optimization formulation should be regarded only as an approximation
• Learn to identify and capture the important issues of a problem
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Types of Problems
3
Optimization
• One approaches a complex decision problem involving
– selection of values for a number of interrelated variables(关联变量) – focus attention on a single objective(单目标) designed to quantify
office handles three types of work: requests for information, new policies,
and claims. There are five workers. Based on a study of office operations,
the average work times (in minutes) for the workers are known (see the Table). The company would like to minimize the overall elapsed time(整
Cabinet
Wood
Labor
Revenue
Bookshelf
10
2
100
With
With Drawers
25
8
200
Custom
20
12
400
8
Linear Programming
Example 2
Consider the assignment of jobs to workers. Suppose that an insurance
Example 1
Suppose that four buildings are to be connected by electrical wires. The positions of the buildings are illustrated in the Figure(如图所示). The first two buildings are circular: one at (1,4)T with radius(半径) 2, the second at (9,5)T with radius 1. The third building is square with sides of length 2 centered at (3,-2)T. The fourth building is rectangular with height 4 and width 2 centered at (7,0)T. The electrical wires will be joined at some central point (x0 , y0)T, and will connect to building i at position (xi , yi)T. The objective(目标) is to minimize the amount of wire used.
performance and measure the quality of the decision(衡量决策品质) – the objective is maximized or minimized subject to the constraints that
may limit the selection of decision variable values(决策变量值)
– running a business to maximize profit, minimize loss, maximize efficiency, or minimize risk
– designing a bridge to minimize weight or maximize strength – selecting a flight plan for an aircraft to minimize time or fuel use
Claim
1
10
28
31
2
15
22
42
3
13
18
35
4
19
25
29
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