Teacher wangshuyao's notes on operations research 01 guidance and introduction

The first 1 speak Guidance and introduction

3 A question

The purpose of operations research ： Find the optimal solution of the general decision （ Optimization problems ）

The source of operations research ： military

What does this course do ： Research on operational research problems from the perspective of application

The relationship between linear programming and operations research

The linear programming of high school learning is a part of operational research , Linear programming requires that the objective function be linear , Constraints are also linear , But the situation of operations research is more general .

meanwhile , Linear programming is also a special nonlinear programming , in other words , Nonlinear programming includes linear programming , Operations research includes nonlinear programming . And the linear programming problem is the core of the whole operation research .

General steps for dealing with nonlinear programming problems

1. mathematical modeling
2. Find the optimal solution or feasible solution （ a key ）
   1. A general method is needed to deal with the maximum value of a function -> The extreme value and maximum value of a function can be obtained by derivative or partial derivative in calculus .（ Generally, the problems encountered are derivable , If you can't find the derivative , You need to use the definition method to find the extreme value and the maximum value of the function .）
      1. For unary functions , The first derivative of shilling is 0, Find suspicious points , Then judge whether the function near the doubtful point is increasing or decreasing, so as to judge whether the doubtful point is the extreme point , Then combined with the endpoint of the closed interval to determine the maximum .（ High school knowledge ）
      2. For multivariate functions , Constraints are strong constraints （ Equal sign ）, use Lag Multiplier method for partial derivation .（ Advanced mathematics volume II knowledge ）
      3. For multivariate functions , Constraints are weak constraints （ Unequal sign ）, use KKT Theorem solving .（Lag Extension of multiplier method ）

Regression from nonlinear programming problem to dealing with programming problem

Using the treatment method of nonlinear programming problem , Combine the mathematical tools of linear algebra to deal with linear programming problems . And there are some more special methods , That is, methods that can only deal with linear programming problems , Such as simplex method .

Natural definition domain and real definition domain of function

The natural domain of definition ： In the context of Pure Mathematics , Definition field of function .（ If the denominator is not zero ）

Real definition domain ： After considering the actual source of the problem , Definition field of function .（ For example, the number of people can only be a non negative integer ）

The real domain is generally a subset of the natural domain .

Consider the solution idea in the real situation ：

1. Find the optimal solution under the natural definition domain
2. Under the guidance of this optimal solution, find the optimal solution in the real domain

summary

The core of operations research is to solve the optimization problem from a general perspective , That is, the problem of finding the maximum value of the function .

To deal with operational research problems , First, mathematical modeling , Then we need to study the function obtained by modeling , Depending on whether the constraint is strong or if , choice Lag Multiplier method or KKT Theorem to solve . After the solution is completed, the optimal solution under the real definition domain can be obtained according to its real definition domain . Specially , For linear programming problems , It can be handled by the general method of dealing with nonlinear programming problems , It can also be handled in some special ways .