Teacher wangshuyao's notes on operations research 02 fundamentals of advanced mathematics

The first 2 speak Fundamentals of Advanced Mathematics

Maximum and extreme values

Definition of maximum value and extreme value

The most value ： Function maxima are divided into function minimum and function maximum , The minimum value is the minimum value of the function value in the definition field , The maximum value is the maximum value of the function value in the definition field .

extremum ： if $x_0$ Is the extreme point , It's in $x_0$ In the neighborhood of $f(x)\ge f\left(x_0\right)$ perhaps $f(x)\le f\left(x_0\right)$ .

difference ： What is worth discussing most is integrity , Extremum discusses locality . The extreme value is not necessarily the maximum value , The maximum value is not necessarily the extreme value .

The maximum and minimum values of a function on a closed interval must exist .

Extreme values cannot be discussed at the boundary points of intervals , The extreme point must be an interior point .

Fermat lemma

$f(x)$ stay $x_0$ Get the extreme value at and $x_0$ Place can lead , Then there are $f^{\prime }(x_0)=0$. Otherwise .

Use Fermat lemma to find the maximum value and extreme value

The maximum value of a function can only be obtained at the end point or extreme point of the interval , There may be two kinds of values , One is the suspicious point of function , Even if $f^{\prime }(x)=0$ The point of , The other is non derivable point .

So find all the extreme points and the end points of the interval , The largest of the corresponding function values is the maximum value , The smallest is the smallest .

Further on , Because operations research is concerned with the most valuable , So for functions that are continuously differentiable on closed intervals , Find out that all derivatives are 0 The point of and the end of the interval , The largest of the corresponding function values is the maximum value , The smallest is the smallest .

The maximum value of a multivariate function

Lag Multiplier method

1. If there are several constraints, just introduce a few Lag Multiplier ${\lambda }_i$.
2. Construct a new function $F=f+{\sum }_i{\lambda }_i{g}_i$, among $f$ It's the objective function ,$g_i$ It's a constraint .
3. Find the partial derivatives of all independent variables , Make it equal to 0, Some solutions are obtained . Compare the function values of these solutions , The largest of the corresponding function values is the maximum value , The smallest is the smallest .

When solving the equations , First choose the linear equation to solve , Then substitute it into other formulas to solve

Extremum of multivariate function

Positive definiteness of Hesse matrix （ The second partial derivative of the function is required to be continuous ）

1. Let the partial derivative of the function be 0, Solve the equations and get several solutions .
2. Verify the positive definiteness of the Hessian matrix of each set of solutions . Positive definite is the maximum point , The negative rule is the minimum point , Uncertainty is not the extreme point , Semi positive or semi negative rules are suspicious （ This method fails ）.

summary

In operations research, the tools of derivatives and partial derivatives in advanced mathematics are needed to solve the maximum value . For unary functions , Find out that all derivatives are 0 The point of and the end of the interval , The largest of the corresponding function values is the maximum value , The smallest is the smallest . For multivariate functions , Use Lag Multiplier method . For the extreme value of multivariate function , We can use Hessian matrix to discuss .