无约束最优化问题是最简单的一类最优化问题，其一般数学描述为

              $\min\limits_{\textbfsymbol{x}\in \mathbb{R}^n} f(\textbfsymbol{x})$

其中 $\textbfsymbol{x}=[x_1,x_2,\cdots,x_n]^T$ 称为优化变量， $f(\cdot)$ 函数称为目标函数。

1、解析法和图解法

令梯度 $\nabla f=\textbfsymbol{0}$ 得到驻点，即为极值可疑点。  图解法只是适合低维的一元、二元和三元函数。

例1 求一元函数最值

$f(t)=\text{e}^{-3t}\sin(4t+2)+4\text{e}^{-0.5t}\cos(2t)-0.5$

解析解方法

>> syms t; y = exp(-3*t)*sin(4*t+2)+4*exp(-0.5*t)*cos(2*t)-0.5;

>> y1 = diff(y,t);t0 = solve(y1),ezplot(y,[0,4])

警告: Cannot solve symbolically. Returning a numeric approximation instead.

> In solve (line 303)

t0 =

-0.22133627468652136234203676475842

>> y1 = diff(y,t);t0 = solve(y1),ezplot(y,[-4,4])

警告: Cannot solve symbolically. Returning a numeric approximation instead.

> In solve (line 303)

t0 =

-0.22133627468652136234203676475842

>> y1 = diff(y,t);t0 = solve(y1),ezplot(y,[-1,4])

警告: Cannot solve symbolically. Returning a numeric approximation instead.

> In solve (line 303)

t0 =

-0.22133627468652136234203676475842

>> y2 = diff(y1);b = subs(y2,t,t0)

b =

-51.712061718150102502340461429533

这说明二阶导数 b<0,

t0 =-0.22133 为极大值点