二次方程优化之 Quadratic Form

Posted 2020-10-12 Mr.ZeroW

=================2017-11-08==============

MATLAB版本低，没有求hessian矩阵的函数，写了一个，需要的拿走

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% --------------------------
%INPUT
%f: funtion;
%X: variables vactor;
%--------------------------
%OUTPUT
%H: Hessian matrix
%--------------------------
%Example 
%syms x y z
%f = x^2 + y^2 + z^2;
%X = [x, y, z];
%H = hessian(f, X)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
funtion H = hessian(f, X)
H = zeros(length(X));
for i = 1: length(X)
  df1 = diff(f, X(i));
  for j = 1: length(X)
      df2 = diff(df1, X(j));
      H(i, j) = df2;
  end
end

 ======================给一个求偏导，也就是第一步解偏导方程等于0的函数===================

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% --------------------------
%INPUT
%f: funtion;
%X: variables vactor;
%--------------------------
%OUTPUT
%op: optimal point
%--------------------------
%Example 
%f = 2*x1^2 - 24 * x1 + 2 * x2^2 - 8*x2 + 2 * x3^2- 12*x3 + 200;
%X = [x1, x2, x3];
%op = first_condition(f, X)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function op = first_condition(f, X)
constraint = [];
for i = 1: length(X)
    df = diff(f, X(i));
    constraint = [constraint, df];
end
op1 = solve(constraint);
op = [];
for i = 1: length(X)
    s = [\'op1.\' char(X(i))];
    a = eval(s);
    op = [op, a];
end
disp(constraint.\')

 ===========================下面给出用法和输出================

比如说求极值，第一步是偏导等于零的点

给出代码如下：

syms x1 x2 x3
f = 2*x1^2 - 24 * x1 + 2 * x2^2 - 8*x2 + 2 * x3^2- 12*x3 + 200;
X = [x1, x2, x3];
op = first_condition(f, X)

 输出结果：

就是说在点(6, 2, 3)处取极值

============================更==========================

我们这一次对二次型进行更深的讨论

讨论多变量的正定型、负定型和半正定、半负定

先给出结论：

判断有两种方法：

1.强推特征值法：如果所有的特征值是正，则是正定型；全是负的就是负定型；全是非负，就是半正定型；全是非正，就是半负定型；

2.子式判断法：分两步：

　　第一步：所有阶的主子式大于0，就是正定型；奇数阶的主子式小于0并且偶数阶的主子式大于0，就是负定型；所有阶的主子式大于等于0或者奇数阶的主子式小于等于0并且偶数阶的主子式大于等于0，转入第二步；除此之外的都是不定型；

　　第二步：所有阶的子式大于等于0，就是半正定型；奇数阶的子式小于等于0并且偶数阶的子式大于等于0，就是半负定型

在继续下面的讨论时，我想先插入一小段关于相关矩阵的一些说明