1 multphr.m
  2 function [x, mu, lambda, output] = multphr(fun, hf, gf, dfun, dhf, dgf, x0)
  3 maxk=500; sigma=2; eta=2; theta=0.8; epsilon=1e-5; k=0; ink=0;
  4 x=x0; he=feval(hf, x); gi=feval(gf, x); l=length(he); m=length(gi);
  5 
  6 mu=0.1*ones(l,1); lambda=0.1*ones(m,1);
  7 btak=10; btakold=10;
  8 
  9 while btak>epsilon && k<maxk
 10     [x, ~, ik] = bfgs('mpsi', 'dmpsi', x0, fun, hf, gf, dfun, dhf, dgf, mu, lambda, sigma);
 11     ink = ink+ik;
 12     he=feval(hf, x); gi=feval(gf, x);
 13     btak=0;
 14     for i=1:l 
 15         btak=btak+he(i)^2;
 16     end
 17     for i=1:m
 18         btak=btak+min(gi(i), lambda(i)/sigma)^2;
 19     end
 20     btak=sqrt(btak);
 21     if btak>epsilon
 22         if k>=2 && btak>theta*btakold
 23             sigma = eta*sigma;
 24         end
 25         for i=1:l
 26             mu(i)=mu(i)-sigma*he(i);
 27         end
 28         for i=1:m
 29             lambda(i)=max(0, lambda(i)-sigma*gi(i)); % lambda(i)=max(0, lambda(i)-gi(i));
 30         end
 31     end
 32     k=k+1;
 33     btakold=btak;
 34     x0=x;
 35     output.fval=feval(fun, x);
 36     output.iter=k;
 37     output.innter_iter=ink;
 38     output.bta=btak;
 39 end
 40 
 41 
 42 bfgs.m
 43 function [x, val, k] = bfgs(fun, gfun, x0, varargin)
 44 maxk=5000; k=0; rho=0.55; sigma=0.4; epsilon=1e-5;
 45 n=length(x0); Bk=eye(n);
 46 while (k<maxk)
 47     gk = feval(gfun, x0, varargin{:});
 48     if norm(gk)<epsilon
 49         break;
 50     end
 51     dk = -Bk\gk;
 52     m=0; mk=0;
 53     while (m<20)
 54         if(feval(fun, x0+rho^m*dk, varargin{:}) < feval(fun, x0, varargin{:})+sigma*rho^m*gk'*dk)
 55             mk=m; break;
 56         end
 57         m=m+1;
 58     end
 59     x=x0+rho^mk*dk;
 60     sk=x-x0;
 61     yk = feval(gfun, x, varargin{:})-gk;
 62     if yk'*sk>0
 63         Bk = Bk-(Bk*sk*sk'*Bk)/(sk'*Bk*sk)+(yk*yk')/(yk'*sk);
 64     end
 65     k=k+1; x0=x;
 66 end
 67 val = feval(fun, x0, varargin{:});
 68 
 69 mpsi.m
 70 function psi = mpsi(x, fun, hf, gf, dfun, dhf, dgf, mu, lambda, sigma)
 71 psi=feval(fun, x); he=feval(hf, x); gi=feval(gf, x); l=length(he); m=length(gi);
 72 for i=1:l
 73     psi=psi-he(i)*mu(i)+0.5*sigma*he(i)^2;
 74 end
 75 for i=1:m
 76     psi=psi +(min(0, sigma*gi(i)-lambda(i))^2 - lambda(i)^2)/(2*sigma);
 77 end
 78 
 79 dmpsi.m
 80 function dpsi=dmpsi(x, fun, hf, gf, dfun, dhf, dgf, mu, lambda, sigma)
 81 dpsi=feval(dfun, x); he=feval(hf, x); gi=feval(gf, x); dhe=feval(dhf, x); dgi=feval(dgf, x);
 82 l=length(he); m=length(gi);
 83 for i=1:l
 84     dpsi = dpsi+(sigma*he(i)-mu(i))*dhe(:,i);
 85 end
 86 for i=1:m
 87     dpsi = dpsi+(sigma*gi(i)-lambda(i))*dgi(:,i);
 88 end
 89 
 90 fun.m
 91 function f=fun(x)
 92 f=(x(1)-2)^2+(x(2)-1)^2;
 93 
 94 dfun.m
 95 function df = dfun(x)
 96 df = [2*x(1)-4, 2*x(2)-2]';
 97 
 98 hf.m
 99 function he=hf(x)
100 he=x(1)-2*x(2)+1;
101 
102 dhf.m
103 function dfe=dhf(x)
104 dfe=[1, -2]';
105 
106 gff.m
107 function gi=gff(x)
108 gi= -0.25*x(1)^2-x(2)^2+1;
109 
110 
111 dgf.m
112 function dgi=dgf(x)
113 dgi=[-0.5*x(1), -2*x(2)]';

运行结果：
>> x0=[3 3]';
>> [x, mu, lambda, output] = multphr('fun', 'hf', 'gff', 'dfun', 'dhf', 'dgf', x0)
x =
    0.8229
    0.9114
mu =
   -1.5945
lambda =
1.8465
output = 
           fval: 1.3934
           iter: 23
innter_iter: 82
        bta: 9.5419e-006