整数规划之分支定界算法

f = [-20,-10];
A = [5,4;2,5];
B = [24;13];
lb = [0;0];
I = [1:length(f)];
I(1);
[x,fval,status] = intprog(f,A,B,I,[],[],lb)

 

function [x,fval,status] = intprog(f,A,B,I,Aeq,Beq,lb,ub,e)
if nargin < 9, e = 0.00001;
   if nargin < 8, ub = []; 
      if nargin < 7, lb = []; 
         if nargin < 6, Beq = []; 
            if nargin < 5, Aeq = [];
               if nargin < 4, I = [1:length(f)];
               end, end, end, end, end, end
               
%求解整数规划对应的线性规划,判断是否有解
options = optimset('display','off');
[x0,fval0,exitflag] = linprog(f,A,B,Aeq,Beq,lb,ub,[],options);
if exitflag < 0
    disp('没有合适整数解');
    x = x0;
    fval = fval0;
    status = exitflag;
    return;
else
    %采用分支定界法求解
    bound = inf;
    [x,fval,status] = branchbound(f,A,B,I,x0,fval0,bound,Aeq,Beq,lb,ub,e);
end

 

 

function [newx,newfval,status,newbound] = branchbound(f,A,B,I,x,fval,bound,Aeq,Beq,lb,ub,e)

% 分支定界法求解整数规划
% f,A,B,Aeq,Beq,lb,ub与线性规划相同
% I为整数限制变量的向量
% x为初始解，fval为初始值
% 
options = optimset('display','off');
[x0,fval0,status0]=linprog(f,A,B,Aeq,Beq,lb,ub,[],options);

% %递归中的最终退出条件
% %无解或者解比现有上界大则返回原解
if status0 <= 0 || fval0 >= bound
    newx = x;
    newfval = fval;
    newbound = bound;
    status = status0;
    return;
end
% 
% %是否为整数解,如果是整数解则返回
intindex = find(abs(x0(I) - round(x0(I)))> e);
if isempty(intindex) %判断是否为空值  1表示为空，0为非空
    newx(I) = round(x0(I));
    newfval = fval0;
    newbound = fval0;
    status = 1;
    return;
end
% 
%当有非整可行解时，则进行分支求解
%此时必定会有整数解或空解
%找到第一个不满足整数要求的变量
n = I(intindex(1));%n=1
addA = zeros(1,length(f));
addA(n) = 1;
%构造第一个分支 x<=floor(x(n))
A = [A;addA];
B = [B;floor(x(n))];%向下取整
[x1,fval1,status1,bound1] = branchbound(f,A,B,I,x0,fval0,bound,Aeq,Beq,lb,ub,e);
%A(end,:) = [];
%B(:,end) = [];
%解得第一个分支，若为更优解则替换，若不是则保持原状
% 

status = status1;
if status1 > 0 && bound1 < bound
    newx = x1;
    newfval = fval1;
    bound = fval1;
    newbound = bound1;
else
    newx = x0;
    newfval = fval0;
    newbound = bound;
end
% 
%构造第二分支
A = [A;-addA];
B = [B;-ceil(x(n))];%向上取整
[x2,fval2,status2,bound2] = branchbound(f,A,B,I,x0,fval0,bound,Aeq,Beq,lb,ub,e);
% A(end,:) = [];
% B(:,end) = [];

%解得第二分支，并与第一分支做比较，如果更优则替换
if status2 > 0 && bound2 < bound    
    status = status2;
    newx = x2;
    newfval = fval2;
    newbound = bound2;
end

 疑问：status2不知代表什么，为什么其结果为-2？

 

 输出结果：

 

>> Test

x =

     4     1


fval =

  -90.0000


status =

     1

 

以上仅供参考，真实数据需重新编写相应函数的代码；

另，也可参考：

https://blog.csdn.net/weixin_53599669/article/details/118978608?ops_request_misc=%257B%2522request%255Fid%2522%253A%2522165891705216782388082506%2522%252C%2522scm%2522%253A%252220140713.130102334.pc%255Fall.%2522%257D&request_id=165891705216782388082506&biz_id=0&utm_medium=distribute.pc_search_result.none-task-blog-2~all~first_rank_ecpm_v1~rank_v31_ecpm-1-118978608-null-null.142^v35^pc_search_v2,185^v2^tag_show&utm_term=function%20%5Bx%2Cfval%2Cstatus%5D%20%3D%20intprog%28f%2CA%2CB%2CI%2CAeq%2CBeq%2Clb%2Cub%2Ce%29&spm=1018.2226.3001.4187