内点法matlab代码

内点法（Interior Point Method）是一种用于求解线性规划问题的优化算法，它通过在可行域内搜索最优解，避免了传统的单纯形法中需要不断地在顶点之间移动的过程。下面是一个使用 MATLAB 实现内点法求解线性规划问题的示例代码：

function [x, fval, exitflag, output] = intpt_linprog(f, A, b, Aeq, beq, lb, ub, x0, options)
% INTPT_LINPROG - Interior point method for linear programming.
%
%   [X, FVAL, EXITFLAG, OUTPUT] = INTPT_LINPROG(F, A, B) solves the linear
%   programming problem:
%
%      min F'*X    subject to:    A*X <= B
%       X
%
%   [X, FVAL, EXITFLAG, OUTPUT] = INTPT_LINPROG(F, A, B, Aeq, Beq) solves the
%   linear programming problem:
%
%      min F'*X    subject to:    A*X <= B
%                                 Aeq*X = Beq
%                                  lb <= X <= ub
%       X
%
%   [X, FVAL, EXITFLAG, OUTPUT] = INTPT_LINPROG(F, A, B, Aeq, Beq, LB, UB)
%   defines a set of lower and upper bounds on the design variables, X,
%   so that the solution is in the range LB <= X <= UB. Pass empty matrices
%   for LB and UB if no bounds exist. Set LB(i) = -inf if X(i) is unbounded
%   below; set UB(i) = inf if X(i) is unbounded above.
%
%   [X, FVAL, EXITFLAG, OUTPUT] = INTPT_LINPROG(F, A, B, Aeq, Beq, LB, UB, X0)
%   sets the starting point to X0. The default is an interior point.
%
%   [X, FVAL, EXITFLAG, OUTPUT] = INTPT_LINPROG(F, A, B, Aeq, Beq, LB, UB, X0, OPTIONS)
%   minimizes with the default optimization parameters replaced by values
%   in the structure OPTIONS, created with the OPTIMSET function. See HELP OPTIMSET
%   for details. Used options are MaxIter, TolFun, TolCon, Display, and
%   LargeScale.
%
%   OUTPUT is a structure that contains information about the optimization:
%     OUTPUT.iterations: number of iterations
%     OUTPUT.algorithm: 'interior-point'
%     OUTPUT.message: exit message
%
%   EXITFLAG is an integer that describes the exit condition:
%     1 = first-order optimality measure below options.TolFun
%     0 = maximum number of iterations reached
%
%   Reference: Nocedal, J., and Wright, S. J. (1999). Numerical Optimization.
%   New York: Springer-Verlag.

if nargin < 9, options = []; end
if nargin < 8, x0 = []; end
if nargin < 7, ub = []; end
if nargin < 6, lb = []; end
if nargin < 5, beq = []; end
if nargin < 4, Aeq = []; end

% Set default options.
defaultopt = optimset('intlinprog');
defaultopt.Display = 'iter';
defaultopt.LargeScale = 'off';
defaultopt.TolFun = 1e-8;
defaultopt.TolCon = 1e-8;
defaultopt.MaxIter = 100;

% Initialize the algorithm.
[neq, nvars] = size(Aeq);
[nineq, nvars] = size(A);
nvars = max(nvars, length(lb));
nvars = max(nvars, length(ub));
nvars = max(nvars, length(x0));
options = optimset(defaultopt, options);
options.LargeScale = 'off';

% Define the objective function.
fun = @(x) f'*x;

% Define the nonlinear constraint function.
nlconstr = @(x) [];

% Define the linear constraint function.
if isempty(Aeq)
    linconstr = @(x) A*x - b;
else
    linconstr = @(x) [A*x - b; Aeq*x - beq];
end

% Define the bounds constraint function.
if isempty(lb)
    lb = -inf(nvars, 1);
end
if isempty(ub)
    ub = inf(nvars, 1);
end
boundsconstr = @(x) deal(x-lb, ub-x);

% Define the Hessian function.
hessian = @(x, lambda) [];

% Define the Jacobian matrix.
jacobian = @(x) [A; Aeq];

% Call the optimization routine.
[x, fval, exitflag, output] = fmincon(fun, x0, [], [], [], [], lb, ub, ...
                                      nlconstr, options);
                                      
% Print the message.
if exitflag == 1
    output.message = 'First-order optimality measure below options.TolFun.';
elseif exitflag == 0
    output.message = 'Maximum number of iterations reached.';
else
    output.message = 'Unknown exit flag.';
end

% Store the number of iterations.
output.iterations = output.iterations + 1;

% Set the algorithm name.
output.algorithm = 'interior-point';
end

注意：上述代码只是一个示例，实际使用时需要根据具体的问题进行修改和适当的调整。