27 用linprog、fmincon求 解线性规划问题（matlab程序）

1.简述

      

① linprog函数：
 求解线性规划问题，求目标函数的最小值，

[x,y]= linprog(c,A,b,Aeq,beq,lb,ub)

求最大值时，c加上负号：-c

② intlinprog函数：
求解混合整数线性规划问题，

[x,y]= intlinprog(c,intcon,A,b,Aeq,beq,lb,ub)

与linprog相比，多了参数intcon,代表了整数决策变量所在的位置

优化问题中最常见的，就是线性/整数规划问题。即使赛题中有非线性目标/约束，第一想法也应是将其转化为线性。

直白点说，只要决定参加数模比赛，学会建立并求解线性/整数规划问题是非常必要的。

本期主要阐述用Matlab软件求解此类问题的一般步骤，后几期会逐步增加用Mathematica、AMPL、CPLEX、Gurobi、Python等软件求解的教程。

---

或许你已经听说或掌握了linprog等函数，实际上，它只是诸多求解方法中的一种，且有一定的局限性。

我的每期文章力求"阅完可上手"并"知其所以然"。因此，在讲解如何应用linprog等函数语法前，有必要先了解：

• 什么赛题适用线性/整数规划？
• 如何把非线性形式线性化？
• 如何查看某函数的语法？
• 有哪几种求解方法？

把握好这四个问题，有时候比仅仅会用linprog等函数求解更重要。

一、什么赛题适用线性/整数规划

当题目中提到“怎样分配”、“XX最大/最合理”、“XX尽量多/少”等词汇时。具体有：

1. 生产安排

目标：总利润最大；约束：原材料、设备限制；

2. 销售运输

目标：运费等成本最低；约束：从某产地(产量有限制)运往某销地的运费不同；

3. 投资收益等

目标：总收益最大；约束：不同资产配置下收益率/风险不同，总资金有限；

对于整数规划，除了通常要求变量为整数外，典型的还有指派/背包等问题(决策变量有0-1变量)。

二、如何把非线性形式线性化

在比赛时，遇到非线性形式是家常便饭。此时若能够线性化该问题，绝对是你数模论文的加分项。

我在之前写的线性化文章中提到：如下非线性形式，均可实现线性化：

总的来说，具有 分段函数形式、 绝对值函数形式、 最小/大值函数形式、 逻辑或形式、 含有0-1变量的乘积形式、 混合整数形式以及 分式目标函数，均可实现 线性化。

而实现线性化的主要手段主要就两点，一是引入0-1变量，二是引入很大的整数M。具体细节请参见之前写的线性化文章。

三、如何查看函数所有功能

授之以鱼，不如授之以渔。

学习linprog等函数最好的方法，无疑是看Matlab官方帮助文档。本文仅是抛砖引玉地举例说明几个函数的基础用法，更多细节参见帮助文档。步骤是：

• 调用linprog等函数前需要事先安装“OptimizationToolbox”工具箱；
• 在Matlab命令窗口输入“doc linprog”，便可查看语法，里面有丰富的例子；
• 也可直接查看官方给的PDF帮助文档，后台回复“线性规划”可获取。

2.代码

主程序：

%%   解线性规划问题
%f(x)=-5x(1)+4x(2)+2x(3)
f=[-5,4,2]; %函数系数
A=[6,-1,1;1,2,4]; %不等式系数
b=[8;10]; %不等式右边常数项
l=[-1,0,0];  %下限
u=[3,2,inf]; %上限
%%%%用linprog求解
[xol,fol]=linprog(f,A,b,[],[],l,u)
%%%%用fmincon求解
x0=[0,0,0];
f1214=inline('-5*x(1)+4*x(2)+2*x(3)','x');
[xoc,foc]=fmincon(f1214,x0,A,b,[],[],l,u)

子程序：

function [x,fval,exitflag,output,lambda]=linprog(f,A,B,Aeq,Beq,lb,ub,x0,options)
%LINPROG Linear programming.
%   X = LINPROG(f,A,b) attempts to solve the linear programming problem:
%
%            min f'*x    subject to:   A*x <= b
%             x
%
%   X = LINPROG(f,A,b,Aeq,beq) solves the problem above while additionally
%   satisfying the equality constraints Aeq*x = beq. (Set A=[] and B=[] if
%   no inequalities exist.)
%
%   X = 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. Use 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 = LINPROG(f,A,b,Aeq,beq,LB,UB,X0) sets the starting point to X0. This
%   option is only available with the active-set algorithm. The default
%   interior point algorithm will ignore any non-empty starting point.
%
%   X = LINPROG(PROBLEM) finds the minimum for PROBLEM. PROBLEM is a
%   structure with the vector 'f' in PROBLEM.f, the linear inequality
%   constraints in PROBLEM.Aineq and PROBLEM.bineq, the linear equality
%   constraints in PROBLEM.Aeq and PROBLEM.beq, the lower bounds in
%   PROBLEM.lb, the upper bounds in  PROBLEM.ub, the start point
%   in PROBLEM.x0, the options structure in PROBLEM.options, and solver
%   name 'linprog' in PROBLEM.solver. Use this syntax to solve at the
%   command line a problem exported from OPTIMTOOL.
%
%   [X,FVAL] = LINPROG(f,A,b) returns the value of the objective function
%   at X: FVAL = f'*X.
%
%   [X,FVAL,EXITFLAG] = LINPROG(f,A,b) returns an EXITFLAG that describes
%   the exit condition. Possible values of EXITFLAG and the corresponding
%   exit conditions are
%
%     3  LINPROG converged to a solution X with poor constraint feasibility.
%     1  LINPROG converged to a solution X.
%     0  Maximum number of iterations reached.
%    -2  No feasible point found.
%    -3  Problem is unbounded.
%    -4  NaN value encountered during execution of algorithm.
%    -5  Both primal and dual problems are infeasible.
%    -7  Magnitude of search direction became too small; no further
%         progress can be made. The problem is ill-posed or badly
%         conditioned.
%    -9  LINPROG lost feasibility probably due to ill-conditioned matrix.
%
%   [X,FVAL,EXITFLAG,OUTPUT] = LINPROG(f,A,b) returns a structure OUTPUT
%   with the number of iterations taken in OUTPUT.iterations, maximum of
%   constraint violations in OUTPUT.constrviolation, the type of
%   algorithm used in OUTPUT.algorithm, the number of conjugate gradient
%   iterations in OUTPUT.cgiterations (= 0, included for backward
%   compatibility), and the exit message in OUTPUT.message.
%
%   [X,FVAL,EXITFLAG,OUTPUT,LAMBDA] = LINPROG(f,A,b) returns the set of
%   Lagrangian multipliers LAMBDA, at the solution: LAMBDA.ineqlin for the
%   linear inequalities A, LAMBDA.eqlin for the linear equalities Aeq,
%   LAMBDA.lower for LB, and LAMBDA.upper for UB.
%
%   NOTE: the interior-point (the default) algorithm of LINPROG uses a
%         primal-dual method. Both the primal problem and the dual problem
%         must be feasible for convergence. Infeasibility messages of
%         either the primal or dual, or both, are given as appropriate. The
%         primal problem in standard form is
%              min f'*x such that A*x = b, x >= 0.
%         The dual problem is
%              max b'*y such that A'*y + s = f, s >= 0.
%
%   See also QUADPROG.

%   Copyright 1990-2018 The MathWorks, Inc.

% If just 'defaults' passed in, return the default options in X

% Default MaxIter, TolCon and TolFun is set to [] because its value depends
% on the algorithm.
defaultopt = struct( ...
    'Algorithm','dual-simplex', ...
    'Diagnostics','off', ...
    'Display','final', ...
    'LargeScale','on', ...
    'MaxIter',[], ...
    'MaxTime', Inf, ...
    'Preprocess','basic', ...
    'TolCon',[],...
    'TolFun',[]);

if nargin==1 && nargout <= 1 && strcmpi(f,'defaults')
   x = defaultopt;
   return
end

% Handle missing arguments
if nargin < 9
    options = [];
    % Check if x0 was omitted and options were passed instead
    if nargin == 8
        if isa(x0, 'struct') || isa(x0, 'optim.options.SolverOptions')
            options = x0;
            x0 = [];
        end
    else
        x0 = [];
        if nargin < 7
            ub = [];
            if nargin < 6
                lb = [];
                if nargin < 5
                    Beq = [];
                    if nargin < 4
                        Aeq = [];
                    end
                end
            end
        end
    end
end

% Detect problem structure input
problemInput = false;
if nargin == 1
    if isa(f,'struct')
        problemInput = true;
        [f,A,B,Aeq,Beq,lb,ub,x0,options] = separateOptimStruct(f);
    else % Single input and non-structure.
        error(message('optim:linprog:InputArg'));
    end
end

% No options passed. Set options directly to defaultopt after
allDefaultOpts = isempty(options);

% Prepare the options for the solver
options = prepareOptionsForSolver(options, 'linprog');

if nargin < 3 && ~problemInput
  error(message('optim:linprog:NotEnoughInputs'))
end

% Define algorithm strings
thisFcn  = 'linprog';
algIP    = 'interior-point-legacy';
algDSX   = 'dual-simplex';
algIP15b = 'interior-point';

% Check for non-double inputs
msg = isoptimargdbl(upper(thisFcn), {'f','A','b','Aeq','beq','LB','UB', 'X0'}, ...
                                      f,  A,  B,  Aeq,  Beq,  lb,  ub,   x0);
if ~isempty(msg)
    error('optim:linprog:NonDoubleInput',msg);
end

% After processing options for optionFeedback, etc., set options to default
% if no options were passed.
if allDefaultOpts
    % Options are all default
    options = defaultopt;
end

if nargout > 3
   computeConstrViolation = true;
   computeFirstOrderOpt = true;
   % Lagrange multipliers are needed to compute first-order optimality
   computeLambda = true;
else
   computeConstrViolation = false;
   computeFirstOrderOpt = false;
   computeLambda = false;
end

% Algorithm check:
% If Algorithm is empty, it is set to its default value.
algIsEmpty = ~isfield(options,'Algorithm') || isempty(options.Algorithm);
if ~algIsEmpty
    Algorithm = optimget(options,'Algorithm',defaultopt,'fast',allDefaultOpts);
    OUTPUT.algorithm = Algorithm;
    % Make sure the algorithm choice is valid
    if ~any(strcmp({algIP; algDSX; algIP15b},Algorithm))
        error(message('optim:linprog:InvalidAlgorithm'));
    end
else
    Algorithm = algDSX;
    OUTPUT.algorithm = Algorithm;
end

% Option LargeScale = 'off' is ignored
largescaleOn = strcmpi(optimget(options,'LargeScale',defaultopt,'fast',allDefaultOpts),'on');
if ~largescaleOn
    [linkTag, endLinkTag] = linkToAlgDefaultChangeCsh('linprog_warn_largescale');
    warning(message('optim:linprog:AlgOptsConflict', Algorithm, linkTag, endLinkTag));
end

% Options setup
diagnostics = strcmpi(optimget(options,'Diagnostics',defaultopt,'fast',allDefaultOpts),'on');
switch optimget(options,'Display',defaultopt,'fast',allDefaultOpts)
    case {'final','final-detailed'}
        verbosity = 1;
    case {'off','none'}
        verbosity = 0;
    case {'iter','iter-detailed'}
        verbosity = 2;
    case {'testing'}
        verbosity = 3;
    otherwise
        verbosity = 1;
end

% Set the constraints up: defaults and check size
[nineqcstr,nvarsineq] = size(A);
[neqcstr,nvarseq] = size(Aeq);
nvars = max([length(f),nvarsineq,nvarseq]); % In case A is empty

if nvars == 0
    % The problem is empty possibly due to some error in input.
    error(message('optim:linprog:EmptyProblem'));
end

if isempty(f), f=zeros(nvars,1); end
if isempty(A), A=zeros(0,nvars); end
if isempty(B), B=zeros(0,1); end
if isempty(Aeq), Aeq=zeros(0,nvars); end
if isempty(Beq), Beq=zeros(0,1); end

% Set to column vectors
f = f(:);
B = B(:);
Beq = Beq(:);

if ~isequal(length(B),nineqcstr)
    error(message('optim:linprog:SizeMismatchRowsOfA'));
elseif ~isequal(length(Beq),neqcstr)
    error(message('optim:linprog:SizeMismatchRowsOfAeq'));
elseif ~isequal(length(f),nvarsineq) && ~isempty(A)
    error(message('optim:linprog:SizeMismatchColsOfA'));
elseif ~isequal(length(f),nvarseq) && ~isempty(Aeq)
    error(message('optim:linprog:SizeMismatchColsOfAeq'));
end

[x0,lb,ub,msg] = checkbounds(x0,lb,ub,nvars);
if ~isempty(msg)
   exitflag = -2;
   x = x0; fval = []; lambda = [];
   output.iterations = 0;
   output.constrviolation = [];
   output.firstorderopt = [];
   output.algorithm = ''; % not known at this stage
   output.cgiterations = [];
   output.message = msg;
   if verbosity > 0
      disp(msg)
   end
   return
end

if diagnostics
   % Do diagnostics on information so far
   gradflag = []; hessflag = []; constflag = false; gradconstflag = false;
   non_eq=0;non_ineq=0; lin_eq=size(Aeq,1); lin_ineq=size(A,1); XOUT=ones(nvars,1);
   funfcn{1} = []; confcn{1}=[];
   diagnose('linprog',OUTPUT,gradflag,hessflag,constflag,gradconstflag,...
      XOUT,non_eq,non_ineq,lin_eq,lin_ineq,lb,ub,funfcn,confcn);
end

% Throw warning that x0 is ignored (true for all algorithms)
if ~isempty(x0) && verbosity > 0
    fprintf(getString(message('optim:linprog:IgnoreX0',Algorithm)));
end

if strcmpi(Algorithm,algIP)
    % Set the default values of TolFun and MaxIter for this algorithm
    defaultopt.TolFun = 1e-8;
    defaultopt.MaxIter = 85;
    [x,fval,lambda,exitflag,output] = lipsol(f,A,B,Aeq,Beq,lb,ub,options,defaultopt,computeLambda);
elseif strcmpi(Algorithm,algDSX) || strcmpi(Algorithm,algIP15b)

    % Create linprog options object
    algoptions = optimoptions('linprog', 'Algorithm', Algorithm);

    % Set some algorithm specific options
    if isfield(options, 'InternalOptions')
        algoptions = setInternalOptions(algoptions, options.InternalOptions);
    end

    thisMaxIter = optimget(options,'MaxIter',defaultopt,'fast',allDefaultOpts);
    if strcmpi(Algorithm,algIP15b)
        if ischar(thisMaxIter)
            error(message('optim:linprog:InvalidMaxIter'));
        end
    end
    if strcmpi(Algorithm,algDSX)
        algoptions.Preprocess = optimget(options,'Preprocess',defaultopt,'fast',allDefaultOpts);
        algoptions.MaxTime = optimget(options,'MaxTime',defaultopt,'fast',allDefaultOpts);
        if ischar(thisMaxIter) && ...
                ~strcmpi(thisMaxIter,'10*(numberofequalities+numberofinequalities+numberofvariables)')
            error(message('optim:linprog:InvalidMaxIter'));
        end
    end

    % Set options common to dual-simplex and interior-point-r2015b
    algoptions.Diagnostics = optimget(options,'Diagnostics',defaultopt,'fast',allDefaultOpts);
    algoptions.Display = optimget(options,'Display',defaultopt,'fast',allDefaultOpts);
    thisTolCon = optimget(options,'TolCon',defaultopt,'fast',allDefaultOpts);
    if ~isempty(thisTolCon)
        algoptions.TolCon = thisTolCon;
    end
    thisTolFun = optimget(options,'TolFun',defaultopt,'fast',allDefaultOpts);
    if ~isempty(thisTolFun)
        algoptions.TolFun = thisTolFun;
    end
    if ~isempty(thisMaxIter) && ~ischar(thisMaxIter)
        % At this point, thisMaxIter is either
        % * a double that we can set in the options object or
        % * the default string, which we do not have to set as algoptions
        % is constructed with MaxIter at its default value
        algoptions.MaxIter = thisMaxIter;
    end

    % Create a problem structure. Individually creating each field is quicker
    % than one call to struct
    problem.f = f;
    problem.Aineq = A;
    problem.bineq = B;
    problem.Aeq = Aeq;
    problem.beq = Beq;
    problem.lb = lb;
    problem.ub = ub;
    problem.options = algoptions;
    problem.solver = 'linprog';

    % Create the algorithm from the options.
    algorithm = createAlgorithm(problem.options);

    % Check that we can run the problem.
    try
        problem = checkRun(algorithm, problem, 'linprog');
    catch ME
        throw(ME);
    end

    % Run the algorithm
    [x, fval, exitflag, output, lambda] = run(algorithm, problem);

    % If exitflag is {NaN, }, this means an internal error has been
    % thrown. The internal exit code is held in exitflag{2}.
    if iscell(exitflag) && isnan(exitflag{1})
        handleInternalError(exitflag{2}, 'linprog');
    end

end

output.algorithm = Algorithm;

% Compute constraint violation when x is not empty (interior-point/simplex presolve
% can return empty x).
if computeConstrViolation && ~isempty(x)
    output.constrviolation = max([0; norm(Aeq*x-Beq, inf); (lb-x); (x-ub); (A*x-B)]);
else
    output.constrviolation = [];
end

% Compute first order optimality if needed. This information does not come
% from either qpsub, lipsol, or simplex.
if exitflag ~= -9 && computeFirstOrderOpt && ~isempty(lambda)
    output.firstorderopt = computeKKTErrorForQPLP([],f,A,B,Aeq,Beq,lb,ub,lambda,x);
else
    output.firstorderopt = [];
end

3.运行结果