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.

% 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, <aString>}, 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