1. Brief description
The linprog function is mainly used to find the minimum value problem in linear programming (the mirror problem of the maximum value, you only need to add a "-" to find the maximum value)
2. Algorithm structure and method of use
For problems where the constraints are Ax=b or Ax≤b
2.1 linprog function
x=linprog(f,A,b)
x=linprog(f,A,b,Aeq,beq)
x=linprog(f,A,b,Aeq,beq,lb,ub)
x=linprog(f ,A,b,Aeq,beq,lb,ub,x0)
2.2 Parameter introduction
f: Objective function
A: Inequality constraint matrix
b: Matrix corresponding to the right side of the inequality
Aeq: Equality constraint matrix
beq: Matrix Aeq on the right side of the inequality
Aeq: Equality constraint matrix
beq: Corresponding to the right side of the equation Matrix
lb: the lower bound of x
ub: the upper bound of x
x0: set the initial point x0, this option is only valid for the medium-scale algorithm. The default large-scale and simple algorithms ignore any initial points. (generally not used)
2.3 Commonly used linprog functions and usage examples
The common forms of linprog functions are:
x=linprog(f,A,b,Aep,beq,lb,ub);
Example: Learning objective: The minimum value of the multivariate function with constraints
is suitable for the solution of the most profitable mode of planned production,
The maximum value solution can be transformed into the minimum value solution algorithm, which is very easy
Finding the maximum value is transformed into finding the maximum value of the minimum value f=70*x1+120*x2. Of course, x1 and x2 are constrained.
This translates to finding the minimum of f=-(70*x1+120*x2).
Constraints: 9*x1+4*x2<=3600;4*x1+5*x2<=2000;3*x1+10*x2<=3000;-x1,-x2<
2. Code
main function:
clc
clear
f=[-70 -120];
A=[9 4;4 5;3 10];
B=[3600;2000;3000];
Aeq=[]; Beq=[];
lb=[0 0];ub=[inf inf];
x0=[1 1];
options=optimset('display','iter','Tolx',1e-8);
[x,f,exitflag]=linprog(f,A,B,Aeq,Beq,lb,ub,x0,options)
%[xmincon,fval,exitflag,output] = fmincon(@(x)-(70*x(1)+120*x(2)),x0,A,B,Aeq,Beq,lb,ub,[],options)
Subfunction:
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; degreeconstflag = false;
non_eq=0;non_eq=0; lin_eq = size ( Aeq , 1 ) ; lin_size=size(A,1); XOUT = ones ( nvars , 1 ) ;
funfcn { 1 } = [ ] ; confcn{1}=[];
diagnose('linprog',OUTPUT,degreeflag,hessflag,constflag,degreeconstflag,...
XOUT,non_eq,non_ine,lin_eq,lin_ine,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. Running results
