MATLAB优化函数fmincon解析_fmincon函数原理

MATLAB，优化函数fmincon解析

[x,fval,exitflag,output,lambda,grad,hessian]=fmincon（fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)；                                                                                      

输入参数：fun要求解的函数值；x0函数fun参数值的初始化；

                   参数值的线性不等式约束A,b

                  参数值的等式线性约束Aeq,beq,

                   参数值的上界和下界lb,ub

                   非线性约束nonlcon

输出参数：X输出最优参数值

                   Fval输出fun在X参数的值

                   Exitflag输出fmincon额外条件值

function [X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD,HESSIAN] = fmincon(FUN,X,A,B,Aeq,Beq,LB,UB,NONLCON,options,varargin)
/*fmincon可以在多元函数中找到最小值
   FMINCON attempts to solve problems of the form:
  min F(X)  subject to:  A*X  <= B, Aeq*X  = Beq (linear constraints)线性约束
    X                     C(X) <= 0, Ceq(X) = 0   (nonlinear constraints)非线性约束
                              LB <= X <= UB        (bounds)
   */
 /*FMINCON implements four different algorithms: interior point, SQP,
%   active set, and trust region reflective. Choose one via the option
%   Algorithm: for instance, to choose SQP, set OPTIONS =
%   optimoptions('fmincon','Algorithm','sqp'), and then pass OPTIONS to
%   FMINCON.
	fmincon函数应用四种不同的算法：内点法（interior point）；序列二次规划算法(SQP)；有效集法（active set）；信赖域有效算法（trust region reflective）。
	如果采用SQP算法可以设置 OPTIONS = optimoptions('fmincon','Algorithm','sqp')，再把OPTIONS赋给fmincon
     */  
	/* 
%   X = FMINCON(FUN,X0,A,B) starts at X0 and finds a minimum X to the 
%   function FUN, subject to the linear inequalities A*X <= B. FUN accepts 
%   input X and returns a scalar function value F evaluated at X. X0 may be
%   a scalar, vector, or matrix. 
%
%   X = FMINCON(FUN,X0,A,B,Aeq,Beq) minimizes FUN subject to the linear 
%   equalities Aeq*X = Beq as well as A*X <= B. (Set A=[] and B=[] if no 
%   inequalities exist.)
%
%   X = FMINCON(FUN,X0,A,B,Aeq,Beq,LB,UB) defines a set of lower and upper
%   bounds on the design variables, X, so that a solution is found 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 = FMINCON(FUN,X0,A,B,Aeq,Beq,LB,UB,NONLCON) subjects the minimization
%   to the constraints defined in NONLCON. The function NONLCON accepts X 
%   and returns the vectors C and Ceq, representing the nonlinear 
%   inequalities and equalities respectively. FMINCON minimizes FUN such 
%   that C(X) <= 0 and Ceq(X) = 0. (Set LB = [] and/or UB = [] if no bounds
%   exist.)
%
%   X = FMINCON(FUN,X0,A,B,Aeq,Beq,LB,UB,NONLCON,OPTIONS) minimizes with
%   the default optimization parameters replaced by values in OPTIONS, an
%   argument created with the OPTIMOPTIONS function. See OPTIMOPTIONS for
%   details. For a list of options accepted by FMINCON refer to the
%   documentation.
%  
%   X = FMINCON(PROBLEM) finds the minimum for PROBLEM. PROBLEM is a
%   structure with the function FUN in PROBLEM.objective, the start point
%   in PROBLEM.x0, 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 nonlinear constraint function in PROBLEM.nonlcon, the
%   options structure in PROBLEM.options, and solver name 'fmincon' in
%   PROBLEM.solver. Use this syntax to solve at the command line a problem 
%   exported from OPTIMTOOL. The structure PROBLEM must have all the fields.
%
%   [X,FVAL] = FMINCON(FUN,X0,...) returns the value of the objective 
%   function FUN at the solution X.
%
%   [X,FVAL,EXITFLAG] = FMINCON(FUN,X0,...) returns an EXITFLAG that 
%   describes the exit condition of FMINCON. Possible values of EXITFLAG 
%   and the corresponding exit conditions are listed below. See the
%   documentation for a complete description.
%   */
 
/*
%   All algorithms:
%     1  First order optimality conditions satisfied.
%     0  Too many function evaluations or iterations.
%    -1  Stopped by output/plot function.
%    -2  No feasible point found.
%   Trust-region-reflective, interior-point, and sqp:
%     2  Change in X too small.
%   Trust-region-reflective:
%     3  Change in objective function too small.
%   Active-set only:
%     4  Computed search direction too small.
%     5  Predicted change in objective function too small.
%   Interior-point and sqp:
%    -3  Problem seems unbounded.
		所有算法中EXITFLAG返回值涵义
			1  满足一阶最优性条件
			0	函数计算或迭代太多。无法求解
			-1 被输出和绘图功能阻止
			-2	找不到可行点
			Trust-region-reflective, interior-point, and sqp:三种算法才有的返回值
			2  X变化太小
			Active-set 算法才有的返回值
			4	计算搜索的方向太小
			5	目标函数的预测变化太小。
			Interior-point and sqp才有的
			-3  问题没有边界
		*/
/*
%   [X,FVAL,EXITFLAG,OUTPUT] = FMINCON(FUN,X0,...) returns a structure 
%   OUTPUT with information such as total number of iterations, and final 
%   objective function value. See the documentation for a complete list.
  返回包含迭代总数和最终目标函数值等信息的结构输出
*/
/*
%   [X,FVAL,EXITFLAG,OUTPUT,LAMBDA] = FMINCON(FUN,X0,...) returns the 
%   Lagrange multipliers at the solution X: LAMBDA.lower for LB, 
%   LAMBDA.upper for UB, LAMBDA.ineqlin is for the linear inequalities, 
%   LAMBDA.eqlin is for the linear equalities, LAMBDA.ineqnonlin is for the
%   nonlinear inequalities, and LAMBDA.eqnonlin is for the nonlinear 
%   equalities.
     返回解x处的拉格朗日乘数：lambda.lower表示lb，lambda.upper表示ub，
	 lambda.ineqlin表示线性不等式，lambda.eqlin表示线性等式，
	 lambda.ineqnonlin表示非线性不等式，lambda.eqnonlin表示非线性不等式。
*/
 
/*
%   [X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD] = FMINCON(FUN,X0,...) returns the 
%   value of the gradient of FUN at the solution X.
返回解决方案x的fun渐变值。
%*/
/*  [X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD,HESSIAN] = FMINCON(FUN,X0,...) 
%   returns the value of the exact or approximate Hessian of the Lagrangian
%   at X. 
返回X的朗格朗日精确解或者近似Hessian矩阵
*/
/* Examples
%     FUN can be specified using @:
%        X = fmincon(@humps,...)
%     In this case, F = humps(X) returns the scalar function value F of 
%     the HUMPS function evaluated at X.
%
%     FUN can also be an anonymous function:
%        X = fmincon(@(x) 3*sin(x(1))+exp(x(2)),[1;1],[],[],[],[],[0 0])
%     returns X = [0;0].
%
%   If FUN or NONLCON are parameterized, you can use anonymous functions to
%   capture the problem-dependent parameters. Suppose you want to minimize 
%   the objective given in the function myfun, subject to the nonlinear 
%   constraint mycon, where these two functions are parameterized by their 
%   second argument a1 and a2, respectively. Here myfun and mycon are 
%   MATLAB file functions such as
%
%        function f = myfun(x,a1)      
%        f = x(1)^2 + a1*x(2)^2;       
%                                      
%        function [c,ceq] = mycon(x,a2)
%        c = a2/x(1) - x(2);
%        ceq = [];
%
%   To optimize for specific values of a1 and a2, first assign the values 
%   to these two parameters. Then create two one-argument anonymous 
%   functions that capture the values of a1 and a2, and call myfun and 
%   mycon with two arguments. Finally, pass these anonymous functions to 
%   FMINCON:
%
%        a1 = 2; a2 = 1.5; % define parameters first
%        options = optimoptions('fmincon','Algorithm','interior-point'); % run interior-point algorithm
%        x = fmincon(@(x) myfun(x,a1),[1;2],[],[],[],[],[],[],@(x) mycon(x,a2),options)
%
%   See also OPTIMOPTIONS, OPTIMTOOL, FMINUNC, FMINBND, FMINSEARCH, @, FUNCTION_HANDLE.
 
%   Copyright 1990-2013 The MathWorks, Inc.
*/
defaultopt = struct( ...
    'Algorithm','interior-point', ...
    'AlwaysHonorConstraints','bounds', ...
    'DerivativeCheck','off', ...
    'Diagnostics','off', ...
    'DiffMaxChange',Inf, ...
    'DiffMinChange',0, ...
    'Display','final', ...
    'FinDiffRelStep', [], ...
    'FinDiffType','forward', ...    
    'FunValCheck','off', ...
    'GradConstr','off', ...
    'GradObj','off', ...
    'HessFcn',[], ...
    'Hessian',[], ...    
    'HessMult',[], ...
    'HessPattern','sparse(ones(numberOfVariables))', ...
    'InitBarrierParam',0.1, ...
    'InitTrustRegionRadius','sqrt(numberOfVariables)', ...
    'MaxFunEvals',[], ...
    'MaxIter',[], ...
    'MaxPCGIter','max(1,floor(numberOfVariables/2))', ...
    'MaxProjCGIter','2*(numberOfVariables-numberOfEqualities)', ...    
    'MaxSQPIter','10*max(numberOfVariables,numberOfInequalities+numberOfBounds)', ...
    'ObjectiveLimit',-1e20, ...
    'OutputFcn',[], ...
    'PlotFcns',[], ...
    'PrecondBandWidth',0, ...
    'RelLineSrchBnd',[], ...
    'RelLineSrchBndDuration',1, ...
    'ScaleProblem','none', ...
    'SubproblemAlgorithm','ldl-factorization', ...
    'TolCon',1e-6, ...
    'TolConSQP',1e-6, ...    
    'TolFun',1e-6, ...
    'TolPCG',0.1, ...
    'TolProjCG',1e-2, ...
    'TolProjCGAbs',1e-10, ...
    'TolX',[], ...
    'TypicalX','ones(numberOfVariables,1)', ...
    'UseParallel',false ...
    );
 
% If just 'defaults' passed in, return the default options in X
if nargin==1 && nargout <= 1 && strcmpi(FUN,'defaults')
   X = defaultopt;
   return
end
 
if nargin < 10
    options = [];
    if nargin < 9
        NONLCON = [];
        if nargin < 8
            UB = [];
            if nargin < 7
                LB = [];
                if nargin < 6
                    Beq = [];
                    if nargin < 5
                        Aeq = [];
                    end
                end
            end
        end
    end
end
 
problemInput = false;
if nargin == 1
    if isa(FUN,'struct')
        problemInput = true;
        [FUN,X,A,B,Aeq,Beq,LB,UB,NONLCON,options] = separateOptimStruct(FUN);
    else % Single input and non-structure.
        error(message('optimlib:fmincon:InputArg'));
    end
end
 
% Prepare the options for the solver
[options, optionFeedback] = prepareOptionsForSolver(options, 'fmincon');
 
if nargin < 4 && ~problemInput
  error(message('optimlib:fmincon:AtLeastFourInputs'))
end
 
if isempty(NONLCON) && isempty(A) && isempty(Aeq) && isempty(UB) && isempty(LB)
   error(message('optimlib:fmincon:ConstrainedProblemsOnly'))
end
 
% Check for non-double inputs
msg = isoptimargdbl('FMINCON', {'X0','A','B','Aeq','Beq','LB','UB'}, ...
                                 X,  A,  B,  Aeq,  Beq,  LB,  UB);
if ~isempty(msg)
    error('optimlib:fmincon:NonDoubleInput',msg);
end
 
if nargout > 4
   computeLambda = true;
else 
   computeLambda = false;
end
 
activeSet = 'medium-scale: SQP, Quasi-Newton, line-search';
sqp = 'sequential quadratic programming';
trustRegionReflective = 'trust-region-reflective';
interiorPoint = 'interior-point';
 
[sizes.xRows,sizes.xCols] = size(X);
XOUT = X(:);
sizes.nVar = length(XOUT);
% Check for empty X
if sizes.nVar == 0
   error(message('optimlib:fmincon:EmptyX'));
end
 
display = optimget(options,'Display',defaultopt,'fast');
flags.detailedExitMsg = ~isempty(strfind(display,'detailed'));
switch display
    case {'off','none'}
        verbosity = 0;
    case {'notify','notify-detailed'}
        verbosity = 1;
    case {'final','final-detailed'}
        verbosity = 2;
    case {'iter','iter-detailed'}
        verbosity = 3;
    case 'testing'
        verbosity = 4;
    otherwise
        verbosity = 2;
end
 
% Set linear constraint right hand sides to column vectors
% (in particular, if empty, they will be made the correct
% size, 0-by-1)
B = B(:);
Beq = Beq(:);
 
% Check for consistency of linear constraints, before evaluating
% (potentially expensive) user functions 
 
% Set empty linear constraint matrices to the correct size, 0-by-n
if isempty(Aeq)
    Aeq = reshape(Aeq,0,sizes.nVar);
end
if isempty(A)
    A = reshape(A,0,sizes.nVar);   
end
 
[lin_eq,Aeqcol] = size(Aeq);
[lin_ineq,Acol] = size(A);
% These sizes checks assume that empty matrices have already been made the correct size
if Aeqcol ~= sizes.nVar
   error(message('optimlib:fmincon:WrongNumberOfColumnsInAeq', sizes.nVar))
end
if lin_eq ~= length(Beq)
    error(message('optimlib:fmincon:AeqAndBeqInconsistent'))
end
if Acol ~= sizes.nVar
   error(message('optimlib:fmincon:WrongNumberOfColumnsInA', sizes.nVar))
end
if lin_ineq ~= length(B)
    error(message('optimlib:fmincon:AeqAndBinInconsistent'))
end
% End of linear constraint consistency check
 
Algorithm = optimget(options,'Algorithm',defaultopt,'fast'); 
 
% Option needed for processing initial guess
AlwaysHonorConstraints = optimget(options,'AlwaysHonorConstraints',defaultopt,'fast'); 
 
% Determine algorithm user chose via options. (We need this now
% to set OUTPUT.algorithm in case of early termination due to 
% inconsistent bounds.) 
if strcmpi(Algorithm,'active-set')
    OUTPUT.algorithm = activeSet;
elseif strcmpi(Algorithm,'sqp')
    OUTPUT.algorithm = sqp;
elseif strcmpi(Algorithm,'interior-point')
    OUTPUT.algorithm = interiorPoint;
elseif strcmpi(Algorithm,'trust-region-reflective')
    OUTPUT.algorithm = trustRegionReflective;
else
    error(message('optimlib:fmincon:InvalidAlgorithm'));
end    
 
[XOUT,l,u,msg] = checkbounds(XOUT,LB,UB,sizes.nVar);
if ~isempty(msg)
   EXITFLAG = -2;
   [FVAL,LAMBDA,GRAD,HESSIAN] = deal([]);
   
   OUTPUT.iterations = 0;
   OUTPUT.funcCount = 0;
   OUTPUT.stepsize = [];
   if strcmpi(OUTPUT.algorithm,activeSet) || strcmpi(OUTPUT.algorithm,sqp)
       OUTPUT.lssteplength = [];
   else % trust-region-reflective, interior-point
       OUTPUT.cgiterations = [];
   end
   if strcmpi(OUTPUT.algorithm,interiorPoint) || strcmpi(OUTPUT.algorithm,activeSet) || ...
      strcmpi(OUTPUT.algorithm,sqp)
       OUTPUT.constrviolation = [];
   end
   OUTPUT.firstorderopt = [];
   OUTPUT.message = msg;
   
   X(:) = XOUT;
   if verbosity > 0
      disp(msg)
   end
   return
end
 
% Get logical list of finite lower and upper bounds
finDiffFlags.hasLBs = isfinite(l);
finDiffFlags.hasUBs = isfinite(u);
 
lFinite = l(finDiffFlags.hasLBs);
uFinite = u(finDiffFlags.hasUBs);
 
% Create structure of flags and initial values, initialize merit function
% type and the original shape of X.
flags.meritFunction = 0;
initVals.xOrigShape = X;
 
diagnostics = strcmpi(optimget(options,'Diagnostics',defaultopt,'fast'),'on');
funValCheck = strcmpi(optimget(options,'FunValCheck',defaultopt,'fast'),'on');
derivativeCheck = strcmpi(optimget(options,'DerivativeCheck',defaultopt,'fast'),'on');
 
% Gather options needed for finitedifferences
% Write checked DiffMaxChange, DiffMinChage, FinDiffType, FinDiffRelStep,
% GradObj and GradConstr options back into struct for later use
options.DiffMinChange = optimget(options,'DiffMinChange',defaultopt,'fast');
options.DiffMaxChange = optimget(options,'DiffMaxChange',defaultopt,'fast');
if options.DiffMinChange >= options.DiffMaxChange
    error(message('optimlib:fmincon:DiffChangesInconsistent', sprintf( '%0.5g', options.DiffMinChange ), sprintf( '%0.5g', options.DiffMaxChange )))
end
% Read in and error check option TypicalX
[typicalx,ME] = getNumericOrStringFieldValue('TypicalX','ones(numberOfVariables,1)', ...
    ones(sizes.nVar,1),'a numeric value',options,defaultopt);
if ~isempty(ME)
    throw(ME)
end
checkoptionsize('TypicalX', size(typicalx), sizes.nVar);
options.TypicalX = typicalx;
options.FinDiffType = optimget(options,'FinDiffType',defaultopt,'fast');
options = validateFinDiffRelStep(sizes.nVar,options,defaultopt);
options.GradObj = optimget(options,'GradObj',defaultopt,'fast');
options.GradConstr = optimget(options,'GradConstr',defaultopt,'fast');
 
flags.grad = strcmpi(options.GradObj,'on');
 
% Notice that defaultopt.Hessian = [], so the variable "hessian" can be empty
hessian = optimget(options,'Hessian',defaultopt,'fast'); 
% If calling trust-region-reflective with an unavailable Hessian option value, 
% issue informative error message
if strcmpi(OUTPUT.algorithm,trustRegionReflective) && ...
        ~( isempty(hessian) || strcmpi(hessian,'on') || strcmpi(hessian,'user-supplied') || ...
           strcmpi(hessian,'off') || strcmpi(hessian,'fin-diff-grads')  )
    error(message('optimlib:fmincon:BadTRReflectHessianValue'))
end
 
if ~iscell(hessian) && ( strcmpi(hessian,'user-supplied') || strcmpi(hessian,'on') )
    flags.hess = true;
else
    flags.hess = false;
end
 
if isempty(NONLCON)
   flags.constr = false;
else
   flags.constr = true;
end
 
% Process objective function
if ~isempty(FUN)  % will detect empty string, empty matrix, empty cell array
   % constrflag in optimfcnchk set to false because we're checking the objective, not constraint
   funfcn = optimfcnchk(FUN,'fmincon',length(varargin),funValCheck,flags.grad,flags.hess,false,Algorithm);
else
   error(message('optimlib:fmincon:InvalidFUN'));
end
% Process constraint function
if flags.constr % NONLCON is non-empty
   flags.gradconst = strcmpi(options.GradConstr,'on');
   % hessflag in optimfcnchk set to false because hessian is never returned by nonlinear constraint 
   % function
   %
   % constrflag in optimfcnchk set to true because we're checking the constraints
   confcn = optimfcnchk(NONLCON,'fmincon',length(varargin),funValCheck,flags.gradconst,false,true);
else
   flags.gradconst = false; 
   confcn = {'','','','',''};
end
 
[rowAeq,colAeq] = size(Aeq);
 
if strcmpi(OUTPUT.algorithm,activeSet) || strcmpi(OUTPUT.algorithm,sqp)
    % See if linear constraints are sparse and if user passed in Hessian
    if issparse(Aeq) || issparse(A)
        warning(message('optimlib:fmincon:ConvertingToFull', Algorithm))
    end
    if flags.hess % conflicting options
        flags.hess = false;
        warning(message('optimlib:fmincon:HessianIgnoredForAlg', Algorithm));
        if strcmpi(funfcn{1},'fungradhess')
            funfcn{1}='fungrad';
        elseif  strcmpi(funfcn{1},'fun_then_grad_then_hess')
            funfcn{1}='fun_then_grad';
        end
    end
elseif strcmpi(OUTPUT.algorithm,trustRegionReflective)
    % Look at constraint type and supplied derivatives, and determine if
    % trust-region-reflective can solve problem
    isBoundedNLP = isempty(NONLCON) && isempty(A) && isempty(Aeq); % problem has only bounds and no other constraints 
    isLinEqNLP = isempty(NONLCON) && isempty(A) && isempty(lFinite) ...
        && isempty(uFinite) && colAeq > rowAeq;
    if isBoundedNLP && flags.grad
        % if only l and u then call sfminbx
    elseif isLinEqNLP && flags.grad
        % if only Aeq beq and Aeq has more columns than rows, then call sfminle
    else
        if ~isBoundedNLP && ~isLinEqNLP
            error(message('optimlib:fmincon:ConstrTRR', ...
                addLink( 'Choosing the Algorithm', 'choose_algorithm' )))            
        else
            % The user has a problem that satisfies the TRR constraint
            % restrictions but they haven't supplied gradients.
            error(message('optimlib:fmincon:GradOffTRR', ...
                addLink( 'Choosing the Algorithm', 'choose_algorithm' )))
        end
    end
end
lenvlb = length(l);
lenvub = length(u);
% Process initial point 
shiftedX0 = false; % boolean that indicates if initial point was shifted
if strcmpi(OUTPUT.algorithm,activeSet)
   %
   % Ensure starting point lies within bounds
   %
   i=1:lenvlb;
   lindex = XOUT(i)<l(i);
   if any(lindex)
      XOUT(lindex)=l(lindex); 
      shiftedX0 = true;
   end
   i=1:lenvub;
   uindex = XOUT(i)>u(i);
   if any(uindex)
      XOUT(uindex)=u(uindex);
      shiftedX0 = true;
   end
   X(:) = XOUT;
elseif strcmpi(OUTPUT.algorithm,trustRegionReflective)
   %
   % If components of initial x not within bounds, set those components  
   % of initial point to a "box-centered" point
   %
   if isempty(Aeq)
       arg = (u >= 1e10); arg2 = (l <= -1e10);
       u(arg) = inf;
       l(arg2) = -inf;
       xinitOutOfBounds_idx = XOUT < l | XOUT > u;
       if any(xinitOutOfBounds_idx)
           shiftedX0 = true;
           XOUT = startx(u,l,XOUT,xinitOutOfBounds_idx);
           X(:) = XOUT;
       end
   else
      % Phase-1 for sfminle nearest feas. pt. to XOUT. Don't print a
      % message for this change in X0 for sfminle. 
       XOUT = feasibl(Aeq,Beq,XOUT);
       X(:) = XOUT;
   end
 
elseif strcmpi(OUTPUT.algorithm,interiorPoint)
    % Variables: fixed, finite lower bounds, finite upper bounds
    xIndices = classifyBoundsOnVars(l,u,sizes.nVar,true);
 
    % If honor bounds mode, then check that initial point strictly satisfies the
    % simple inequality bounds on the variables and exactly satisfies fixed variable
    % bounds.
    if strcmpi(AlwaysHonorConstraints,'bounds') || strcmpi(AlwaysHonorConstraints,'bounds-ineqs')
        violatedFixedBnds_idx = XOUT(xIndices.fixed) ~= l(xIndices.fixed);
        violatedLowerBnds_idx = XOUT(xIndices.finiteLb) <= l(xIndices.finiteLb);
        violatedUpperBnds_idx = XOUT(xIndices.finiteUb) >= u(xIndices.finiteUb);
        if any(violatedLowerBnds_idx) || any(violatedUpperBnds_idx) || any(violatedFixedBnds_idx)
            XOUT = shiftInitPtToInterior(sizes.nVar,XOUT,l,u,Inf);
            X(:) = XOUT;
            shiftedX0 = true;
        end
    end
else % SQP
    % Classify variables: finite lower bounds, finite upper bounds
    xIndices = classifyBoundsOnVars(l,u,sizes.nVar,false);
    
    % SQP always honors the bounds. Check that initial point
    % strictly satisfies the bounds on the variables.
    violatedLowerBnds_idx = XOUT(xIndices.finiteLb) < l(xIndices.finiteLb);
    violatedUpperBnds_idx = XOUT(xIndices.finiteUb) > u(xIndices.finiteUb);
    if any(violatedLowerBnds_idx) || any(violatedUpperBnds_idx)
        finiteLbIdx = find(xIndices.finiteLb);
        finiteUbIdx = find(xIndices.finiteUb);
        XOUT(finiteLbIdx(violatedLowerBnds_idx)) = l(finiteLbIdx(violatedLowerBnds_idx));
        XOUT(finiteUbIdx(violatedUpperBnds_idx)) = u(finiteUbIdx(violatedUpperBnds_idx));
        X(:) = XOUT;
        shiftedX0 = true;
    end
end
 
% Display that x0 was shifted in order to honor bounds
if shiftedX0
    if verbosity >= 3
        if strcmpi(OUTPUT.algorithm,interiorPoint) 
            fprintf(getString(message('optimlib:fmincon:ShiftX0StrictInterior')));
            fprintf('\n');
        else
            fprintf(getString(message('optimlib:fmincon:ShiftX0ToBnds')));
            fprintf('\n');
        end
    end
end
    
% Evaluate function
initVals.g = zeros(sizes.nVar,1);
HESSIAN = []; 
 
switch funfcn{1}
case 'fun'
   try
      initVals.f = feval(funfcn{3},X,varargin{:});
   catch userFcn_ME
        optim_ME = MException('optimlib:fmincon:ObjectiveError', ...
            getString(message('optimlib:fmincon:ObjectiveError')));
        userFcn_ME = addCause(userFcn_ME,optim_ME);
        rethrow(userFcn_ME)
   end
case 'fungrad'
   try
      [initVals.f,initVals.g] = feval(funfcn{3},X,varargin{:});
   catch userFcn_ME
        optim_ME = MException('optimlib:fmincon:ObjectiveError', ...
            getString(message('optimlib:fmincon:ObjectiveError')));
        userFcn_ME = addCause(userFcn_ME,optim_ME);
        rethrow(userFcn_ME)
   end
case 'fungradhess'
   try
      [initVals.f,initVals.g,HESSIAN] = feval(funfcn{3},X,varargin{:});
   catch userFcn_ME
        optim_ME = MException('optimlib:fmincon:ObjectiveError', ...
            getString(message('optimlib:fmincon:ObjectiveError')));
        userFcn_ME = addCause(userFcn_ME,optim_ME);
        rethrow(userFcn_ME)
   end
case 'fun_then_grad'
   try
      initVals.f = feval(funfcn{3},X,varargin{:});
   catch userFcn_ME
        optim_ME = MException('optimlib:fmincon:ObjectiveError', ...
            getString(message('optimlib:fmincon:ObjectiveError')));
        userFcn_ME = addCause(userFcn_ME,optim_ME);
        rethrow(userFcn_ME)
   end
   try
      initVals.g = feval(funfcn{4},X,varargin{:});
   catch userFcn_ME
        optim_ME = MException('optimlib:fmincon:GradientError', ...
            getString(message('optimlib:fmincon:GradientError')));
        userFcn_ME = addCause(userFcn_ME,optim_ME);
        rethrow(userFcn_ME)
   end
case 'fun_then_grad_then_hess'
   try
      initVals.f = feval(funfcn{3},X,varargin{:});
   catch userFcn_ME
        optim_ME = MException('optimlib:fmincon:ObjectiveError', ...
            getString(message('optimlib:fmincon:ObjectiveError')));
        userFcn_ME = addCause(userFcn_ME,optim_ME);
        rethrow(userFcn_ME)
   end
   try
      initVals.g = feval(funfcn{4},X,varargin{:});
   catch userFcn_ME
        optim_ME = MException('optimlib:fmincon:GradientError', ...
            getString(message('optimlib:fmincon:GradientError')));
        userFcn_ME = addCause(userFcn_ME,optim_ME);
        rethrow(userFcn_ME)
   end
   try
      HESSIAN = feval(funfcn{5},X,varargin{:});
   catch userFcn_ME
        optim_ME = MException('optimlib:fmincon:HessianError', ...
            getString(message('optimlib:fmincon:HessianError')));            
        userFcn_ME = addCause(userFcn_ME,optim_ME);
        rethrow(userFcn_ME)
   end
otherwise
   error(message('optimlib:fmincon:UndefinedCallType'));
end
 
% Check that the objective value is a scalar
if numel(initVals.f) ~= 1
   error(message('optimlib:fmincon:NonScalarObj'))
end
 
% Check that the objective gradient is the right size
initVals.g = initVals.g(:);
if numel(initVals.g) ~= sizes.nVar
   error('optimlib:fmincon:InvalidSizeOfGradient', ...
       getString(message('optimlib:commonMsgs:InvalidSizeOfGradient',sizes.nVar)));
end
 
% Evaluate constraints
switch confcn{1}
case 'fun'
    try
        [ctmp,ceqtmp] = feval(confcn{3},X,varargin{:});
    catch userFcn_ME
        if strcmpi('MATLAB:maxlhs',userFcn_ME.identifier)
                error(message('optimlib:fmincon:InvalidHandleNonlcon'))
        else
            optim_ME = MException('optimlib:fmincon:NonlconError', ...
                getString(message('optimlib:fmincon:NonlconError')));
            userFcn_ME = addCause(userFcn_ME,optim_ME);
            rethrow(userFcn_ME)
        end
    end
    initVals.ncineq = ctmp(:);
    initVals.nceq = ceqtmp(:);
    initVals.gnc = zeros(sizes.nVar,length(initVals.ncineq));
    initVals.gnceq = zeros(sizes.nVar,length(initVals.nceq));
case 'fungrad'
   try
      [ctmp,ceqtmp,initVals.gnc,initVals.gnceq] = feval(confcn{3},X,varargin{:});
   catch userFcn_ME
       optim_ME = MException('optimlib:fmincon:NonlconError', ...
           getString(message('optimlib:fmincon:NonlconError')));           
       userFcn_ME = addCause(userFcn_ME,optim_ME);
       rethrow(userFcn_ME)
   end
   initVals.ncineq = ctmp(:);
   initVals.nceq = ceqtmp(:);
case 'fun_then_grad'
    try
        [ctmp,ceqtmp] = feval(confcn{3},X,varargin{:});
    catch userFcn_ME
        optim_ME = MException('optimlib:fmincon:NonlconError', ...
            getString(message('optimlib:fmincon:NonlconError')));
        userFcn_ME = addCause(userFcn_ME,optim_ME);
        rethrow(userFcn_ME)
    end
    initVals.ncineq = ctmp(:);
    initVals.nceq = ceqtmp(:);
    try
        [initVals.gnc,initVals.gnceq] = feval(confcn{4},X,varargin{:});
    catch userFcn_ME
        optim_ME = MException('optimlib:fmincon:NonlconFunOrGradError', ...
            getString(message('optimlib:fmincon:NonlconFunOrGradError')));
        userFcn_ME = addCause(userFcn_ME,optim_ME);
        rethrow(userFcn_ME)
    end
case ''
   % No nonlinear constraints. Reshaping of empty quantities is done later
   % in this file, where both cases, (i) no nonlinear constraints and (ii)
   % nonlinear constraints that have one type missing (equalities or
   % inequalities), are handled in one place
   initVals.ncineq = [];
   initVals.nceq = [];
   initVals.gnc = [];
   initVals.gnceq = [];
otherwise
   error(message('optimlib:fmincon:UndefinedCallType'));
end
 
% Check for non-double data typed values returned by user functions 
if ~isempty( isoptimargdbl('FMINCON', {'f','g','H','c','ceq','gc','gceq'}, ...
   initVals.f, initVals.g, HESSIAN, initVals.ncineq, initVals.nceq, initVals.gnc, initVals.gnceq) )
    error('optimlib:fmincon:NonDoubleFunVal',getString(message('optimlib:commonMsgs:NonDoubleFunVal','FMINCON')));
end
 
sizes.mNonlinEq = length(initVals.nceq);
sizes.mNonlinIneq = length(initVals.ncineq);
 
% Make sure empty constraint and their derivatives have correct sizes (not 0-by-0):
if isempty(initVals.ncineq)
    initVals.ncineq = reshape(initVals.ncineq,0,1);
end
if isempty(initVals.nceq)
    initVals.nceq = reshape(initVals.nceq,0,1);
end
if isempty(initVals.gnc)
    initVals.gnc = reshape(initVals.gnc,sizes.nVar,0);
end
if isempty(initVals.gnceq)
    initVals.gnceq = reshape(initVals.gnceq,sizes.nVar,0);
end
[cgrow,cgcol] = size(initVals.gnc);
[ceqgrow,ceqgcol] = size(initVals.gnceq);
 
if cgrow ~= sizes.nVar || cgcol ~= sizes.mNonlinIneq
   error(message('optimlib:fmincon:WrongSizeGradNonlinIneq', sizes.nVar, sizes.mNonlinIneq))
end
if ceqgrow ~= sizes.nVar || ceqgcol ~= sizes.mNonlinEq
   error(message('optimlib:fmincon:WrongSizeGradNonlinEq', sizes.nVar, sizes.mNonlinEq))
end
 
if diagnostics
   % Do diagnostics on information so far
   diagnose('fmincon',OUTPUT,flags.grad,flags.hess,flags.constr,flags.gradconst,...
      XOUT,sizes.mNonlinEq,sizes.mNonlinIneq,lin_eq,lin_ineq,l,u,funfcn,confcn);
end
 
% Create default structure of flags for finitedifferences:
% This structure will (temporarily) ignore some of the features that are
% algorithm-specific (e.g. scaling and fault-tolerance) and can be turned
% on later for the main algorithm.
finDiffFlags.fwdFinDiff = strcmpi(options.FinDiffType,'forward');
finDiffFlags.scaleObjConstr = false; % No scaling for now
finDiffFlags.chkFunEval = false;     % No fault-tolerance yet
finDiffFlags.chkComplexObj = false;  % No need to check for complex values
finDiffFlags.isGrad = true;          % Scalar objective
 
% Check derivatives
if derivativeCheck && ...               % User wants to check derivatives...
   (flags.grad || ...                   % of either objective or ...
   flags.gradconst && sizes.mNonlinEq+sizes.mNonlinIneq > 0) % nonlinear constraint function.
    validateFirstDerivatives(funfcn,confcn,X, ...
        l,u,options,finDiffFlags,sizes,varargin{:});
end
 
% call algorithm
if strcmpi(OUTPUT.algorithm,activeSet) % active-set
    defaultopt.MaxIter = 400; defaultopt.MaxFunEvals = '100*numberofvariables'; defaultopt.TolX = 1e-6;
    defaultopt.Hessian = 'off';
    problemInfo = []; % No problem related data
    [X,FVAL,LAMBDA,EXITFLAG,OUTPUT,GRAD,HESSIAN]=...
        nlconst(funfcn,X,l,u,full(A),B,full(Aeq),Beq,confcn,options,defaultopt, ...
        finDiffFlags,verbosity,flags,initVals,problemInfo,optionFeedback,varargin{:});
elseif strcmpi(OUTPUT.algorithm,trustRegionReflective) % trust-region-reflective
   if (strcmpi(funfcn{1}, 'fun_then_grad_then_hess') || strcmpi(funfcn{1}, 'fungradhess'))
      Hstr = [];
   elseif (strcmpi(funfcn{1}, 'fun_then_grad') || strcmpi(funfcn{1}, 'fungrad'))
      n = length(XOUT); 
      Hstr = optimget(options,'HessPattern',defaultopt,'fast');
      if ischar(Hstr) 
         if strcmpi(Hstr,'sparse(ones(numberofvariables))')
            Hstr = sparse(ones(n));
         else
            error(message('optimlib:fmincon:InvalidHessPattern'))
         end
      end
      checkoptionsize('HessPattern', size(Hstr), n);
   end
   
   defaultopt.MaxIter = 400; defaultopt.MaxFunEvals = '100*numberofvariables'; defaultopt.TolX = 1e-6;
   defaultopt.Hessian = 'off';
   % Trust-region-reflective algorithm does not compute constraint
   % violation as it progresses. If the user requests the output structure,
   % we need to calculate the constraint violation at the returned
   % solution.
   if nargout > 3
       computeConstrViolForOutput = true;
   else
       computeConstrViolForOutput = false;
   end
 
   if isempty(Aeq)
      [X,FVAL,LAMBDA,EXITFLAG,OUTPUT,GRAD,HESSIAN] = ...
         sfminbx(funfcn,X,l,u,verbosity,options,defaultopt,computeLambda,initVals.f,initVals.g, ...
         HESSIAN,Hstr,flags.detailedExitMsg,computeConstrViolForOutput,optionFeedback,varargin{:});
   else
      [X,FVAL,LAMBDA,EXITFLAG,OUTPUT,GRAD,HESSIAN] = ...
         sfminle(funfcn,X,sparse(Aeq),Beq,verbosity,options,defaultopt,computeLambda,initVals.f, ...
         initVals.g,HESSIAN,Hstr,flags.detailedExitMsg,computeConstrViolForOutput,optionFeedback,varargin{:});
   end
elseif strcmpi(OUTPUT.algorithm,interiorPoint)
    defaultopt.MaxIter = 1000; defaultopt.MaxFunEvals = 3000; defaultopt.TolX = 1e-10;
    defaultopt.Hessian = 'bfgs';
    mEq = lin_eq + sizes.mNonlinEq + nnz(xIndices.fixed); % number of equalities
    % Interior-point-specific options. Default values for lbfgs memory is 10, and 
    % ldl pivot threshold is 0.01
    options = getIpOptions(options,sizes.nVar,mEq,flags.constr,defaultopt,10,0.01); 
 
    [X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD,HESSIAN] = barrier(funfcn,X,A,B,Aeq,Beq,l,u,confcn,options.HessFcn, ...
        initVals.f,initVals.g,initVals.ncineq,initVals.nceq,initVals.gnc,initVals.gnceq,HESSIAN, ...
        xIndices,options,optionFeedback,finDiffFlags,varargin{:});
else % sqp
    defaultopt.MaxIter = 400; defaultopt.MaxFunEvals = '100*numberofvariables'; 
    defaultopt.TolX = 1e-6; defaultopt.Hessian = 'bfgs';
    % Validate options used by sqp
    options = getSQPOptions(options,defaultopt,sizes.nVar);
    optionFeedback.detailedExitMsg = flags.detailedExitMsg;
    % Call algorithm
    [X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD,HESSIAN] = sqpLineSearch(funfcn,X,full(A),full(B),full(Aeq),full(Beq), ...
        full(l),full(u),confcn,initVals.f,full(initVals.g),full(initVals.ncineq),full(initVals.nceq), ...
        full(initVals.gnc),full(initVals.gnceq),xIndices,options,finDiffFlags,verbosity,optionFeedback,varargin{:});
end