prima.m

function [x, fx, exitflag, output] = prima(varargin)
%PRIMA is a package for solving the following generic continuous
%   optimization problem without using derivatives:
%
%   minimize    fun(x)
%       s.t.    Aineq * x <= bineq,
%               Aeq * x = beq,
%               lb <= x <= ub,
%               cineq(x) <= 0,
%               ceq(x) = 0.
%
%   In the backend, PRIMA calls the late Professor M. J. D. Powell's algorithms
%   UOBYQA, NEWUOA, BOBYQA, LINCOA, and COBYLA.
%
%   The interface of PRIMA is the same as that of function FMINCON included
%   in the Optimization Toolbox of MATLAB. So PRIMA can be called in the same
%   way as calling FMINCON. In addition, PRIMA can be called in some more
%   flexible ways that are not allowed by FMINCON.
%
%   1. Basic syntax
%
%   The same as FMINCON, the command
%
%   x = prima(fun, x0, Aineq, bineq, Aeq, beq, lb, ub, nonlcon)
%
%   solves the problem formulated above, where
%
%   *** fun is the name or function handle of the objective function; if
%       there is no objective function (i.e., we have a feasibility problem),
%       then set fun = []
%   *** x0 is the starting point; x0 CANNOT be omitted or set to []
%   *** Aineq and bineq are the coefficient matrix and right-hand side of
%       the linear inequality constraint Aineq * x <= bineq; if there is
%       no such constraint, set Aineq = [], bineq = []
%   *** Aeq and beq are the coefficient matrix and right-hand side of the
%       linear equality constraint Aeq * x = beq; if there is no such
%       constraint, set Aeq = [], beq = []
%   *** lb and ub, which are vectors of the same length as x, are the
%       lower and upper bound in the bound constraint lb <= x <= ub;
%       set lb = [] if no lower bound, and ub = [] if no upper bound
%   *** nonlcon is a function that has 1 input x and 2 outputs [cineq, ceq];
%       it calculates cineq(x) and ceq(x) for any given x; if the first
%       output of nonlcon is [], then there is no inequality constraint
%       cineq(x) <= 0; if the second output of nonlcon is [], then there
%       is no equality constraint ceq(x) = 0. If there is no nonlinear
%       constraint, set nonlcon = []
%
%   The function can also be called with more outputs, e.g.,
%
%   [x, fx, exitflag, output] = prima(INPUTS)
%
%   See "3. Outputs" below for explanations on these outputs.
%
%   2. Flexible syntax
%
%   x = prima(fun, x0) solves
%       minimize fun(x)
%   x = prima(fun, x0, Aineq, bineq) solves
%       minimize fun(x) s.t. Aineq * x <= bineq
%   x = prima(fun, x0, Aineq, bineq, Aeq, beq) solves
%       minimize fun(x) s.t. Aineq * x <= bineq, Aeq * x = beq
%   x = prima(fun, x0, Aineq, bineq, Aeq, beq, lb) solves
%       minimize fun(x) s.t. Aineq * x <= bineq, Aeq * x = beq, lb <= x
%   x = prima(fun, x0, Aineq, bineq, Aeq, beq, lb, ub) solves
%       minimize fun(x) s.t. Aineq * x <= bineq, Aeq * x = beq, lb <= x <= ub
%   x = prima(fun, x0, nonlcon) solves
%       minimize fun(x) s.t. cineq(x) <= 0, ceq(x) = 0
%   x = prima(fun, x0, Aineq, bineq, nonlcon) solves
%       minimize fun(x) s.t. Aineq * x <= bineq, cineq(x) <= 0, ceq(x) = 0
%   x = prima(fun, x0, Aineq, bineq, Aeq, beq, nonlcon) solves
%       minimize fun(x) s.t. Aineq * x <= bineq, Aeq * x = beq, cineq(x) <= 0, ceq(x) = 0
%   x = prima(fun, x0, Aineq, bineq, Aeq, beq, lb, nonlcon) solves
%       minimize fun(x) s.t. Aineq * x <= bineq, Aeq * x = beq, lb <= x, cineq(x) <= 0, ceq(x) = 0
%
%   information = prima(request) returns information about the package
%       according to the information-requesting string "request", which can
%       be 'about', 'author', 'email', 'url', 'maintainer', 'credits',
%       'copyright', 'license', 'version', 'date', 'status', 'message',
%       or 'information'.
%
%   3. Outputs
%
%   *** x is the approximate solution to the optimization problem
%   *** fx is fun(x)
%   *** exitflag is an integer indicating why PRIMA or its backend solver returns;
%       the possible values are
%       0: the lower bound for the trust region radius is reached
%       1: the target function value is achieved
%       2: a trust region step failed to reduce the quadratic model (possible only in classical mode)
%       3: the objective function has been evaluated maxfun times
%       7: rounding errors become severe in the Fortran code
%       8: a linear constraint has zero gradient
%       13: all variables are fixed by the constraints
%       14: a linear feasibility problem received and solved
%       15: a linear feasibility problem received but not solved
%       20: the trust region iteration has been performed for 2*maxfun times
%       -1: NaN occurs in x (possible only in the classical mode)
%       -2: the objective/constraint function returns an Inf/NaN value (possible only in classical
%       mode)
%       -3: NaN occurs in the models (possible only in classical mode)
%       -4: constraints are infeasible
%   *** output is a structure with the following fields:
%       funcCount: number of function evaluations
%       nlcineq: cineq(x) (if there is nonlcon)
%       nlceq: ceq(x) (if there is nonlcon)
%       constrviolation: constrviolation of x (if problem is constrained)
%       xhist: history of iterates (if options.output_xhist = true)
%       fhist: history of function values
%       chist: history of constraint violations
%       nlcihist: history of nonlinear inequality constraint values (if
%       options.output_nlchist = true)
%       nlcehist: history of nonlinear equality constraint values (if
%       options.output_nlchist = true)
%       solver: backend solver that does the computation
%       message: return message
%       warnings: a cell array that records all the warnings raised
%       during the computation
%
%   4. Options
%
%   The same as FMINCON, PRIMA accepts options passed by a structure.
%   Such a structure should be passed as an additional input appended to
%   the end of the input list in the basic syntax or the flexible syntax.
%
%   The options include
%   *** maxfun: maximal number of function evaluations; default: 500*length(x0)
%   *** ftarget: target function value; default: -Inf
%   *** ctol (only if classical = false; see below): tolerance for the constraint
%       validation for constrained problems; default: machine epsilon
%   *** rhobeg: initial trust region radius; typically, rhobeg should be in
%       the order of one tenth of the greatest expected change to a variable;
%       rhobeg should be positive; default: 1 if the problem is not scaled
%       (but min(1, min(ub-lb)/4) if the solver is BOBYQA), 0.5 if the problem
%       is scaled
%   *** rhoend: final trust region radius; rhoend reflects the precision
%       of the approximate solution obtained by PRIMA; rhoend should be
%       positive and not larger than rhobeg; default: 1e-6
%   *** npt: (only for NEWUOA, BOBYQA, LINCOA) number of interpolation
%       points for constructing a model; default: 2*length(x0)+1
%   *** solver: a string indicating which solver to use; possible values are:
%       'uobyqa', 'newuoa' (for unconstrained problems),
%       'bobyqa' (for bound-constrained or unconstrained problems),
%       'lincoa' (for linearly-constrained or bound-constrained or
%       unconstrained problems),
%       'cobyla' (for general constrained or unconstrained problems)
%   *** fortran: a boolean value indicating whether to call Fortran code or
%       not; default: true
%   *** classical: a boolean value indicating whether to call the classical
%       version of Powell's Fortran code or not; default: false
%   *** eta1, eta2, gamma1, gamma2 (only if classical = false)
%       eta1, eta2, gamma1, and gamma2 are parameters in the updating scheme
%       of the trust region radius. Roughly speaking, the trust region radius
%       is contracted by a factor of gamma1 when the reduction ratio is below
%       eta1, and  enlarged by a factor of gamma2 when the reduction ratio is
%       above eta2. It is required that 0 <= eta1 <= eta2 < 1 and
%       0 < gamma1 < 1 <= gamma2. Normally, eta1 <= 0.25. It is not recommended
%       to set eta1 >= 0.5. Default: eta1 = 0.1, eta2 = 0.7, gamma1 = 0.5,
%       and gamma2 = 2.
%   *** scale: (only for BOBYQA, LINCOA, and COBYLA) a boolean value
%       indicating whether to scale the problem according to bounds or not;
%       default: false; if the problem is to be scaled, then rhobeg and rhoend
%       mentioned above will be used as the initial and final trust region
%       radii for the scaled problem
%   *** honour_x0: (only for BOBYQA) a boolean value indicating whether to
%       respect the user-defined x0 or not; default: false if the user provides
%       a rhobeg in (0, Inf), and true otherwise.
%   *** iprint: a flag deciding how much information will be printed during
%       the computation; possible values are value 0 (default), 1, -1, 2,
%       -2, 3, or -3.
%       0: there will be no printing; this is the default;
%       1: a message will be printed to the screen at the return, showing
%          the best vector of variables found and its objective function value;
%       2: in addition to 1, at each "new stage" of the computation, a message
%          is printed to the screen with the best vector of variables so far
%          and its objective function value;
%       3: in addition to 2, each function evaluation with its variables will
%          be printed to the screen;
%       -1, -2, -3: the same information as 1, 2, 3 will be printed, not to
%          the screen but to a file named SOLVER_output.txt; the file will be
%          created if it does not exist; the new output will be appended to
%          the end of this file if it already exists.
%       N.B.:
%          iprint = +/-3 can be costly in terms of time and/or space.
%          When quiet = true (see below), setting iprint = 1, 2, or 3 is
%          the same as setting it to -1, -2, or -3, respectively.
%          When classical = true, only iprint = 0 is supported.
%   *** quiet: a boolean value indicating whether to keep quiet or not;
%       if this flag is set to false or not set, then it affects nothing;
%       if it is set to true and iprint = 1, 2, or 3, the effect is the
%       same as setting iprint to -1, -2, or -3, respectively; default: true
%   *** maxhist: a nonnegative integer controlling how much history will
%       be included in the output structure; default: maxfun
%       *******************************************************************
%       IMPORTANT NOTICE:
%       If maxhist is so large that recording the history takes too much memory,
%       the Fortran code will reset maxhist to a smaller value. The maximal
%       amount of memory defined the Fortran code is 2GB.
%       Let L = length(x) + 2*(number of nonlinear constraints). Assume
%       that maxfun <= 500*L. Then any problem with L <= 400 is not affected.
%       *******************************************************************
%   *** output_xhist: a boolean value indicating whether to output the
%       history of the iterates; if it is set to true, then the output
%       structure will include a field "xhist", which contains the last
%       maxhist iterates of the algorithm; default: false
%   *** output_nlchist: a boolean value indicating whether to output the
%       history of the function values; if it is set to true; then the
%       output structure will include fields "nlcihist" and "nlcehist",
%       which respectively contain the inequality and equality constraint
%       values of the last maxhist iterates of the algorithm; default: false
%   *** maxfilt: a nonnegative integer indicating the maximal length of the
%       "filter" used for selecting the returned solution; default: 2000
%   *** debug: a boolean value indicating whether to debug or not; default: false
%   *** chkfunval: a boolean value indicating whether to verify the returned
%       function and constraint (if applicable) values or not; default: false
%       (if it is true, PRIMA will check whether the returned values of fun and
%       nonlcon match fun(x) and nonlcon(x) or not, which costs function/constraint
%       evaluations; designed only for debugging)
%
%   For example, the following code
%
%   options = struct();
%   options.maxfun = 50;
%   x = prima(@cos, -1, 2, 3, options);
%
%   solves
%       min cos(x) s.t. 2 * x <= 3
%   starting from x0 = -1 with at most 50 function evaluations.
%
%   5. Problem defined by a structure
%
%   The same as FMINCON, a problem can be passed to PRIMA by a structure
%   PROBLEM containing the following fields:
%   PROBLEM.objective, PROBLEM.x0, PROBLEM.Aineq, PROBLEM.bineq,
%   PROBLEM.Aeq, PROBLEM.beq, PROBLEM.lb, PROBLEM.ub, PROBLEM.nonlcon,
%   PROBLEM.options, where PROBLEM.objective is the function name or
%   function handle of the objective function (corresponding to the input
%   'fun' mentioned above), and all the other fields correspond to the
%   inputs introduced above with the same names. The backend solver can
%   be indicated by either PROBLEM.solver or PROBLEM.options.solver; if
%   both fields are defined, then PROBLEM.solver will be followed.
%
%   For example, the following code
%
%   problem = struct();
%   problem.objective = @cos;
%   problem.x0 = -1;
%   problem.Aineq = 2;
%   problem.bineq = 3;
%   problem.options.maxfun = 50;
%   x = prima(problem);
%
%   solves
%       min cos(x) s.t. 2 * x <= 3
%   starting from x0 = -1 with at most 50 function evaluations.
%
%   See also uobyqa, newuoa, bobyqa, lincoa, cobyla.
%
%   See www.libprima.net for more information.
%
%   Dedicated to the late Professor M. J. D. Powell FRS (1936--2015).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Attribute: public (can  be called directly by users)
%
% Remarks:
%
% 1. Public function v.s. private function
% 1.1. Public functions are functions that can be directly called by users.
% They should be either prima or a solver.
% 1.2. Private functions are functions that are not supposed to be called
% by users. They are the preprocessing/postprocessing functions and
% auxiliary functions.
%
% 1. Errors that may be generated by prima
%
% 1.1. Normal error v.s. unexpected error
% A. Normal errors are usually caused by incorrect inputs.
% B. Unexpected errors usually imply bugs in the code. Such errors are
% displayed with 'UNEXPECTED ERROR:', except those generated by mex files.
%
% 1.2. Public errors v.s. private errors
% A. Public errors are designed to be shown to users. The identifier
% of an public error starts with 'PUB_FUN:', where PUB_FUN is a
% public function, i.e., the function that is being called by the user
% (should be either prima or a solver). Thanks to the error-handling
% code in public functions, public errors are displayed in a 'friendly'
% manner as follows:
%
% Error using PUB_FUN (line XXX)
% error message
%
% where the error message is a string starts with 'PUB_FUN:'. If the
% error is not generated in the file of PUB_FUN, then an additional line
% as follows is displayed:
%
% (error generated in FILE, line YYY)
%
% From the user's view point, a public error is an error raised by the
% function that he/she is calling. Therefore, if the user calls prima,
% then PUB_FUN should be prima; if the user calls a solver directly, then
% PUB_FUN should be the solver.
%
% B. Private errors are not expected to be seen by users unless there
% is a bug. All private errors are unexpected errors (but not vice versa).
% The identifier of a private error starts with 'PRI_FUN:', where PRI_FUN
% represents a private function. Private errors are displayed in the
% default manner, showing the detailed call stack trace, which is not so
% friendly but easy to debug.
%
% C. Private functions display errors as is. Public functions catch
% the exceptions thrown by private functions and display public/private
% errors properly.
%
% 2. Warnings that may be raised by prima
% 2.1. All the  warnings are considered to be public, i.e., they are
% designed to be shown to the users.
% 2.2. Warnings are displayed without the call stack trace.
%
% 3. probinfo
% !!! TREAT probinfo AS A READONLY VARIABLE AFTER PREPRIMA !!!
% !!! DO NOT CHANGE probinfo AFTER PREPRIMA !!!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% prima starts

callstack = dbstack;
funname = callstack(1).name; % Name of the current function

% OUTPUT records the information that is produced by the solver and
% intended to pass to postprima.
% OUTPUT should contain at least x, fx, exitflag, funcCount, and constrviolation;
% for internal solvers (solvers from PRIMA), it should also contain fhist, chist, warnings;
% for nonlinearly constrained internal solvers, it should also contain nlcineq and nlceq.
output = struct();

output.warnings = {}; % A cell that records all the warnings
% This version of prima.m produces no warning. However, initializing output.warnings
% is still necessary, as output.warnings is required by postprima.
warning('off', 'backtrace'); % Do not display the stack trace of a warning

maxarg = 10; % Maximal number of inputs
nvararg = length(varargin); % Number of inputs

% Interpret the input.
% Expected inputs: [fun, x0, Aineq, bineq, Aeq, beq, lb, ub, nonlcon, options],
% yet some of them may be omitted
if (nvararg < 1) % Public/normal error
    error(sprintf('%s:TooFewInputs', funname), '%s: at least 1 input.', funname);
elseif (nvararg == 1)
    % The only input should be either a information-requesting string or
    % a problem-defining structure.
    args = varargin;
elseif (nvararg >= 2 && nvararg <= maxarg)
    % If 2 <= nvararg <= 10 and the last input is a structure or [], then it is the 'options'
    if isempty(varargin{end}) || isa(varargin{end}, 'struct')
        % If nvararg >= 4 and the second last input is a function, then it is the 'nonlcon'
        if (nvararg >= 4) && (ischarstr(varargin{end-1}) || isa(varargin{end-1}, 'function_handle'))
            args = [varargin(1:end-2), cell(1, maxarg-nvararg), varargin(end-1:end)]; % 'augment' the inputs to 10 by adding []; args{:} (should have 10 entries) will be the inputs for preprima
            % cell(m,n) returns an mxn array of []
        else
            args = [varargin(1:end-1), cell(1, maxarg-nvararg), varargin(end)];
        end
    % if nvararg >= 3 and the last input is a function, then it is the 'nonlcon'
    elseif (nvararg >= 3) && (ischarstr(varargin{end}) || isa(varargin{end}, 'function_handle'))
        args = [varargin(1:end-1), cell(1, maxarg-nvararg-1), varargin(end), {[]}];
    else
        args = [varargin, cell(1, maxarg-nvararg)];
    end
else % Public/normal error
    error(sprintf('%s:TooManyInputs', funname), '%s: at most %d inputs.', funname, maxarg);
end

% Preprocess the input
try % preprima and package_info are private functions that may generate public errors; error-handling needed
    if (nvararg == 1) && ischarstr(varargin{1})
        % If there is only 1 input and it is a string, then it should be
        % a string requesting information about the package.
        x = package_info(varargin{1});
        return % Return immediately
    else
        [fun, x0, Aineq, bineq, Aeq, beq, lb, ub, nonlcon, options, probinfo] = preprima(args{:});
    end
catch exception
    if ~isempty(regexp(exception.identifier, sprintf('^%s:', funname), 'once')) % Public error; displayed friendly
        error(exception.identifier, '%s\n(error generated in %s, line %d)', exception.message, exception.stack(1).file, exception.stack(1).line);
    else % Private error; displayed as is
        rethrow(exception);
    end
end

if probinfo.infeasible % The problem turned out infeasible during preprima
    output.x = x0;
    output.fx = fun(output.x);
    output.exitflag = -4;
    output.funcCount = 1;
    if options.output_xhist
        output.xhist = output.x;
    end
    output.fhist = output.fx;
    output.constrviolation = probinfo.constrv_x0;
    output.chist = output.constrviolation;
    output.nlcineq = probinfo.nlcineq_x0;
    output.nlceq = probinfo.nlceq_x0;
    if options.output_nlchist
        output.nlcihist = output.nlcineq;
        output.nlcehist = output.nlceq;
    end
elseif probinfo.nofreex % x was fixed by the bound constraints during preprima
    output.x = probinfo.fixedx_value;
    output.fx = fun(output.x);
    output.exitflag = 13;
    output.funcCount = 1;
    if options.output_xhist
        output.xhist = output.x;
    end
    output.fhist = output.fx;
    output.constrviolation = probinfo.constrv_fixedx;
    output.chist = output.constrviolation;
    output.nlcineq = probinfo.nlcineq_fixedx;
    output.nlceq = probinfo.nlceq_fixedx;
    if options.output_nlchist
        output.nlcihist = output.nlcineq;
        output.nlcehist = output.nlceq;
    end
elseif probinfo.feasibility_problem && ~strcmp(probinfo.refined_type, 'nonlinearly-constrained')
    output.x = x0;  % preprima has tried to set x0 to a feasible point (but may have failed)
    % We could set fx = [], funcCount = 0, and fhist = [] since no function evaluation
    % occurred. But then we will have to modify the validation of fx, funcCount,
    % and fhist in postprima. To avoid such a modification, we set fx, funcCount,
    % and fhist as below and then revise them in postprima.
    output.fx = fun(output.x);  % preprima has defined a fake objective function
    output.funcCount = 1;
    if options.output_xhist
        output.xhist = output.x;
    end
    output.fhist = output.fx;
    output.constrviolation = probinfo.constrv_x0;
    output.chist = output.constrviolation;
    output.nlcineq = [];  % No nonlinear constraints
    output.nlceq = [];
    if output.constrviolation <= eps  % Did preprima find a feasible point?
        output.exitflag = 14;
    else
        output.exitflag = 15;
    end
    if options.output_nlchist
        output.nlcihist = output.nlcineq;
        output.nlcehist = output.nlceq;
    end
else
    % The problem turns out 'normal' during preprima. Solve it by
    % options.solver, which has been defined in preprima.
    try
        switch lower(options.solver)
        case 'uobyqa'
            [x, fx, exitflag, output] = uobyqa(fun, x0, options);
        case 'newuoa'
            [x, fx, exitflag, output] = newuoa(fun, x0, options);
        case 'bobyqa'
            [x, fx, exitflag, output] = bobyqa(fun, x0, lb, ub, options);
        case 'lincoa'
            [x, fx, exitflag, output] = lincoa(fun, x0, Aineq, bineq, Aeq, beq, lb, ub, options);
        case 'cobyla'
            [x, fx, exitflag, output] = cobyla(fun, x0, Aineq, bineq, Aeq, beq, lb, ub, nonlcon, options);
        end
    catch exception
        if ~isempty(regexp(exception.identifier, sprintf('^%s:', funname), 'once')) % Public error; displayed friendly
            error(exception.identifier, '%s\n(error generated in %s, line %d)', exception.message, exception.stack(1).file, exception.stack(1).line);
        else % Private error; displayed as is
            rethrow(exception);
        end
    end
    % Record the results of the solver in OUTPUT
    output.x = x;
    output.fx = fx;
    output.exitflag = exitflag;
end

% Postprocess the result
try % postprima is a private function that may generate public errors; error-handling needed
    [x, fx, exitflag, output] = postprima(probinfo, output);
catch exception
    if ~isempty(regexp(exception.identifier, sprintf('^%s:', funname), 'once')) % Public error; displayed friendly
        error(exception.identifier, '%s\n(error generated in %s, line %d)', exception.message, exception.stack(1).file, exception.stack(1).line);
    else % Private error; displayed as is
        rethrow(exception);
    end
end

% prima ends
return