# optim

non-linear optimization routine

## Calling Sequence

                fopt = optim(costf, x0)
                fopt = optim(costf [,<contr>],x0 [,algo] [,df0 [,mem]] [,work] [,<stop>] [,<params>] [,imp=iflag])
                [fopt, xopt] = optim(...)
                [fopt, xopt, gopt] = optim(...)
                [fopt, xopt, gopt, work] = optim(...)
                [fopt, xopt, gopt, work, iters] = optim(...)
                [fopt, xopt, gopt, work, iters, evals] = optim(...)
                [fopt, xopt, gopt, work, iters, evals, err] = optim(...)
                [fopt, xopt, [gopt, work, iters, evals, err], ti, td] = optim(..., params="si","sd")

## Arguments

costf

  a function, a list or a string, the objective function.

x0

  real vector, the initial guess for `x`.

<contr>

  an optional sequence of arguments containing the lower and upper bounds on `x`. If bounds are required, this sequence of arguments must be `"b",binf,bsup` where `binf` and `bsup` are real vectors with same dimension as `x0`.

algo

  a string, the algorithm to use (default `algo="qn"`).

  The available algorithms are:

  - `"qn"`: Quasi-Newton with BFGS

  - `"gc"`: limited memory BFGS

  - `"nd"`: non-differentiable.

    The `"nd"` algorithm does not accept bounds on `x`.

    The `"qn"` cannot be used if `size(x0)>46333`.

df0

  real scalar, a guess of the decreasing of `f` at first iteration. (default `df0=1`).

mem

  integer, the number of variables used to approximate the Hessian (default `mem=10`). This feature is available for the `"gc"` algorithm without constraints and the non-smooth algorithm `"nd"` without constraints.

<stop>

  a sequence of arguments containing the parameters controlling the convergence of the algorithm. The following sequences are available:

                    "ar",nap
                                  "ar",nap,iter
                                  "ar",nap,iter,epsg
                                  "ar",nap,iter,epsg,epsf
                                  "ar",nap,iter,epsg,epsf,epsx

  where:

  nap

    maximum number of calls to `costf` allowed (default `nap=100`).

  iter

    maximum number of iterations allowed (default `iter=100`).

  epsg

    threshold on gradient norm (default `epsg= %eps`).

  epsf

    threshold controlling decreasing of `f` (default `epsf=0`).

  epsx

    threshold controlling variation of `x` (default `epsx=0`). This vector (possibly matrix) with same size as `x0` can be used to scale `x`.

<params>

  in the case where the objective function is a C or Fortran routine, a sequence of arguments containing the method to communicate with the objective function. This option has no meaning when the cost function is a Scilab script.

  The available values for <params> are the following.

  - `"in"`

    That mode allows to allocate memory in the internal Scilab workspace so that the objective function can get arrays with the required size, but without directly allocating the memory. The `"in"` value stands for "initialization". In that mode, before the value and derivative of the objective function is to be computed, there is a dialog between the `optim` Scilab primitive and the objective function `costf`. In this dialog, the objective function is called two times, with particular values of the `ind` parameter. The first time, `ind` is set to 10 and the objective function is expected to set the `nizs`, `nrzs` and `ndzs` integer parameters of the `nird` common, which is defined as:

        common /nird/ nizs,nrzs,ndzs

    This allows Scilab to allocate memory inside its internal workspace. The second time the objective function is called, `ind` is set to 11 and the objective function is expected to set the `ti`, `tr` and `tz` arrays. After this initialization phase, each time it is called, the objective function is ensured that the `ti`, `tr` and `tz` arrays which are passed to it have the values that have been previously initialized.

  - `"ti",valti`

    In this mode, `valti` is expected to be a Scilab vector variable containing integers. Whenever the objective function is called, the `ti` array it receives contains the values of the Scilab variable.

  - `"td", valtd`

    In this mode, `valtd` is expected to be a Scilab vector variable containing double values. Whenever the objective function is called, the `td` array it receives contains the values of the Scilab variable.

  - `"ti",valti,"td",valtd`

    This mode combines the two previous modes.

  - `"si"`

    In this mode, `valti` is saved in the output variable `ti`.

  - `"sd"`

    In this mode, `valtd` is saved in the output variable `td`.

  - `"si","sd"`

    This mode combines the two previous modes.

  The `ti, td` arrays may be used so that the objective function can be computed. For example, if the objective function is a polynomial, the ti array may may be used to store the coefficients of that polynomial.

  Users should choose carefully between the `"in"` mode and the `"ti"` and `"td"` mode, depending on the fact that the arrays are Scilab variables or not. If the data is available as Scilab variables, then the `"ti", valti, "td", valtd` mode should be chosen. If the data is available directly from the objective function, the `"in"` mode should be chosen. Notice that there is no `"tr"` mode, since, in Scilab, all real values are doubles.

  If neither the "in" mode, nor the "ti", "td" mode is chosen, that is, if <params> is not present as an option of the optim primitive, the user may should not assume that the ti,tr and td arrays can be used : reading or writing the arrays may generate unpredictable results.

"imp=iflag"

  named argument used to set the trace mode (default `imp=0`, which prints no messages). If `imp` is greater or equal to 1, more information are printed, depending on the algorithm chosen. More precisely:

  - `"qn"` without constraints: from `iflag=1` to `iflag=3`.

    - `iflag>=1`: initial and final print,

    - `iflag>=2`: one line per iteration (number of iterations, number of calls to f, value of f),

    - `iflag>=3`: extra information on line searches.

  - `"qn"` with bounds constraints: from `iflag=1` to `iflag=4`.

    - `iflag>=1`: initial and final print,

    - `iflag>=2`: one line per iteration (number of iterations, number of calls to f, value of f),

    - `iflag>=3`: extra information on line searches.

  - `"gc"` without constraints: from `iflag=1` to `iflag=5`.

    - `iflag>=1` and `iflag>=2`: initial and final print,

    - `iflag=3`: one line per iteration (number of iterations, number of calls to f, value of f),

    - `iflag>=4`: extra information on lines searches.

  - `"gc"` with bounds constraints: from `iflag=1` to `iflag=3`.

    - `iflag>=1`: initial and final print,

    - `iflag>=2`: one print per iteration,

    - `iflag=3`: extra information.

  - `"nd"` with bounds constraints: from `iflag=1` to `iflag=8`.

    - `iflag>=1`: initial and final print,

    - `iflag>=2`: one print on each convergence,

    - `iflag>=3`: one print per iteration,

    - `iflag>=4`: line search,

    - `iflag>=5`: various tolerances,

    - `iflag>=6`: weight and information on the computation of direction.

  If `imp` is lower than 0, then the cost function is evaluated every `-m` iterations, with `ind=1`.

fopt

  the value of the objective function at the point `xopt`

xopt

  best value of `x` found.

gopt

  the gradient of the objective function at the point `xopt`

work

  working array for hot restart for quasi-Newton method. This array is automatically initialized by `optim` when `optim` is invoked. It can be used as input parameter to speed-up the calculations.

iters

  scalar, the number of iterations that is displayed when `imp=2`.

evals

  scalar, the number of `cost` function evaluations that is displayed when `imp=2`.

err

  scalar, a termination indicator. The success flag is `9`. `err=1`: Norm of projected gradient lower than... `err=2`: At last iteration f decreases by less than... `err=3`: Optimization stops because of too small variations for x. `err=4`: Optim stops: maximum number of calls to f is reached. `err=5`: Optim stops: maximum number of iterations is reached. `err=6`: Optim stops: too small variations in gradient direction. `err=7`: Stop during calculation of descent direction. `err=8`: Stop during calculation of estimated hessian. `err=9`: End of optimization, successful completion. `err=10`: End of optimization (linear search fails).

## Description

This function solves unconstrained nonlinear optimization problems:

    min f(x)

where `x` is a vector and `f(x)` is a function that returns a scalar. This function can also solve bound constrained nonlinear optimization problems:

    min f(x)
                binf <= x <= bsup

where `binf` is the lower bound and `bsup` is the upper bound on `x`.

The `costf` argument can be a Scilab function, a list or a string giving the name of a C or Fortran routine (see "external"). This external must return the value `f` of the cost function at the point `x` and the gradient `g` of the cost function at the point `x`.

Scilab function case

  If `costf` is a Scilab function, its calling sequence must be:

      [f, g, ind] = costf(x, ind)

  where `x` is the current point, `ind` is an integer flag described below, `f` is the real value of the objective function at the point `x` and `g` is a vector containing the gradient of the objective function at `x`. The variable `ind` is described below.

List case

  It may happen that objective function requires extra arguments. In this case, we can use the following feature. The `costf` argument can be the list `(real_costf, arg1,...,argn)`. In this case, `real_costf`, the first element in the list, must be a Scilab function with calling sequence:

              [f,g,ind]=real_costf(x,ind,arg1,...,argn)

  The `x`, `f`, `g`, `ind` arguments have the same meaning as before. In this case, each time the objective function is called back, the arguments `arg1,...,argn` are automatically appended at the end of the calling sequence of `real_costf`.
String case

  If `costf` is a string, it refers to the name of a C or Fortran routine which must be linked to Scilab

  Fortran case

    The calling sequence of the Fortran subroutine computing the objective must be:

        subroutine costf(ind,n,x,f,g,ti,tr,td)

    with the following declarations:

        integer ind,n ti(*)
                                            double precision x(n),f,g(n),td(*)
                                            real tr(*)

    The argument `ind` is described below.

    If ind = 2, 3 or 4, the inputs of the routine are : `x, ind, n, ti, tr,td`.

    If ind = 2, 3 or 4, the outputs of the routine are : `f` and `g`.

  C case

    The calling sequence of the C function computing the objective must be:

        void costf(int *ind, int *n, double *x, double *f, double *g, int *ti, float *tr, double *td)

    The argument `ind` is described below.

    The inputs and outputs of the function are the same as in the fortran case.

On output, `ind<0` means that `f` cannot be evaluated at `x` and `ind=0` interrupts the optimization.

## Termination criteria

Each algorithm has its own termination criteria, which may use the parameters given by the user, that is `nap`, `iter`, `epsg`, `epsf` and `epsx`. Not all the parameters are taken into account. In the table below, we present the specific termination parameters which are taken into account by each algorithm. The unconstrained solver is identified by "UNC" while the bound constrained solver is identified by "BND". An empty entry means that the parameter is ignored by the algorithm.

Solver | nap | iter | epsg | epsf | epsx
optim/"qn" UNC | X | X | X |  |
optim/"qn" BND | X | X | X | X | X
optim/"gc" UNC | X | X | X |  | X
optim/"gc" BND | X | X | X | X | X
optim/"nd" UNC | X | X |  | X | X

## Example: Scilab function

The following is an example with a Scilab function. Notice, for simplifications reasons, the Scilab function "cost" of the following example computes the objective function f and its derivative no matter of the value of ind. This allows to keep the example simple. In practical situations though, the computation of "f" and "g" may raise performances issues so that a direct optimization may be to use the value of "ind" to compute "f" and "g" only when needed.

    function [f, g, ind] = cost(x, ind)
                xref = [1; 2; 3];
                f = 0.5 * norm(x - xref)^2;
                g = x - xref;
                endfunction

                // Simplest call
                x0 = [1; -1; 1];
                [fopt, xopt] = optim(cost, x0)

                // Use "gc" algorithm
                [fopt, xopt, gopt] = optim(cost, x0, "gc")

                // Use "nd" algorithm
                [fopt, xopt, gopt] = optim(cost, x0, "nd")

                // Upper and lower bounds on x
                [fopt, xopt, gopt] = optim(cost, "b", [-1;0;2], [0.5;1;4], x0)

                // Upper and lower bounds on x and setting up the algorithm to "gc"
                [fopt, xopt, gopt] = optim(cost, "b", [-1; 0; 2], [0.5; 1; 4], x0, "gc")

                // Bound on the number of calls to the objective function
                [fopt, xopt, gopt] = optim(cost, "b", [-1; 0; 2], [0.5; 1; 4], x0, "gc", "ar", 3)

                // Set max number of calls to the objective function (3)
                // Set max number of iterations (100)
                // Set stopping threshold on the value of f (1e-6),
                // on the value of the norm of the gradient of the objective function (1e-6)
                // on the improvement on the parameters x_opt (1e-6;1e-6;1e-6)
                [fopt, xopt, gopt] = optim(cost, "b", [-1; 0; 2], [0.5; 1; 4], x0, "gc", "ar", 3, 100, 1e-6, 1e-6, [1e-3; 1e-3; 1e-3])

                // Additional messages are printed in the console.
                [fopt, xopt] = optim(cost, x0, imp = 3)

## Example: Print messages

The `imp` flag may take negative integer values, say k. In that case, the cost function is called once every -k iterations. This allows to draw the function value or write a log file.

This feature is available only with the `"qn"` algorithm without constraints.

In the following example, we solve the Rosenbrock test case. For each iteration of the algorithm, we print the value of x, f and g.

    function [f, g, ind] = cost(x, ind)
                xref = [1; 2; 3];
                f = 0.5 * norm(x - xref)^2;
                g = x - xref;
                if (ind == 1) then
                mprintf("f(x) = %s, |g(x)|=%s\n", string(f), string(norm(g)))
                end
                endfunction

                x0 = [1; -1; 1];
                [fopt, xopt] = optim(cost, x0, imp = -1)

The previous script produces the following output.

    -->[fopt, xopt] = optim(cost, x0, imp = -1)
                f(x) = 6.5, |g(x)|=3.6055513
                f(x) = 2.8888889, |g(x)|=2.4037009
                f(x) = 9.861D-31, |g(x)|=1.404D-15
                f(x) = 0, |g(x)|=0
                Norm of projected gradient lower than   0.0000000D+00.
                xopt  =
                1.
                2.
                3.
                fopt  =
                0.

In the following example, we solve the Rosenbrock test case. For each iteration of the algorithm, we plot the current value of x into a 2D graph containing the contours of Rosenbrock's function. This allows to see the progress of the algorithm while the algorithm is performing. We could as well write the value of x, f and g into a log file if needed.

    // 1. Define Rosenbrock for optimization
                function [f , g , ind] = rosenbrock (x , ind)
                f = 100.0 *(x(2) - x(1)^2)^2 + (1 - x(1))^2;
                g(1) = - 400. * ( x(2) - x(1)**2 ) * x(1) -2. * ( 1. - x(1) )
                g(2) = 200. * ( x(2) - x(1)**2 )
                endfunction

                // 2. Define rosenbrock for contouring
                function f = rosenbrockC ( x1 , x2 )
                x = [x1 x2]
                ind = 4
                [ f , g , ind ] = rosenbrock ( x , ind )
                endfunction

                // 3. Define Rosenbrock for plotting
                function [ f , g , ind ] = rosenbrockPlot ( x , ind )
                [ f , g , ind ] = rosenbrock ( x , ind )
                if (ind == 1) then
                plot ( x(1) , x(2) , "g." )
                end
                endfunction

                // 4. Draw the contour of Rosenbrock's function
                x0 = [-1.2 1.0];
                xopt = [1.0 1.0];
                xdata = linspace(-2,2,100);
                ydata = linspace(-2,2,100);
                contour ( xdata , ydata , rosenbrockC , [1 10 100 500 1000])
                plot(x0(1) , x0(2) , "b.")
                plot(xopt(1) , xopt(2) , "r*")

                // 5. Plot the optimization process, during optimization
                [fopt, xopt] = optim ( rosenbrockPlot , x0 , imp = -1)

function [f, g, ind]=rosenbrock(x, ind) f = 100.0 *(x(2) - x(1)^2)^2 + (1 - x(1))^2; g(1) = - 400. * ( x(2) - x(1)**2 ) * x(1) -2. * ( 1. - x(1) ) g(2) = 200. * ( x(2) - x(1)**2 ) endfunction

function f=rosenbrockC(x1, x2) x = [x1 x2] ind = 4 [ f , g , ind ] = rosenbrock ( x , ind ) endfunction

function [f, g, ind]=rosenbrockPlot(x, ind) [ f , g , ind ] = rosenbrock ( x , ind ) if (ind == 1) then plot ( x(1) , x(2) , "g." ) end endfunction

x0 = [-1.2 1.0]; xopt = [1.0 1.0]; xdata = linspace(-2,2,100); ydata = linspace(-2,2,100); contour ( xdata , ydata , rosenbrockC , [1 10 100 500 1000]) plot(x0(1) , x0(2) , "b.") plot(xopt(1) , xopt(2) , "r*") [fopt, xopt] = optim ( rosenbrockPlot , x0 , imp = -1)

## Example: Optimizing with numerical derivatives

It is possible to optimize a problem without an explicit knowledge of the derivative of the cost function. For this purpose, we can use the `numderivative` function to compute a numerical derivative of the cost function.

In the following example, we use the `numderivative` function to solve Rosenbrock's problem.

    function f = rosenbrock ( x )
                f = 100.0 *(x(2)-x(1)^2)^2 + (1-x(1))^2;
                endfunction

                function [ f , g , ind ] = rosenbrockCost ( x , ind )
                f = rosenbrock ( x );
                g = numderivative ( rosenbrock , x );
                endfunction

                x0 = [-1.2 1.0];

                [ fopt , xopt ] = optim ( rosenbrockCost , x0 )

## Example: Counting function evaluations and number of iterations

The `imp` option can take negative values. If the `imp` is equal to `m` where `m` is a negative integer, then the cost function is evaluated every -`m` iterations, with the `ind` input argument equal to 1. The following example uses this feature to compute the number of iterations. The global variable `mydata` is used to store the number of function evaluations as well as the number of iterations.

    function [f, g, ind] = cost(x, ind)
        global _MYDATA_
        if ( ind == 1 )
            _MYDATA_.niter = _MYDATA_.niter + 1;
        end
        _MYDATA_.nfevals = _MYDATA_.nfevals + 1;
        xref = [1; 2; 3];
        if ( ind == 2 | ind == 4 ) then
            f = 0.5*norm(x-xref)^2;
        else
            f = 0;
        end
        if ( ind == 3 | ind == 4 ) then
            g = x-xref;
        else
            g = zeros(3, 1);
        end
    endfunction

    x0 = [1; -1; 1];
    global _MYDATA_
    _MYDATA_ = tlist ( ["MYDATA", "niter", "nfevals"]);
    _MYDATA_.niter = 0;
    _MYDATA_.nfevals = 0;

    [f, xopt] = optim(cost, x0, imp=-1);
    mprintf ( "Number of function evaluations: %d\n", _MYDATA_.nfevals );
    mprintf ( "Number of iterations: %d\n", _MYDATA_.niter );

While the previous example perfectly works, there is a risk that the same variable `_MYDATA_` is used by some internal function used by `optim`. In this case, the value may be wrong. This is why a sufficiently weird variable name has been used.

## Example : Passing extra parameters

In most practical situations, the cost function depends on extra parameters which are required to evaluate the cost function. There are several methods to achieve this goal.

In the following example, the cost function uses 4 parameters `a, b, c` and `d`. We define the cost function with additional input arguments, which are declared after the index argument. Then we pass a list as the first input argument of the `optim` solver. The first element of the list is the cost function. The additional variables are directly passed to the cost function.

    function [ f , g , ind ] = costfunction ( x , ind , a , b , c , d )
                f = a * ( x(1) - c ) ^2 + b * ( x(2) - d )^2
                g(1) = 2 * a * ( x(1) - c )
                g(2) = 2 * b * ( x(2) - d )
                endfunction

                x0 = [1 1];
                a = 1.0;
                b = 2.0;
                c = 3.0;
                d = 4.0;
                costf = list ( costfunction , a , b , c, d );
                [fopt , xopt] = optim ( costf , x0 , imp = 2)

In complex cases, the cost function may have so many parameters, that having a function which takes all arguments as inputs is not convenient. For example, consider the situation where the cost function needs 12 parameters. Then, designing a function with 14 input arguments (x, index and the 12 parameters) is difficult to manage. Instead, we can use a more complex data structure to store our data. In the following example, we use a tlist to store the 4 input arguments. This method can easily be expanded to an arbitrary number of parameters.

    function [f , g , ind] = costfunction ( x , ind , parameters)
                // Get the parameters
                a = parameters.a
                b = parameters.b
                c = parameters.c
                d = parameters.d
                f = a * ( x(1) - c ) ^2 + b * ( x(2) - d )^2
                g(1) = 2 * a * ( x(1) - c )
                g(2) = 2 * b * ( x(2) - d )
                endfunction

                x0 = [1 1];
                a = 1.0;
                b = 2.0;
                c = 3.0;
                d = 4.0;
                // Store the parameters
                parameters = tlist ( [
                "T_MYPARAMS"
                "a"
                "b"
                "c"
                "d"
                ]);

                parameters.a = a;
                parameters.b = b;
                parameters.c = c;
                parameters.d = d;
                costf = list ( costfunction , parameters );
                [fopt , xopt] = optim ( costf , x0 , imp = 2)

In the following example, the parameters are defined before the optimizer is called. They are directly used in the cost function.

    // The example NOT to follow
                function [ f , g , ind ] = costfunction ( x , ind )
                f = a * ( x(1) - c ) ^2 + b * ( x(2) - d )^2
                g(1) = 2 * a * ( x(1) - c )
                g(2) = 2 * b * ( x(2) - d )
                endfunction
                x0 = [1 1];
                a = 1.0;
                b = 2.0;
                c = 3.0;
                d = 4.0;
                [ fopt , xopt ] = optim ( costfunction , x0 , imp = 2 )

While the previous example perfectly works, there is a risk that the same variables are used by some internal function used by `optim`. In this case, the value of the parameters are not what is expected and the optimization can fail or, worse, give a wrong result. It is also difficult to manage such a function, which requires that all the parameters are defined in the calling context.

In the following example, we define the cost function with the classical header. Inside the function definition, we declare that the parameters `a, b, c` and `d` are global variables. Then we declare and set the global variables.

    // Another example NOT to follow
                function [ f , g , ind ] = costfunction ( x , ind )
                global a b c d
                f = a * ( x(1) - c ) ^2 + b * ( x(2) - d )^2
                g(1) = 2 * a * ( x(1) - c )
                g(2) = 2 * b * ( x(2) - d )
                endfunction
                x0 = [1 1];
                global a b c d
                a = 1.0;
                b = 2.0;
                c = 3.0;
                d = 4.0;
                [ fopt , xopt ] = optim ( costfunction , x0 , imp = 2 )

While the previous example perfectly works, there is a risk that the same variables are used by some internal function used by `optim`. In this case, the value of the parameters are not what is expected and the optimization can fail or, worse, give a wrong result.

## Example : Checking that derivatives are correct

Many optimization problem can be avoided if the derivatives are computed correctly. One common reason for failure in the step-length procedure is an error in the calculation of the cost function and its gradient. Incorrect calculation of derivatives is by far the most common user error.

In the following example, we give a false implementation of Rosenbrock's gradient. In order to check the computation of the derivatives, we use the `numderivative` function. We define the `simplified` function, which delegates the computation of `f` to the rosenbrock function. The `simplified` function is passed as an input argument of the `numderivative` function.

    function [ f , g , index ] = rosenbrock ( x , index )
                f = 100.0 *(x(2)-x(1)^2)^2 + (1-x(1))^2;
                // Exact :
                g(1) = - 400. * ( x(2) - x(1)**2 ) * x(1) -2. * ( 1. - x(1) )
                // Wrong :
                g(1) = - 1200. * ( x(2) - x(1)**2 ) * x(1) -2. * ( 1. - x(1) )
                g(2) = 200. * ( x(2) - x(1)**2 )
                endfunction

                function f = simplified ( x )
                index = 1;
                [ f , g , index ] = rosenbrock ( x , index )
                endfunction

                x0 = [-1.2 1];
                index = 1;
                [ f , g , index ] = rosenbrock ( x0 , index );
                gnd = numderivative ( simplified , x0.' );
                mprintf("Exact derivative:[%s]\n" , strcat ( string(g) , " " ));
                mprintf("Numerical derivative:[%s]\n" , strcat ( string(gnd) , " " ));

The previous script produces the following output. Obviously, the difference between the two gradient is enormous, which shows that the wrong formula has been used in the gradient.

          Exact derivative:[-638 -88]
                Numerical derivative:[-215.6 -88]

## Example: C function

The following is an example with a C function, where a C source code is written into a file, dynamically compiled and loaded into Scilab, and then used by the "optim" solver. The interface of the "rosenc" function is fixed, even if the arguments are not really used in the cost function. This is because the underlying optimization solvers must assume that the objective function has a known, constant interface. In the following example, the arrays ti and tr are not used, only the array "td" is used, as a parameter of the Rosenbrock function. Notice that the content of the arrays ti and td are the same that the content of the Scilab variable, as expected.

    // External function written in C (C compiler required)
                // write down the C code (Rosenbrock problem)
                C=['#include <math.h>'
                'double sq(double x)'
                '{ return x*x;}'
                'void rosenc(int *ind, int *n, double *x, double *f, double *g, '
                '                                int *ti, float *tr, double *td)'
                '{'
                '  double p;'
                '  int i;'
                '  p=td[0];'
                '  if (*ind==2||*ind==4) {'
                '    *f=1.0;'
                '    for (i=1;i<*n;i++)'
                '      *f+=p*sq(x[i]-sq(x[i-1]))+sq(1.0-x[i]);'
                '  }'
                '  if (*ind==3||*ind==4) {'
                '    g[0]=-4.0*p*(x[1]-sq(x[0]))*x[0];'
                '    for (i=1;i<*n-1;i++)'
                '      g[i]=2.0*p*(x[i]-sq(x[i-1]))-4.0*p*(x[i+1]-sq(x[i]))*x[i]-2.0*(1.0-x[i]);'
                '    g[*n-1]=2.0*p*(x[*n-1]-sq(x[*n-2]))-2.0*(1.0-x[*n-1]);'
                '  }'
                '}'];
                cd TMPDIR;
                mputl(C, TMPDIR+'/rosenc.c')

                // compile the C code
                l = ilib_for_link('rosenc', 'rosenc.c', [], 'c');

                // incremental linking
                link(l, 'rosenc', 'c')

                //solve the problem
                x0 = [40; 10; 50];
                p = 100;
                [f, xo, go] = optim('rosenc', x0, 'td', p)

## Example: Fortran function

The following is an example with a Fortran function.

    // External function written in Fortran (Fortran compiler required)
                // write down the Fortran  code (Rosenbrock problem)
                F = [ '      subroutine rosenf(ind, n, x, f, g, ti, tr, td)'
                '      integer ind,n,ti(*)'
                '      double precision x(n),f,g(n),td(*)'
                '      real tr(*)'
                'c'
                '      double precision y,p'
                '      p=td(1)'
                '      if (ind.eq.2.or.ind.eq.4) then'
                '        f=1.0d0'
                '        do i=2,n'
                '          f=f+p*(x(i)-x(i-1)**2)**2+(1.0d0-x(i))**2'
                '        enddo'
                '      endif'
                '      if (ind.eq.3.or.ind.eq.4) then'
                '        g(1)=-4.0d0*p*(x(2)-x(1)**2)*x(1)'
                '        if(n.gt.2) then'
                '          do i=2,n-1'
                '            g(i)=2.0d0*p*(x(i)-x(i-1)**2)-4.0d0*p*(x(i+1)-x(i)**2)*x(i)'
                '     &           -2.0d0*(1.0d0-x(i))'
                '          enddo'
                '        endif'
                '        g(n)=2.0d0*p*(x(n)-x(n-1)**2)-2.0d0*(1.0d0-x(n))'
                '      endif'
                '      return'
                '      end'];
                cd TMPDIR;
                mputl(F, TMPDIR+'/rosenf.f')

                // compile the Fortran code
                l = ilib_for_link('rosenf', 'rosenf.f', [], 'f');

                // incremental linking
                link(l, 'rosenf', 'f')

                //solve the problem
                x0 = [40; 10; 50];
                p = 100;
                [f, xo, go] = optim('rosenf', x0, 'td', p)

## Example: Fortran function with initialization

The following is an example with a Fortran function in which the "in" option is used to allocate memory inside the Scilab environment. In this mode, there is a dialog between Scilab and the objective function. The goal of this dialog is to initialize the parameters of the objective function. Each part of this dialog is based on a specific value of the "ind" parameter.

At the beginning, Scilab calls the objective function, with the ind parameter equal to 10. This tells the objective function to initialize the sizes of the arrays it needs by setting the nizs, nrzs and ndzs integer parameters of the "nird" common. Then the objective function returns. At this point, Scilab creates internal variables and allocate memory for the variable izs, rzs and dzs. Scilab calls the objective function back again, this time with ind equal to 11. This tells the objective function to initialize the arrays izs, rzs and dzs. When the objective function has done so, it returns. Then Scilab enters in the real optimization mode and calls the optimization solver the user requested. Whenever the objective function is called, the izs, rzs and dzs arrays have the values that have been previously initialized.

    //
                // Define a fortran source code and compile it (fortran compiler required)
                //
                fortransource = ['      subroutine rosenf(ind,n,x,f,g,izs,rzs,dzs)'
                'C     -------------------------------------------'
                'c     Example of cost function given by a subroutine'
                'c     if n<=2 returns ind=0'
                'c     f.bonnans, oct 86'
                '      implicit double precision (a-h,o-z)'
                '      real rzs(1)'
                '      double precision dzs(*)'
                '      dimension x(n),g(n),izs(*)'
                '      common/nird/nizs,nrzs,ndzs'
                '      if (n.lt.3) then'
                '        ind=0'
                '        return'
                '      endif'
                '      if(ind.eq.10) then'
                '         nizs=2'
                '         nrzs=1'
                '         ndzs=1'
                '         return'
                '      endif'
                '      if(ind.eq.11) then'
                '         izs(1)=5'
                '         izs(2)=10'
                '         dzs(1)=100.0d+0'
                '         return'
                '      endif'
                '      if(ind.eq.2)go to 5'
                '      if(ind.eq.3)go to 20'
                '      if(ind.eq.4)go to 5'
                '      ind=-1'
                '      return'
                '5     f=1.0d+0'
                '      do 10 i=2,n'
                '        im1=i-1'
                '10      f=f + dzs(1)*(x(i)-x(im1)**2)**2 + (1.0d+0-x(i))**2'
                '      if(ind.eq.2)return'
                '20    g(1)=-4.0d+0*dzs(1)*(x(2)-x(1)**2)*x(1)'
                '      nm1=n-1'
                '      do 30 i=2,nm1'
                '        im1=i-1'
                '        ip1=i+1'
                '        g(i)=2.0d+0*dzs(1)*(x(i)-x(im1)**2)'
                '30      g(i)=g(i) -4.0d+0*dzs(1)*(x(ip1)-x(i)**2)*x(i) - '
                '     &        2.0d+0*(1.0d+0-x(i))'
                '      g(n)=2.0d+0*dzs(1)*(x(n)-x(nm1)**2) - 2.0d+0*(1.0d+0-x(n))'
                '      return'
                '      end'];
                cd TMPDIR;
                mputl(fortransource, TMPDIR + '/rosenf.f')

                // compile the C code
                libpath = ilib_for_link('rosenf', 'rosenf.f', [], 'f');

                // incremental linking
                linkid = link(libpath, 'rosenf', 'f');

                x0 = 1.2 * ones(1, 5);
                //
                // Solve the problem
                //
                [f, x, g] = optim('rosenf', x0, 'in');

## Example: Fortran function with initialization on Windows with Intel Fortran Compiler

Under the Windows operating system with Intel Fortran Compiler, one must carefully design the fortran source code so that the dynamic link works properly. On Scilab's side, the optimization component is dynamically linked and the symbol "nird" is exported out of the optimization dll. On the cost function's side, which is also dynamically linked, the "nird" common must be imported in the cost function dll.

The following example is a re-writing of the previous example, with special attention for the Windows operating system with Intel Fortran compiler as example. In that case, we introduce additional compiling instructions, which allows the compiler to import the "nird" symbol.

    fortransource = ['subroutine rosenf(ind,n,x,f,g,izs,rzs,dzs)'
                'cDEC$ IF DEFINED (FORDLL)'
                'cDEC$ ATTRIBUTES DLLIMPORT:: /nird/'
                'cDEC$ ENDIF'
                'C     -------------------------------------------'
                'c     Example of cost function given by a subroutine'
                'c     if n<=2 returns ind=0'
                'c     f.bonnans, oct 86'
                '      implicit double precision (a-h,o-z)'
                [etc...]

## Example: Fortran function with tab saving

Adding the "si" and/or "sd" options to `optim` implies the addition of the output arguments "ti" and/or "td", which will represent the working tables of the objective function.

Those output arguments must be placed at the end of the output list.

                // Move into the temporary directory to create the temporary files there
                cur_dir = pwd();
                chdir(TMPDIR);
                fortransource = ['      subroutine rosenf(ind,n,x,f,g,izs,rzs,dzs)'
                'C     -------------------------------------------'
                'c (DLL Digital Visual Fortran)'
                'c On Windows , we need to import common nird from scilab'
                'cDEC$ IF DEFINED (FORDLL)'
                'cDEC$ ATTRIBUTES DLLIMPORT:: /nird/'
                'cDEC$ ENDIF'
                'C     -------------------------------------------'
                'c     Example of cost function given by a subroutine'
                'c     if n.le.2 returns ind=0'
                'c     f.bonnans, oct 86'
                '      implicit double precision (a-h,o-z)'
                '      real rzs(1)'
                '      double precision dzs(*)'
                '      dimension x(n),g(n),izs(*)'
                '      common/nird/nizs,nrzs,ndzs'
                '      if (n.lt.3) then'
                '        ind=0'
                '        return'
                '      endif'
                '      if(ind.eq.10) then'
                '         nizs=2'
                '         nrzs=1'
                '         ndzs=1'
                '         return'
                '      endif'
                '      if(ind.eq.11) then'
                '         izs(1)=5'
                '         izs(2)=10'
                '         dzs(1)=100.0d+0'
                '         return'
                '      endif'
                '      if(ind.eq.2)go to 5'
                '      if(ind.eq.3)go to 20'
                '      if(ind.eq.4)go to 5'
                '      ind=-1'
                '      return'
                '5     f=1.0d+0'
                '      do 10 i=2,n'
                '        im1=i-1'
                '10      f=f + dzs(1)*(x(i)-x(im1)**2)**2 + (1.0d+0-x(i))**2'
                '      if(ind.eq.2)return'
                '20    g(1)=-4.0d+0*dzs(1)*(x(2)-x(1)**2)*x(1)'
                '      nm1=n-1'
                '      do 30 i=2,nm1'
                '        im1=i-1'
                '        ip1=i+1'
                '        g(i)=2.0d+0*dzs(1)*(x(i)-x(im1)**2)'
                '30      g(i)=g(i) -4.0d+0*dzs(1)*(x(ip1)-x(i)**2)*x(i) - '
                '     &        2.0d+0*(1.0d+0-x(i))'
                '      g(n)=2.0d+0*dzs(1)*(x(n)-x(nm1)**2) - 2.0d+0*(1.0d+0-x(n))'
                '      return'
                '      end'];
                mputl(fortransource, TMPDIR + '/rosenf.f');
                ilib_for_link('rosenf', 'rosenf.f', [], 'f');
                exec loader.sce;
                chdir(cur_dir);
                //
                // Define some constants
                //
                Leps = 10e3 * 8.e-5;
                bs = 10.*ones(1, 5);
                bi = -bs;
                x0 = 0.12 * bs;
                epsx = 1.e-15 * x0;
                xopt = .1*bs;

                // 'ti' and 'td' always at the end of the output sequence
                [f, x, ti, td]                           = optim('rosenf', x0, 'in', 'si', 'sd')
                [f, x, g, ti, td]                        = optim('rosenf', x0, 'in', 'si', 'sd')
                [f, x, g, w, ti, td]                     = optim('rosenf', x0, 'in', 'si', 'sd')
                [f, x, g, w, niter, nevals, ti, td]      = optim('rosenf', x0, 'in', 'si', 'sd')
                [f, x, g, w, niter, nevals, err, ti, td] = optim('rosenf', x0, 'in', 'si', 'sd')

                // With input argument 'in', ti and td will be initialized by rosenf function.
                [f, x, ti, td] = optim('rosenf', x0, 'in', 'si', 'sd')
                // Reuses the last ti and td for the next call and return it again.
                [f, x, ti, td] = optim('rosenf', x0, 'ti', ti, 'td', td, 'si', 'sd')

                // Initializes ti and td but return only ti
                [f, x, ti] = optim('rosenf', x0, 'in', 'si')
                [f, x, ti] = optim('rosenf', x0, 'ti', ti, 'si')

                // Initializes ti and td but return only td
                [f, x, td] = optim('rosenf', x0, 'in', 'sd')
                [f, x, td] = optim('rosenf', x0, 'td', td, 'sd')

## See Also

- external
- qpsolve
- datafit
- leastsq
- numderivative
- NDcost

## References

The following is a map from the various options to the underlying solvers.

"qn" without constraints

  n1qn1 : a quasi-Newton method with a Wolfe-type line search

"qn" with bounds constraints

  qnbd : a quasi-Newton method with projection

  RR-0242 - A variant of a projected variable metric method for bound constrained optimization problems, Bonnans Frederic, Rapport de recherche de l'INRIA - Rocquencourt, Octobre 1983

"gc" without constraints

  n1qn3 : a Quasi-Newton limited memory method with BFGS.

"gc" with bounds constraints

  gcbd : a BFGS-type method with limited memory and projection

"nd" without constraints

  n1fc1 : a bundle method

"nd" with bounds constraints

  not available