# sqpSolve#

## Purpose#

Solves the nonlinear programming problem using a sequential quadratic programming method.

## Format#

{ x, f, lagr, retcode } = sqpSolve(&fct, start[, ...])#
Parameters:
• &fct (function pointer) – pointer to a procedure that computes the function to be minimized. This procedure must have one input argument, a vector of parameter values, and one output argument, the value of the function evaluated at the input vector of parameter values. Additional optional arguments may be passed in to the objective function fct using dots (...).

• start (Kx1 vector) – start values

• ... (any) – Optional. A variable number of extra arguments to pass to the user function. These arguments will be passed to the user function untouched.

Returns:
• x (Kx1 vector) – of parameters at minimum.

• f (scalar) – function evaluated at x.

• lagr (vector) –

created using vput(). Contains the Lagrangian for the constraints. They may be extracted with the vread() command using the following strings:

 "lineq" Lagrangians of linear equality constraints "nlineq" Lagrangians of nonlinear equality constraints "linineq" Lagrangians of linear inequality constraints "nlinineq" Lagrangians of nonlinear inequality constraints "bounds" Lagrangians of bounds

Whenever a constraint is active, its associated Lagrangian will be nonzero.

• retcode (scalar) –

return code:

 0 normal convergence 1 forced exit 2 maximum number of iterations exceeded 3 function calculation failed 4 gradient calculation failed 5 Hessian calculation failed 6 line search failed 7 error with constraints

## Global Input#

_sqp_A#

MxK matrix, linear equality constraint coefficients.

_sqp_B#

Mx1 vector, linear equality constraint constants. These globals are used to specify linear equality constraints of the following type:

$\_sqp\_A * x = \_sqp\_B$

where x is the Kx1 unknown parameter vector.

_sqp_EqProc#

scalar, pointer to a procedure that computes the nonlinear equality constraints. For example, the statement:

_sqp_EqProc = &eqproc;


tells sqpSolve() that nonlinear equality constraints are to be placed on the parameters and where the procedure computing them is to be found. The procedure must have one input argument, the Kx1 vector of parameters, and one output argument, the Rx1 vector of computed constraints that are to be equal to zero. For example, suppose that you wish to place the following constraint:

p[1] * p[2] = p[3]


The procedure for this is:

proc eqproc(p);
retp(p[1]*p[2]-p[3]);
endp;

_sqp_C#

MxK matrix, linear inequality constraint coefficients.

_sqp_D#

Mx1 vector, linear inequality constraint constants. These globals are used to specify linear inequality constraints of the following type:

$\_sqp\_C * X \geq \_sqp\_D$

where x is the Kx1 unknown parameter vector.

_sqp_IneqProc#

scalar, pointer to a procedure that computes the nonlinear inequality constraints. For example the statement:

_sqp_EqProc = &ineqproc;


tells sqpSolve() that nonlinear equality constraints are to be placed on the parameters and where the procedure computing them is to be found. The procedure must have one input argument, the Kx1 vector of parameters, and one output argument, the Rx1 vector of computed constraints that are to be equal to zero. For example, suppose that you wish to place the following constraint:

$p[1] * p[2] \geq p[3]$

The procedure for this is:

proc ineqproc(p);
retp(p[1]*[2]-p[3]);
endp;

_sqp_Bounds#

Kx2 matrix, bounds on parameters. The first column contains the lower bounds, and the second column the upper bounds. If the bounds for all the coefficients are the same, a 1x2 matrix may be used. Default is:

[1] -1e256     [2] 1e256


scalar, pointer to a procedure that computes the gradient of the function with respect to the parameters. For example, the statement:

_sqp_GradProc = &gradproc;


tells sqpSolve() that a gradient procedure exists and where to find it. The user-provided procedure has two input arguments, a Kx1 vector of parameter values and an NxP matrix of data. The procedure returns a single output argument, an NxK matrix of gradients of the log-likelihood function with respect to the parameters evaluated at the vector of parameter values.

Default = 0, i.e., no gradient procedure has been provided.

_sqp_HessProc#

scalar, pointer to a procedure that computes the Hessian, i.e., the matrix of second order partial derivatives of the function with respect to the parameters. For example, the instruction:

_sqp_HessProc = &hessproc;


will tell sqpSolve() that a procedure has been provided for the computation of the Hessian and where to find it. The procedure that is provided by the user must have two input arguments, a Px1 vector of parameter values and an NxK data matrix. The procedure returns a single output argument, the PxP symmetric matrix of second order derivatives of the function evaluated at the parameter values.

_sqp_MaxIters#

scalar, maximum number of iterations. Default = 1e+5. Termination can be forced by pressing C on the keyboard.

_sqp_DirTol#

scalar, convergence tolerance for gradient of estimated coefficients. Default = 1e-5. When this criterion has been satisfied, sqpSolve() will exit the iterations.

_sqp_ParNames#

Kx1 character vector, parameter names.

_sqp_PrintIter#

scalar, if nonzero, prints iteration information. Default = 0. Can be toggled during iterations by pressing P on the keyboard.

_sqp_FeasibleT#

scalar, if nonzero, parameters are tested for est feasibility before computing function in line search. If function is defined outside inequality boundaries, then this test can be turned off.

scalar, if zero, no random search is attempted. If nonzero it is the radius of random search which is invoked whenever the usual line search fails. Default = .01.

__output#

scalar, if nonzero, results are printed. Default = 0.

## Examples#

// Reset all sqpSolve global variables

sqpSolveSet;

proc fct(x);
retp( (x[1] + 3*x[2] + x[3])^2 + 4*(x[1] - x[2])^2);
endp;

proc ineqp(x);
retp(6*x[2] + 4*x[3] - x[1]^3 - 3);
endp;

proc eqp(x);
retp(1-sumc(x));
endp;

_sqp_Bounds = { 0 1e256 };

start = { .1, .7, .2 };

_sqp_IneqProc = &ineqp;
_sqp_EqProc = &eqp;

{ x,f,lagr,ret } = sqpSolve(&fct,start);


## Remarks#

Pressing C on the keyboard will terminate iterations, and pressing P will toggle iteration output.

sqpSolve() is recursive, that is, it can call itself with another function and set of global variables,

sqpsolve.src