e04kbc is a comprehensive quasi-Newton algorithm for finding:
–an unconstrained minimum of a function of several variables;
–a minimum of a function of several variables subject to fixed upper and/or lower bounds on the variables.
First derivatives are required. e04kbc is intended for objective functions which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).
The function may be called by the names: e04kbc or nag_opt_bounds_deriv.
e04kbc is applicable to problems of the form:
Special provision is made for unconstrained minimization (i.e., problems which actually have no bounds on the ), problems which have only non-negativity bounds, and problems in which and . It is possible to specify that a particular should be held constant. You must supply a starting point and a function objfun to calculate the value of and its first derivatives at any point .
A typical iteration starts at the current point where (say) variables are free from both their bounds. The vector , whose elements are the derivatives of with respect to the free variables, is known. A unit lower triangular matrix and a diagonal matrix (both of dimension ), such that is a positive definite approximation to the matrix of second derivatives with respect to the free variables, are also stored. The equations
are solved to give a search direction , which is expanded to an -vector by the insertion of appropriate zero elements. Then is found such that is approximately a minimum (subject to the fixed bounds) with respect to ; is replaced by , and the matrices and are updated so as to be consistent with the change produced in the gradient by the step . If any variable actually reaches a bound during the search along , it is fixed and is reduced for the next iteration.
There are two sets of convergence criteria – a weaker and a stronger. Whenever the weaker criteria are satisfied, the Lagrange-multipliers are estimated for all the active constraints. If any Lagrange-multiplier estimate is significantly negative, then one of the variables associated with a negative Lagrange-multiplier estimate is released from its bound and the next search direction is computed in the extended subspace (i.e., is increased). Otherwise minimization continues in the current subspace provided that this is practicable. When it is not, or when the stronger convergence criteria is already satisfied, then, if one or more Lagrange-multiplier estimates are close to zero, a slight perturbation is made in the values of the corresponding variables in turn until a lower function value is obtained. The normal algorithm is then resumed from the perturbed point.
If a saddle point is suspected, a local search is carried out with a view to moving away from the saddle point. In addition, e04kbc gives you the option of specifying that a local search should be performed when a point is found which is thought to be a constrained minimum.
If you specify that the problem is unconstrained, e04kbc sets the to and the to . Thus, provided that the problem has been sensibly scaled, no bounds will be encountered during the minimization process and e04kbc will act as an unconstrained minimization algorithm.
Gill P E and Murray W (1972) Quasi-Newton methods for unconstrained optimization J. Inst. Math. Appl.9 91–108
Gill P E and Murray W (1973) Safeguarded steplength algorithms for optimization using descent methods NPL Report NAC 37 National Physical Laboratory
Gill P E and Murray W (1976) Minimization subject to bounds on the variables NPL Report NAC 72 National Physical Laboratory
Gill P E, Murray W and Pitfield R A (1972) The implementation of two revised quasi-Newton algorithms for unconstrained optimization NPL Report NAC 11 National Physical Laboratory
1: – IntegerInput
On entry: the number of independent variables.
2: – function, supplied by the userExternal Function
objfun must evaluate the function and its first derivatives at any point . (However, if you do not wish to calculate or its first derivatives at a particular , there is the option of setting an argument to cause e04kbc to terminate immediately.)
On entry: the point at which the value of , or and , are required.
3: – double *Output
On exit: objfun must set objf to the value of the objective function at the current point . If it is not possible to evaluate , then objfun should assign a negative value to ; e04kbc will then terminate.
4: – doubleOutput
On exit: if on entry, then objfun must set to the value of the first derivative at the current point, for . If it is not possible to evaluate the first derivatives then objfun should assign a negative value to ; e04kbc will then terminate.
(If on entry, objfun must not change the elements of g.)
5: – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to objfun.
flag – IntegerInput/Output
On entry: will be set to 0 or . The value 0 indicates that only itself needs to be evaluated. The value 2 indicates that both and its first derivatives must be calculated.
On exit: if objfun resets to some negative number then e04kbc will terminate immediately with the error indicator NE_USER_STOP. If fail is supplied to e04kbc, will be set to your setting of .
first – Nag_BooleanInput
On entry: will be set to Nag_TRUE on the first call to objfun and Nag_FALSE for all subsequent calls.
nf – IntegerInput
On entry: the number of calculations of the objective function; this value will be equal to the number of calls made to objfun, including the current one.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void * with a C compiler that defines void * and char * otherwise.
Before calling e04kbc these pointers may be allocated memory and initialized with various quantities for use by objfun when called from e04kbc.
Note:objfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04kbc. If your code inadvertently does return any NaNs or infinities, e04kbc is likely to produce unexpected results.
Note:objfun should be tested separately before being used in conjunction with e04kbc. The array x must not be changed by objfun.
3: – Nag_BoundTypeInput
On entry: indicates whether the problem is unconstrained or bounded and, if it is bounded, whether the facility for dealing with bounds of special forms is to be used. bound should be set to one of the following values:
If the variables are bounded and you will be supplying all the and individually.
If the problem is unconstrained.
If the variables are bounded, but all the bounds are of the form .
If all the variables are bounded, and and .
, , or .
4: – doubleInput/Output
On entry: the lower bounds .
If , you must set to , for . (If a lower bound is not required for any , the corresponding should be set to a large negative number, e.g., .)
If , you must set to ; e04kbc will then set the remaining elements of bl equal to .
On exit: the first derivative vector corresponding to the final point in x. The elements of g corresponding to free variables should normally be close to zero.
9: – Nag_E04_Opt *Input/Output
On entry/exit: a pointer to a structure of type Nag_E04_Opt whose members are optional parameters for e04kbc. These structure members offer the means of adjusting some of the argument values of the algorithm and on output will supply further details of the results. A description of the members of options is given below in Section 11. Some of the results returned in options can be used by e04kbc to perform a ‘warm start’ if it is re-entered (see the member in Section 11.2).
If any of these optional parameters are required then the structure options should be declared and initialized by a call to e04xxc and supplied as an argument to e04kbc. However, if the optional parameters are not required the NAG defined null pointer, E04_DEFAULT, can be used in the function call.
10: – Nag_Comm *Input/Output
Note:comm is a NAG defined type (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
On entry/exit: structure containing pointers for communication with user-supplied functions; see the above description of objfun for details. If you do not need to make use of this communication feature the null pointer NAGCOMM_NULL may be used in the call to e04kbc; comm will then be declared internally for use in calls to user-supplied functions.
11: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
An exit of , NW_COND_MIN and NW_LOCAL_SEARCH may also be caused by mistakes in objfun, by the formulation of the problem or by an awkward function. If there are no such mistakes, it is worth restarting the calculations from a different starting point (not the point at which the failure occurred) in order to avoid the region which caused the failure.
The lower bound for variable (array element ) is greater than the upper bound.
An overflow would have occurred during the updating of the Cholesky factors if the calculations had been allowed to continue. Restart from the current point with .
Large errors were found in the derivatives of the objective function.
The initial values of the supplied has some value(s) which is negative or too small or the ratio of the largest element of to the smallest is too large.
Option but the pointer in the option structure has not been allocated memory.
On entry, .
Value given to is not valid. Correct range is .
Value given to not valid. Correct range is .
Value given to not valid. Correct range is .
Option but at least one of the pointers in the option structure has not been allocated memory.
Cannot open file for appending.
Cannot close file .
Options structure not initialized.
User requested termination, user flag value .
This exit occurs if you set to a negative value in objfun. If fail is supplied the value of will be the same as your setting of .
Error occurred when writing to file .
The conditions for a minimum have not all been satisfied, but a lower point could not be found.
Provided that, on exit, the first derivatives of with respect to the free variables are sufficiently small, and that the estimated condition number of the second derivative matrix is not too large, this error exit may simply mean that, although it has not been possible to satisfy the specified requirements, the algorithm has in fact found the minimum as far as the accuracy of the machine permits. This could be because has been set so small that rounding error in objfun makes attainment of the convergence conditions impossible.
If the estimated condition number of the approximate Hessian matrix at the final point is large, it could be that the final point is a minimum but that the smallest eigenvalue of the second derivative matrix is so close to zero that it is not possible to recognize the point as a minimum.
The local search has failed to find a feasible point which gives a significant change of function value.
If the problem is a genuinely unconstrained one, this type of exit indicates that the problem is extremely ill conditioned or that the function has no minimum. If the problem has bounds which may be close to the minimum, it may just indicate that steps in the subspace of free variables happened to meet a bound before they changed the function value.
The maximum number of iterations, , have been performed.
If steady reductions in , were monitored up to the point where this exit occurred, then the exit probably occurred simply because was set too small, so the calculations should be restarted from the final point held in x. This exit may also indicate that has no minimum.
A successful exit is made from e04kbc when (B1, B2 and B3) or B4 hold, and the local search (if used) confirms a minimum, where
(Quantities with superscript are the values at the th iteration of the quantities mentioned in Section 3; is the machine precision, denotes the Euclidean norm and is described in Section 11.)
If , then the vector in x on exit, , is almost certainly an estimate of the position of the minimum, , to the accuracy specified by .
If or NW_LOCAL_SEARCH, may still be a good estimate of , but the following checks should be made. Let the largest of the first elements of be , let the smallest be , and define . The scalar is usually a good estimate of the condition number of the projected Hessian matrix at . If
(a)the sequence converges to at a superlinear or a fast linear rate,
then it is almost certain that is a close approximation to the position of a minimum. When (b) is true, then usually is a close approximation to . The quantities needed for these checks are all available in the results printout from e04kbc; in particular the final value of Cond H gives .
The number of iterations required depends on the number of variables, the behaviour of , the accuracy demanded and the distance of the starting point from the solution. The number of multiplications performed in an iteration of e04kbc is roughly proportional to . In addition, each iteration makes at least one call of objfun with if is used or one call of objfun with if is chosen. So, unless can be evaluated very quickly, the run time will be dominated by the time spent in objfun.
Ideally, the problem should be scaled so that, at the solution, and the corresponding values of the are each in the range , and so that at points one unit away from the solution, differs from its value at the solution by approximately one unit. This will usually imply that the Hessian matrix at the solution is well conditioned. It is unlikely that you will be able to follow these recommendations very closely, but it is worth trying (by guesswork), as sensible scaling will reduce the difficulty of the minimization problem, so that e04kbc will take less computer time.
If a problem is genuinely unconstrained and has been scaled sensibly, the following points apply:
(a) will always be ,
(b)if or on entry, has simply to be set to , for ,
(c) and will be factors of the full approximate second derivative matrix with elements stored in the natural order,
(d)the elements of g should all be close to zero at the final point,
(e)the Status values given in the printout from e04kbc and in on exit are unlikely to be of interest (unless they are negative, which would indicate that the modulus of one of the has reached for some reason),
(f)Norm g simply gives the norm of the first derivative vector.
This example minimizes the function
subject to the bounds
starting from the initial guess .
The options structure is declared and initialized by e04xxc. Four option values are read from a data file by use of e04xyc. The memory freeing function e04xzc is used to free the memory assigned to the pointers in the option structure. You must not use the standard C function free() for this purpose.
A number of optional input and output arguments to e04kbc are available through the structure argument options, type Nag_E04_Opt. An argument may be selected by assigning an appropriate value to the relevant structure member; those arguments not selected will be assigned default values. If no use is to be made of any of the optional parameters you should use the NAG defined null pointer, E04_DEFAULT, in place of options when calling e04kbc; the default settings will then be used for all arguments.
Before assigning values to options directly the structure must be initialized by a call to the function e04xxc. Values may then be assigned to the structure members in the normal C manner.
After return from e04kbc, the options structure may only be re-used for future calls of e04kbc if the dimensions of the new problem are the same. Otherwise, the structure must be cleared by a call of e04xzc) and re-initialized by a call of e04xxc before future calls. Failure to do this will result in unpredictable behaviour.
Option settings may also be read from a text file using the function e04xyc in which case initialization of the options structure will be performed automatically if not already done. Any subsequent direct assignment to the options structure must not be preceded by initialization.
If assignment of functions and memory to pointers in the options structure is required, then this must be done directly in the calling program; they cannot be assigned using e04xyc.
11.1Optional Parameter Checklist and Default Values
For easy reference, the following list shows the members of options which are valid for e04kbc together with their default values where relevant. The number is a generic notation for machine precision (see X02AJC).
On entry: if a check of the derivatives defined by objfun will be made at the starting point x. The derivative check is carried out by a call to e04hcc. If is set to a value other than its default value () then the default of will be Nag_FALSE. A starting point of or should be avoided if this test is to be meaningful, if either of these starting points is necessary then e04hcc should be used to check objfun at a different point prior to calling e04kbc.
init_state – Nag_InitType
On entry: specifies which of the arguments objf, g, , and are actually being initialized. Such information will generally reduce the time taken by e04kbc.
No values are assumed to have been set in any of objf, g, , or . (e04kbc will use the unit matrix as the initial estimate of the Hessian matrix.)
The arguments objf and g must contain the value of and its first derivatives at the starting point. The elements must have been set to estimates of the derivatives at the starting point. No values are assumed to have been set in or .
The arguments objf and g must contain the value of and its first derivatives at the starting point. All elements of must have been set to indicate which variables are on their bounds and which are free. and must contain the Cholesky factors of a positive definite approximation to the Hessian matrix for the subspace of free variables. (This option is useful for restarting the minimization process if is reached.)
No values are assumed to have been set in objf or g, but , and must have been set as for . (This option is useful for starting off a minimization run using second derivative information from a previous, similar, run.)
, , or .
max_iter – Integer
On entry: the limit on the number of iterations allowed before termination.
optim_tol – double
On entry: the accuracy in to which the solution is required. If is the true value of at the minimum, then , the estimated position prior to a normal exit, is such that
where . For example, if the elements of are not much larger than in modulus and if is set to , then is usually accurate to about 5 decimal places. (For further details see Section 9.) If the problem is scaled roughly as described in Section 9 and is the machine precision, then is probably the smallest reasonable choice for . (This is because, normally, to machine accuracy, where is any column of the identity matrix.)
minlin – Nag_LinFun
On entry: specifies whether the linear minimizations (i.e., minimizations of with respect to ) are to be performed by a function which just requires the evaluation of , , or by a function which also requires the first derivatives of , .
It will often be possible to evaluate the first derivatives of in about the same amount of computer time that is required for the evaluation of itself – if this is so then e04kbc should be called with set to . However, if the evaluation of the derivatives takes more than about 4 times as long as the evaluation of , then a setting of will usually be preferable. If in doubt, use the default setting as it is slightly more robust.
linesearch_tol – double
Default if , and otherwise
If then the default value of will be changed from to if .
On entry: every iteration of e04kbc involves a linear minimization (i.e., minimization of with respect to ). specifies how accurately these linear minimizations are to be performed. The minimum with respect to will be located more accurately for small values of (say 0.01) than for large values (say 0.9).
Although accurate linear minimizations will generally reduce the number of iterations performed by e04kbc, they will increase the number of function evaluations required for each iteration. On balance, it is usually more efficient to perform a low accuracy linear minimization.
A smaller value such as may be worthwhile:
(a)if objfun takes so little computer time that it is worth using extra calls of objfun to reduce the number of iterations and associated matrix calculations
(b)if is a penalty or barrier function arising from a constrained minimization problem (since such problems are very difficult to solve)
(c)if and the calculation of first derivatives takes so much computer time (relative to the time taken to evaluate the function) that it is worth using extra function evaluations to reduce the number of derivative evaluations.
If , the default for (if the problem is effectively one-dimensional then should be set to even though ; i.e., if for all except one of the variables the lower and upper bounds are equal).
step_max – double
On entry: an estimate of the Euclidean distance between the solution and the starting point supplied. (For maximum efficiency a slight overestimate is preferable.) e04kbc will ensure that, for each iteration,
where is the iteration number. Thus, if the problem has more than one solution, e04kbc is most likely to find the one nearest the starting point. On difficult problems, a realistic choice can prevent the sequence of entering a region where the problem is ill-behaved and can also help to avoid possible overflow in the evaluation of . However an underestimate of can lead to inefficiency.
f_est – double
On entry: an estimate of the function value at the minimum. This estimate is just used for calculating suitable step lengths for starting linear minimizations off, so the choice is not too critical. However, it is better for to be set to an underestimate rather than to an overestimate. If no value is supplied then an initial step length of , subject to the variable bounds, will be used.
local_search – Nag_Boolean
On entry: must specify whether or not you wish a ‘local search’ to be performed when a point is found which is thought to be a constrained minimum.
If and either the quasi-Newton direction of search fails to produce a lower function value or the convergence criteria are satisfied, then a local search will be performed. This may move the search away from a saddle point or confirm that the final point is a minimum.
If there will be no local search when a point is found which is thought to be a minimum.
The amount of work involved in a local search is comparable to twice that required in a normal iteration to minimize with respect to . For most problems this will be small (relative to the total time required for the minimization). could be set Nag_FALSE if:
–it is known from the physical properties of a problem that a stationary point will be the required minimum;
–a point which is not a minimum could be easily recognized, for example if the value of at the minimum is known.
state – Integer *
On entry: need not be set if the default option of is used as n values of memory will be automatically allocated by e04kbc.
If or has been chosen, must point to a minimum of n elements of memory. This memory will already be available if the calling program has used the options structure in a previous call to e04kbc with and the same value of n. If a previous call has not been made you must allocate sufficient memory.
When or then must specify information about which variables are currently on their bounds and which are free. If is:
(a)fixed on its upper bound, is ;
(b)fixed on its lower bound, is ;
(c)effectively a constant (i.e., ), is ;
(d)free, gives its position in the sequence of free variables.
If or , will be initialized by e04kbc.
If or , must be initialized before e04kbc is called.
On exit: gives information as above about the final point given in x.
hesl – double *
hesd – double *
On entry: and need not be set if the default of is used as sufficient memory will be automatically allocated by e04kbc.
If or has been set then must point to a minimum of elements of memory.
must point to at least n elements of memory if , or has been chosen.
The appropriate amount of memory will already be available for and if the calling program has used the options structure in a previous call to e04kbc with and the same value of n. If a previous call has not been made, you must allocate sufficient memory.
and are used to store the factors and of the current approximation to the matrix of second derivatives with respect to the free variables (see Section 3). (The elements of the matrix are assumed to be ordered according to the permutation specified by the positive elements of , see above.) holds the lower triangle of , omitting the unit diagonal, stored by rows. stores the diagonal elements of . Thus if elements of are positive, the strict lower triangle of will be held in the first elements of and the diagonal elements of in the first elements of .
If (the default), and will be initialized within e04kbc to the factors of the unit matrix.
If you set , must contain on entry an approximation to the second derivative with respect to , for . need not be set.
If or , and must contain on entry the Cholesky factors of a positive definite approximation to the matrix of second derivatives for the subspace of free variables as specified by your setting of .
On exit: and hold the factors and corresponding to the final point given in x. The elements of are useful for deciding whether to accept the result produced by e04kbc (see Section 9).
iter – Integer
On exit: the number of iterations which have been performed in e04kbc.
nf – Integer
On exit: the number of times the residuals have been evaluated (i.e., number of calls of objfun).
11.3Description of Printed Output
The level of printed output can be controlled with the structure members and (see Section 11.2). If then the argument values to e04kbc are listed, whereas the printout of results is governed by the value of . The default of provides a single line of output at each iteration and the final result. This section describes all of the possible levels of results printout available from e04kbc.
When or a single line of output is produced on completion of each iteration, this gives the following values:
the Euclidean norm of the projected gradient vector, .
the Euclidean norm of .
the Euclidean norm of .
the step taken along the computed search direction .
the ratio of the largest to the smallest element of the diagonal factor of the projected Hessian matrix. This quantity is usually a good estimate of the condition number of the projected Hessian matrix. (If no variables are currently free, this value will be zero.)
When , or this single line of output is also produced for the final solution.
When more detailed results are given at each iteration. Additional values output are:
the current point .
the current projected gradient vector, .
the current state of the variable with respect to its bound(s).
If , or the final result is printed out. This consists of:
the final point, .
the final projected gradient vector, .
the final state of the variable with respect to its bound(s).
If then printout will be suppressed; you can print the final solution when e04kbc returns to the calling program.
11.3.1Output of results via a user-defined printing function
You may also specify your own print function for output of iteration results and the final solution by use of the function pointer, which has prototype
The rest of this section can be skipped if the default printing facilities provide the required functionality.
When a user-defined function is assigned to this will be called in preference to the internal print function of e04kbc. Calls to the user-defined function are again controlled by means of the member. Information is provided through st and comm, the two structure arguments to .
The results contained in the members of st are those on completion of the last iteration or those after a local search. (An iteration may be followed by a local search (see , Section 11.2) in which case is called with the results of the last iteration () and then again when the local search has been completed ().)
If then the results on completion of an iteration of e04kbc are contained in the members of st. If then the final results from e04kbc, including details of the final iteration, are contained in the members of st. In both cases, the same members of st are set, as follows:
iter – Integer
The current iteration count, , if ; the final iteration count, , if .
n – Integer
The number of variables.
x – double *
The coordinates of the point .
f – double *
The value of the current objective function.
g – double *
Points to the n memory locations holding the first derivatives of at the current point .
gpj_norm – double *
The Euclidean norm of the current projected gradient .
step – double *
The step taken along the search direction .
cond – double *
The estimate of the condition number of the Hessian matrix.
xk_norm – double *
The Euclidean norm of .
state – Integer
The status of variables , , with respect to their bounds. See Section 3 for a description of the possible status values.
Will be Nag_TRUE when the print function is called with the results of the current iteration.
sol_prt – Nag_Boolean
Will be Nag_TRUE when the print function is called with the final result.
user – double *
iuser – Integer *
p – Pointer
Pointers for communication of user information. If used they must be allocated memory either before entry to e04kbc or during a call to objfun or . The type Pointer will be void * with a C compiler that defines void * and char * otherwise.