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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_opt_bounds_quasi_func_easy (e04jy)

Purpose

nag_opt_bounds_quasi_func_easy (e04jy) is an easy-to-use quasi-Newton algorithm for finding a minimum of a function F(x1,x2,,xn)F(x1,x2,,xn), subject to fixed upper and lower bounds of the independent variables x1,x2,,xnx1,x2,,xn, using function values only.
It is intended for functions which are continuous and which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).

Syntax

[bl, bu, x, f, iw, w, user, ifail] = e04jy(ibound, funct1, bl, bu, x, 'n', n, 'liw', liw, 'lw', lw, 'user', user)
[bl, bu, x, f, iw, w, user, ifail] = nag_opt_bounds_quasi_func_easy(ibound, funct1, bl, bu, x, 'n', n, 'liw', liw, 'lw', lw, 'user', user)

Description

nag_opt_bounds_quasi_func_easy (e04jy) is applicable to problems of the form:
Minimize F(x1,x2,,xn)  subject to  ljxjuj,  j = 1,2,,n
Minimize F(x1,x2,,xn)  subject to  ljxjuj,  j=1,2,,n
when derivatives of F(x)F(x) are unavailable.
Special provision is made for problems which actually have no bounds on the xjxj, problems which have only non-negativity bounds and problems in which l1 = l2 = = lnl1=l2==ln and u1 = u2 = = unu1=u2==un. You must supply a function to calculate the value of F(x)F(x) at any point xx.
From a starting point you supplied there is generated, on the basis of estimates of the gradient and the curvature of F(x)F(x), a sequence of feasible points which is intended to converge to a local minimum of the constrained function. An attempt is made to verify that the final point is a minimum.
A typical iteration starts at the current point xx where nznz (say) variables are free from both their bounds. The projected gradient vector gzgz, whose elements are finite difference approximations to the derivatives of F(x)F(x) with respect to the free variables, is known. A unit lower triangular matrix LL and a diagonal matrix DD (both of dimension nznz), such that LDLTLDLT is a positive definite approximation of the matrix of second derivatives with respect to the free variables (i.e., the projected Hessian) are also held. The equations
LDLTpz = gz
LDLTpz=-gz
are solved to give a search direction pzpz, which is expanded to an nn-vector pp by an insertion of appropriate zero elements. Then αα is found such that F(x + αp)F(x+αp) is approximately a minimum (subject to the fixed bounds) with respect to αα; xx is replaced by x + αpx+αp, and the matrices LL and DD are updated so as to be consistent with the change produced in the estimated gradient by the step αpαp. If any variable actually reaches a bound during the search along pp, it is fixed and nznz is reduced for the next iteration. Most iterations calculate gzgz using forward differences, but central differences are used when they seem necessary.
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., nznz is increased). Otherwise minimization continues in the current subspace provided that this is practicable. When it is not, or when the stronger convergence criteria are 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. A local search is also performed when a point is found which is thought to be a constrained minimum.

References

Gill P E and Murray W (1976) Minimization subject to bounds on the variables NPL Report NAC 72 National Physical Laboratory

Parameters

Compulsory Input Parameters

1:     ibound – int64int32nag_int scalar
Indicates whether the facility for dealing with bounds of special forms is to be used.
It must be set to one of the following values:
ibound = 0ibound=0
If you are supplying all the ljlj and ujuj individually.
ibound = 1ibound=1
If there are no bounds on any xjxj.
ibound = 2ibound=2
If all the bounds are of the form 0xj0xj.
ibound = 3ibound=3
If l1 = l2 = = lnl1=l2==ln and u1 = u2 = = unu1=u2==un.
2:     funct1 – function handle or string containing name of m-file
You must supply funct1 to calculate the value of the function F(x)F(x) at any point xx. It should be tested separately before being used with nag_opt_bounds_quasi_func_easy (e04jy) (see the E04 Chapter Introduction).
[fc, user] = funct1(n, xc, user)

Input Parameters

1:     n – int64int32nag_int scalar
The number nn of variables.
2:     xc(n) – double array
The point xx at which the function value is required.
3:     user – Any MATLAB object
funct1 is called from nag_opt_bounds_quasi_func_easy (e04jy) with the object supplied to nag_opt_bounds_quasi_func_easy (e04jy).

Output Parameters

1:     fc – double scalar
The value of the function FF at the current point xx.
2:     user – Any MATLAB object
3:     bl(n) – double array
n, the dimension of the array, must satisfy the constraint n1n1.
The lower bounds ljlj.
If ibound is set to 00, you must set bl(j)blj to ljlj, for j = 1,2,,nj=1,2,,n. (If a lower bound is not specified for a particular xjxj, the corresponding bl(j)blj should be set to 106-106.)
If ibound is set to 33, you must set bl(1)bl1 to l1l1; nag_opt_bounds_quasi_func_easy (e04jy) will then set the remaining elements of bl equal to bl(1)bl1.
4:     bu(n) – double array
n, the dimension of the array, must satisfy the constraint n1n1.
The upper bounds ujuj.
If ibound is set to 00, you must set bu(j)buj to ujuj, for j = 1,2,,nj=1,2,,n. (If an upper bound is not specified for a particular xjxj, the corresponding bu(j)buj should be set to 106106.)
If ibound is set to 33, you must set bu(1)bu1 to u1u1; nag_opt_bounds_quasi_func_easy (e04jy) will then set the remaining elements of bu equal to bu(1)bu1.
5:     x(n) – double array
n, the dimension of the array, must satisfy the constraint n1n1.
x(j)xj must be set to an estimate of the jjth component of the position of the minimum, for j = 1,2,,nj=1,2,,n.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the arrays bl, bu, x. (An error is raised if these dimensions are not equal.)
The number nn of independent variables.
Constraint: n1n1.
2:     liw – int64int32nag_int scalar
Default: n + 2n+2
The dimension of the array iw as declared in the (sub)program from which nag_opt_bounds_quasi_func_easy (e04jy) is called.
Constraint: liwn + 2liwn+2.
3:     lw – int64int32nag_int scalar
Default: max (n × (n1) / 2 + 12 × n,13)max(n×(n-1)/2+12×n,13)
The dimension of the array w as declared in the (sub)program from which nag_opt_bounds_quasi_func_easy (e04jy) is called.
Constraint: lwmax (n × (n1) / 2 + 12 × n,13)lwmax(n×(n-1)/2+12×n,13).
4:     user – Any MATLAB object
user is not used by nag_opt_bounds_quasi_func_easy (e04jy), but is passed to funct1. Note that for large objects it may be more efficient to use a global variable which is accessible from the m-files than to use user.

Input Parameters Omitted from the MATLAB Interface

iuser ruser

Output Parameters

1:     bl(n) – double array
The lower bounds actually used by nag_opt_bounds_quasi_func_easy (e04jy).
2:     bu(n) – double array
The upper bounds actually used by nag_opt_bounds_quasi_func_easy (e04jy).
3:     x(n) – double array
The lowest point found during the calculations. Thus, if ifail = 0ifail=0 on exit, x(j)xj is the jjth component of the position of the minimum.
4:     f – double scalar
The value of F(x)F(x) corresponding to the final point stored in x.
5:     iw(liw) – int64int32nag_int array
If ifail = 0ifail=0, 33 or 55, the first n elements of iw contain information about which variables are currently on their bounds and which are free. Specifically, if xixi is:
fixed on its upper bound, iw(i)iwi is 1-1;
fixed on its lower bound, iw(i)iwi is 2-2;
effectively a constant (i.e., lj = ujlj=uj), iw(i)iwi is 3-3;
free, iw(i)iwi gives its position in the sequence of free variables.
In addition, iw(n + 1)iwn+1 contains the number of free variables (i.e., nznz). The rest of the array is used as workspace.
6:     w(lw) – double array
If ifail = 0ifail=0, 33 or 55, w(i)wi contains a finite difference approximation to the iith element of the projected gradient vector gzgz, for i = 1,2,,ni=1,2,,n. In addition, w(n + 1)wn+1 contains an estimate of the condition number of the projected Hessian matrix (i.e., kk). The rest of the array is used as workspace.
7:     user – Any MATLAB object
8:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Note: nag_opt_bounds_quasi_func_easy (e04jy) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

  ifail = 1ifail=1
On entry,n < 1n<1,
oribound < 0ibound<0,
oribound > 3ibound>3,
oribound = 0ibound=0 and bl(j) > bu(j)blj>buj for some jj,
oribound = 3ibound=3 and bl(1) > bu(1)bl1>bu1,
orliw < n + 2liw<n+2,
or lw < max (13,12 × n + n × (n1) / 2) lw < max(13,12×n+n×(n-1)/2) .
  ifail = 2ifail=2
There have been 400 × n400×n function evaluations, yet the algorithm does not seem to be converging. The calculations can be restarted from the final point held in x. The error may also indicate that F(x)F(x) has no minimum.
W ifail = 3ifail=3
The conditions for a minimum have not all been met but a lower point could not be found and the algorithm has failed.
  ifail = 4ifail=4
An overflow has occurred during the computation. This is an unlikely failure, but if it occurs you should restart at the latest point given in x.
W ifail = 5ifail=5
W ifail = 6ifail=6
W ifail = 7ifail=7
W ifail = 8ifail=8
There is some doubt about whether the point xx found by nag_opt_bounds_quasi_func_easy (e04jy) is a minimum. The degree of confidence in the result decreases as ifail increases. Thus, when ifail = 5ifail=5 it is probable that the final xx gives a good estimate of the position of a minimum, but when ifail = 8ifail=8 it is very unlikely that the function has found a minimum.
  ifail = 9ifail=9
In the search for a minimum, the modulus of one of the variables has become very large (106)(106). This indicates that there is a mistake in funct1, that your problem has no finite solution, or that the problem needs rescaling (see Section [Further Comments]).
  ifail = 10ifail=10
The computed set of forward-difference intervals (stored in w(9 × n + 1),w(9 × n + 2),, w(10 × n)w9×n+1,w9×n+2,, w10×n) is such that x(i) + w(9 × n + i)x(i)xi+w9×n+ixi for some ii.
This is an unlikely failure, but if it occurs you should attempt to select another starting point.
If you are dissatisfied with the result (e.g., because ifail = 5ifail=5, 66, 77 or 88), 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. If persistent trouble occurs and the gradient can be calculated, it may be advisable to change to a function which uses gradients (see the E04 Chapter Introduction).

Accuracy

A successful exit (ifail = 0ifail=0) is made from nag_opt_bounds_quasi_func_easy (e04jy) when (B1B1, B2B2 and B3B3) or B4B4 hold, and the local search confirms a minimum, where (Quantities with superscript kk are the values at the kkth iteration of the quantities mentioned in Section [Description], xtol = 100sqrt(ε)xtol=100ε, εε is the machine precision and . . denotes the Euclidean norm. The vector gzgz is returned in the array w.)
If ifail = 0ifail=0, then the vector in x on exit, xsolxsol, is almost certainly an estimate of the position of the minimum, xtruextrue, to the accuracy specified by xtolxtol.
If ifail = 3ifail=3 or 55, xsolxsol may still be a good estimate of xtruextrue, but the following checks should be made. Let kk denote an estimate of the condition number of the projected Hessian matrix at xsolxsol. (The value of kk is returned in w(n + 1)wn+1). If
(i) the sequence {F(x(k))}{F(x (k) )} converges to F(xsol)F(xsol) at a superlinear or a fast linear rate,
(ii) gz(xxol)2 < 10.0 × εgz(xxol)2<10.0×ε, and
(iii) k < 1.0 / gz(xsol)k<1.0/gz(xsol),
then it is almost certain that xsolxsol is a close approximation to the position of a minimum. When (ii) is true, then usually F(xsol)F(xsol) is a close approximation to F(xtrue)F(xtrue)
When a successful exit is made then, for a computer with a mantissa of tt decimals, one would expect to get about t / 21t/2-1 decimals accuracy in xx and about t1t-1 decimals accuracy in FF, provided the problem is reasonably well scaled.

Further Comments

The number of iterations required depends on the number of variables, the behaviour of F(x)F(x) and the distance of the starting point from the solution. The number of operations performed in an iteration of nag_opt_bounds_quasi_func_easy (e04jy) is roughly proportional to n2n2. In addition, each iteration makes at least m + 1m+1 calls of funct1, where mm is the number of variables not fixed on bounds. So, unless F(x)F(x) can be evaluated very quickly, the run time will be dominated by the time spent in funct1.
Ideally the problem should be scaled so that at the solution the value of F(x)F(x) and the corresponding values of x1,x2,,xnx1,x2,,xn are each in the range (1, + 1)(-1,+1), and so that at points a unit distance away from the solution, FF is approximately a unit value greater than at the minimum. 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 nag_opt_bounds_quasi_func_easy (e04jy) will take less computer time.

Example

function nag_opt_bounds_quasi_func_easy_example
ibound = int64(0);
bl = [1;
     -2;
     -1000000;
     1];
bu = [3;
     0;
     1000000;
     3];
x = [3;
     -1;
     0;
     1];
[blOut, buOut, xOut, f, iw, w, user, ifail] = ...
    nag_opt_bounds_quasi_func_easy(ibound, @funct1, bl, bu, x)

function [fc, user] = funct1(n, xc, user)
  fc = (xc(1)+10*xc(2))^2 + 5*(xc(3)-xc(4))^2 + (xc(2)-2*xc(3))^4 + 10*(xc(1)-xc(4))^4;
 
Warning: nag_opt_bounds_quasi_func_easy (e04jy) returned a warning indicator (5) 

blOut =

           1
          -2
    -1000000
           1


buOut =

           3
           0
     1000000
           3


xOut =

    1.0000
   -0.0852
    0.4093
    1.0000


f =

    2.4338


iw =

                   -2
                    1
                    2
                   -2
                    2
                   -2


w =

         0
   -0.0000
   -0.0000
         0
    1.0000
    1.5832
    4.3435
         0
         0
         0
         0
         0
    1.0000
   -0.0852
    0.4093
    1.0000
    1.0000
   -0.0852
    0.4093
    1.0000
         0
   -0.0015
   -0.0050
         0
   -0.0000
   -0.0000
    0.0622
         0
   -0.0000
   -0.0000
    0.3462
         0
   -0.0000
   -0.0000
    0.3462
         0
    0.0000
    0.0000
    0.0000
    0.0000
         0
         0
         0
         0
         0
         0
    1.0000
    1.0000
         0
         0
    0.2953
   -0.0000
   -0.0000
    5.9070


user =

     0


ifail =

                    5


function e04jy_example
ibound = int64(0);
bl = [1;
     -2;
     -1000000;
     1];
bu = [3;
     0;
     1000000;
     3];
x = [3;
     -1;
     0;
     1];
[blOut, buOut, xOut, f, iw, w, user, ifail] = ...
    e04jy(ibound, @funct1, bl, bu, x)

function [fc, user] = funct1(n, xc, user)
  fc = (xc(1)+10*xc(2))^2 + 5*(xc(3)-xc(4))^2 + (xc(2)-2*xc(3))^4 + 10*(xc(1)-xc(4))^4;
 
Warning: nag_opt_bounds_quasi_func_easy (e04jy) returned a warning indicator (5) 

blOut =

           1
          -2
    -1000000
           1


buOut =

           3
           0
     1000000
           3


xOut =

    1.0000
   -0.0852
    0.4093
    1.0000


f =

    2.4338


iw =

                   -2
                    1
                    2
                   -2
                    2
                   -2


w =

         0
   -0.0000
   -0.0000
         0
    1.0000
    1.5832
    4.3435
         0
         0
         0
         0
         0
    1.0000
   -0.0852
    0.4093
    1.0000
    1.0000
   -0.0852
    0.4093
    1.0000
         0
   -0.0015
   -0.0050
         0
   -0.0000
   -0.0000
    0.0622
         0
   -0.0000
   -0.0000
    0.3462
         0
   -0.0000
   -0.0000
    0.3462
         0
    0.0000
    0.0000
    0.0000
    0.0000
         0
         0
         0
         0
         0
         0
    1.0000
    1.0000
         0
         0
    0.2953
   -0.0000
   -0.0000
    5.9070


user =

     0


ifail =

                    5



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