NAG Library Routine Document
E04MFF/E04MFA
Note: this routine uses optional parameters to define choices in the problem specification and in the details of the algorithm. If you wish to use default
settings for all of the optional parameters, you need only read Sections 1 to 9 of this document.
If, however, you wish to reset some or all of the settings please refer to
Section 10 for a detailed description of the algorithm, to
Section 11 for a detailed description of the specification of the optional parameters and to
Section 12 for a detailed description of the monitoring information produced by the routine.
1 Purpose
E04MFF/E04MFA solves general linear programming problems. It is not intended for large sparse problems.
E04MFA is a version of E04MFF that has additional parameters in order to make it safe for use in multithreaded applications (see
Section 5). The initialization routine
E04WBF must have been called before calling E04MFA.
2 Specification
2.1 Specification for E04MFF
| SUBROUTINE E04MFF ( | N, NCLIN, A, LDA, BL, BU, CVEC, ISTATE, X, ITER, OBJ, AX, CLAMDA, IWORK, LIWORK, WORK, LWORK, IFAIL) |
| INTEGER | N, NCLIN, LDA, ISTATE(N+NCLIN), ITER, IWORK(LIWORK), LIWORK, LWORK, IFAIL |
| double precision | A(LDA,*), BL(N+NCLIN), BU(N+NCLIN), CVEC(*), X(N), OBJ, AX(max(1,NCLIN)), CLAMDA(N+NCLIN), WORK(LWORK) |
2.2 Specification for E04MFA
| SUBROUTINE E04MFA ( | N, NCLIN, A, LDA, BL, BU, CVEC, ISTATE, X, ITER, OBJ, AX, CLAMDA, IWORK, LIWORK, WORK, LWORK, LWSAV, IWSAV, RWSAV, IFAIL) |
| INTEGER | N, NCLIN, LDA, ISTATE(N+NCLIN), ITER, IWORK(LIWORK), LIWORK, LWORK, IWSAV(610), IFAIL |
| double precision | A(LDA,*), BL(N+NCLIN), BU(N+NCLIN), CVEC(*), X(N), OBJ, AX(max(1,NCLIN)), CLAMDA(N+NCLIN), WORK(LWORK), RWSAV(475) |
| LOGICAL | LWSAV(120) |
Before calling E04MFA, or either of the option setting routines
E04MGA or
E04MHA,
E04WBF
must be called. The specification for
E04WBF is:
| SUBROUTINE E04WBF ( | RNAME, CWSAV, LCWSAV, LWSAV, LLWSAV, IWSAV, LIWSAV, RWSAV, LRWSAV, IFAIL) |
| INTEGER | LCWSAV, LLWSAV, IWSAV(LIWSAV), LIWSAV, LRWSAV, IFAIL |
| double precision | RWSAV(LRWSAV) |
| LOGICAL | LWSAV(LLWSAV) |
| CHARACTER*6 | RNAME |
| CHARACTER*80 | CWSAV(LCWSAV) |
E04WBF should be called with
RNAME='E04MFA'.
LCWSAV,
LLWSAV,
LIWSAV and
LRWSAV, the declared lengths of
CWSAV,
LWSAV,
IWSAV and
RWSAV respectively, must satisfy:
- LCWSAV≥1
- LLWSAV≥120
- LIWSAV≥610
- LRWSAV≥475
The contents of the arrays
CWSAV,
LWSAV,
IWSAV and
RWSAV
must not be altered between calling routines
E04MFA,
E04MGA,
E04MHA and
E04WBF.
3 Description
E04MFF/E04MFA is designed to solve linear programming (LP) problems of the form
where
c is an
n element vector and
A is an
mL by
n matrix.
This is the default type of problem, referred to as type LP. The optional parameter Problem Type may be used to specify an alternative problem type FP, in which the objective function is omitted and the routine attempts to find a feasible point for the set of constraints.
The constraints involving A are called the general constraints. Note that upper and lower bounds are specified for all the variables and for all the general constraints. An equality constraint can be specified by setting li=ui. If certain bounds are not present, the associated elements of l or u can be set to special values that will be treated as -∞ or +∞. (See the description of the optional parameter Infinite Bound Size.)
You must supply an initial estimate of the solution.
The method used by E04MFF/E04MFA is described in detail in
Section 10.
4 References
Gill P E, Hammarling S, Murray W, Saunders M A and Wright M H (1986) Users' guide for LSSOL (Version 1.0)
Report SOL 86-1 Department of Operations Research, Stanford University
Gill P E and Murray W (1978) Numerically stable methods for quadratic programming
Math. Programming 14 349–372
Gill P E, Murray W, Saunders M A and Wright M H (1984) Procedures for optimization problems with a mixture of bounds and general linear constraints
ACM Trans. Math. Software 10 282–298
Gill P E, Murray W, Saunders M A and Wright M H (1989) A practical anti-cycling procedure for linearly constrained optimization
Math. Programming 45 437–474
Gill P E, Murray W, Saunders M A and Wright M H (1991) Inertia-controlling methods for general quadratic programming
SIAM Rev. 33 1–36
Gill P E, Murray W and Wright M H (1981)
Practical Optimization Academic Press
5 Parameters
- 1: N – INTEGERInput
-
On entry:
n, the number of variables.
Constraint:
N>0.
- 2: NCLIN – INTEGERInput
-
On entry:
mL, the number of general linear constraints.
Constraint:
NCLIN≥0.
- 3: A(LDA,*) – double precision arrayInput
Note: the second dimension of the array
A
must be at least
N if
NCLIN>0 and at least
1 if
NCLIN=0.
On entry: the
ith row of
A must contain the coefficients of the
ith general linear constraint, for
i=1,2,…,mL.
If
NCLIN=0,
A is not referenced.
- 4: LDA – INTEGERInput
-
On entry: the first dimension of the array
A as declared in the (sub)program from which E04MFF/E04MFA is called.
Constraint:
LDA≥max1,NCLIN.
- 5: BL(N+NCLIN) – double precision arrayInput
- 6: BU(N+NCLIN) – double precision arrayInput
On entry: must contain the lower bounds and
BU the upper bounds, for all the constraints in the following order. The first
n elements of each array must contain the bounds on the variables, and the next
mL elements the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e.,
lj=-∞), set
BLj≤-bigbnd, and to specify a nonexistent upper bound (i.e.,
uj=+∞), set
BUj≥bigbnd; the default value of
bigbnd is
1020, but this may be changed by the optional parameter
Infinite Bound Size. To specify the
jth constraint as an
equality, set
BLj=BUj=β, say, where
β<bigbnd.
Constraints:
- BLj≤BUj, for j=1,2,…,N+NCLIN;
- if BLj=BUj=β, β<bigbnd.
- 7: CVEC(*) – double precision arrayInput
Note: the dimension of the array
CVEC
must be at least
N if when the problem is of type LP (the default), and at least
1 otherwise.
On entry: the coefficients of the objective function when the problem is of type LP.
If the problem is of type FP,
CVEC is not referenced.
- 8: ISTATE(N+NCLIN) – INTEGER arrayInput/Output
On entry: need not be set if the (default) optional parameter
Cold Start is used.
If the optional parameter
Warm Start has been chosen,
ISTATE specifies the desired status of the constraints at the start of the feasibility phase. More precisely, the first
n elements of
ISTATE refer to the upper and lower bounds on the variables, and the next
mL elements refer to the general linear constraints (if any). Possible values for
ISTATEj are as follows:
| ISTATEj |
Meaning |
| 0 |
The corresponding constraint should not be in the initial working set. |
| 1 |
The constraint should be in the initial working set at its lower bound. |
| 2 |
The constraint should be in the initial working set at its upper bound. |
| 3 |
The constraint should be in the initial working set as an equality. This value must not be specified unless BLj=BUj. |
The values
-2,
-1 and
4 are also acceptable but will be reset to zero by the routine. If E04MFF/E04MFA has been called previously with the same values of
N and
NCLIN,
ISTATE already contains satisfactory information. (See also the description of the optional parameter
Warm Start.) The routine also adjusts (if necessary) the values supplied in
X to be consistent with
ISTATE.
Constraint:
-2≤ISTATEj≤4, for j=1,2,…,N+NCLIN.
On exit: the status of the constraints in the working set at the point returned in
X. The significance of each possible value of
ISTATEj is as follows:
| ISTATEj |
Meaning |
| -2 |
The constraint violates its lower bound by more than the feasibility tolerance. |
| -1 |
The constraint violates its upper bound by more than the feasibility tolerance. |
| -0 |
The constraint is satisfied to within the feasibility tolerance, but is not in the working set. |
| -1 |
This inequality constraint is included in the working set at its lower bound. |
| -2 |
This inequality constraint is included in the working set at its upper bound. |
| -3 |
This constraint is included in the working set as an equality. This value of ISTATE can occur only when BLj=BUj. |
| -4 |
This corresponds to optimality being declared with Xj being temporarily fixed at its current value. This value of ISTATE can occur only when IFAIL=1 on exit. |
- 9: X(N) – double precision arrayInput/Output
On entry: an initial estimate of the solution.
On exit: the point at which E04MFF/E04MFA terminated. If
IFAIL=0,
1 or
4,
X contains an estimate of the solution.
- 10: ITER – INTEGEROutput
On exit: the total number of iterations performed.
- 11: OBJ – double precisionOutput
-
On exit: the value of the objective function at
x if
x is feasible, or the sum of infeasibiliites at
x otherwise. If the problem is of type FP and
x is feasible,
OBJ is set to zero.
- 12: AX(max1,NCLIN) – double precision arrayOutput
On exit: the final values of the linear constraints
Ax.
If
NCLIN=0,
AX is not referenced.
- 13: CLAMDA(N+NCLIN) – double precision arrayOutput
On exit: the values of the Lagrange multipliers for each constraint with respect to the current working set. The first n elements contain the multipliers for the bound constraints on the variables, and the next mL elements contain the multipliers for the general linear constraints (if any). If ISTATEj=0 (i.e., constraint j is not in the working set), CLAMDAj is zero. If x is optimal, CLAMDAj should be non-negative if ISTATEj=1, nonpositive if ISTATEj=2 and zero if ISTATEj=4.
- 14: IWORK(LIWORK) – INTEGER arrayWorkspace
- 15: LIWORK – INTEGERInput
On entry: the dimension of the array
IWORK as declared in the (sub)program from which E04MFF/E04MFA is called.
Constraint:
LIWORK≥2×N+3.
- 16: WORK(LWORK) – double precision arrayWorkspace
- 17: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which E04MFF/E04MFA is called.
Constraints:
- if the problem is of type LP (the default),
- if NCLIN=0, LWORK≥7×N+1;
- if NCLIN≥N, LWORK≥2×N2+7×N+5×NCLIN;
- otherwise LWORK≥2×NCLIN+12+7×N+5×NCLIN;
- if the problem is of type FP,
- if NCLIN=0, LWORK≥6×N+1;
- if NCLIN≥N, LWORK≥2×N2+6×N+5×NCLIN;
- otherwise LWORK≥2×NCLIN+12+6×N+5×NCLIN.
The amounts of workspace provided and required are (by default) output on the current advisory message unit (as defined by
X04ABF). As an alternative to computing
LIWORK and
LWORK from the formulas given above, you may prefer to obtain appropriate values from the output of a preliminary run with
LIWORK and
LWORK set to
1. (E04MFF/E04MFA will then terminate with
IFAIL=6.)
- 18: IFAIL – INTEGERInput/Output
-
Note: for E04MFA, IFAIL does not occur in this position in the parameter list. See the additional parameters described below.
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
IFAIL≠0 on exit, the recommended value is
-1.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit. E04MFF/E04MFA returns with
IFAIL=0 if
x is a strong local minimizer, i.e., the reduced gradient (
Norm Gz; see
Section 8.2) is negligible and the Lagrange multipliers (
Lagr Mult; see
Section 8.2) are optimal.
- Note: the following are additional parameters for specific use with E04MFA. Users of E04MFF therefore need not read the remainder of this description.
- 18: LWSAV(120) – LOGICAL arrayCommunication Array
- 19: IWSAV(610) – INTEGER arrayCommunication Array
- 20: RWSAV(475) – double precision arrayCommunication Array
The arrays
LWSAV,
IWSAV and
RWSAV must not be altered between calls to any of the routines E04MFA,
E04MGA,
E04MHA or
E04WBF.
- 21: IFAIL – INTEGERInput/Output
Note: see the parameter description for
IFAIL above.
6 Error Indicators and Warnings
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Note: E04MFF/E04MFA may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
- IFAIL=1
-
x is a weak local minimum (the projected gradient is negligible and the Lagrange multipliers are optimal but there is a small multiplier). This means that the solution is not unique.
- IFAIL=2
-
The solution appears to be unbounded, i.e., the objective function is not bounded below in the feasible region. This value of
IFAIL occurs if a step larger than
Infinite Step Size (
default value=1020) would have to be taken in order to continue the algorithm, or the next step would result in an element of
x having magnitude larger than optional parameter
Infinite Bound Size (
default value=1020).
- IFAIL=3
-
No feasible point was found, i.e., it was not possible to satisfy all the constraints to within the feasibility tolerance. In this case, the constraint violations at the final
x will reveal a value of the tolerance for which a feasible point will exist – for example, when the feasibility tolerance for each violated constraint exceeds its
Slack (see
Section 8.2) at the final point. The modified problem (with an altered feasibility tolerance) may then be solved using a
Warm Start. You should check that there are no constraint redundancies. If the data for the constraints are accurate only to the absolute precision
σ, you should ensure that the value of the optional parameter
Feasibility Tolerance (
default value=ε, where
ε is the
machine precision) is
greater than
σ. For example, if all elements of
A are of order unity and are accurate only to three decimal places, the
Feasibility Tolerance should be at least
10-3.
- IFAIL=4
-
The limiting number of iterations was reached before normal termination occurred.
The value of the optional parameter Iteration Limit (
default value
=
max50,5n+mL
) may be too small. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), either rerun E04MFF/E04MFA with a larger value of Iteration Limit or, alternatively, rerun E04MFF/E04MFA using the Warm Start facility to specify the initial working set.
- IFAIL=5
Not used by this routine.
- IFAIL=6
An input parameter is invalid.
- IFAIL=7
-
The designated problem type was not FP or LP. Rerun E04MFF/E04MFA with the optional parameter Problem Type set to one of these values.
- Overflow
If the printed output before the overflow error contains a warning about serious ill-conditioning in the working set when adding the jth constraint, it may be possible to avoid the difficulty by increasing the magnitude of the optional parameter Feasibility Tolerance and rerunning the program. If the message recurs even after this change, the offending linearly dependent constraint (with index ‘j’) must be removed from the problem.
7 Accuracy
E04MFF/E04MFA implements a numerically stable active set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.
8 Further Comments
This section contains some comments on scaling and a description of the printed output.
8.1 Scaling
Sensible scaling of the problem is likely to reduce the number of iterations required and make the problem less sensitive to perturbations in the data, thus improving the condition of the problem. In the absence of better information it is usually sensible to make the Euclidean lengths of each constraint of comparable magnitude. See the
E04 Chapter Introduction and
Gill et al. (1981) for further information and advice.
8.2 Description of the Printed Output
This section describes the intermediate printout and final printout produced by E04MFF/E04MFA. The intermediate printout is a subset of the monitoring information produced by the routine at every iteration (see
Section 12). You can control the level of printed output (see the description of the optional parameter
Print Level). Note that the intermediate printout and final printout are produced only if
Print Level≥10 (the default for E04MFF, by default no output is produced by E04MFA).
The following line of summary output (
<80 characters) is produced at every iteration. In all cases, the values of the quantities printed are those in effect
on completion of the given iteration.
| Itn |
is the iteration count.
|
| Step |
is the step taken along the computed search direction. If a constraint is added during the current iteration, Step will be the step to the nearest constraint. When the problem is of type LP, the step can be greater than one during the optimality phase.
|
| Ninf |
is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
|
| Sinf/Objective |
is the value of the current objective function. If x is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If x is feasible, Objective is the value of the objective function of (1). The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point. During the optimality phase the value of the objective function will be nonincreasing. During the feasibility phase the number of constraint infeasibilities will not increase until either a feasible point is found or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.
|
| Norm Gz |
is
ZRT
gFR
, the Euclidean norm of the reduced gradient with respect to ZR. During the optimality phase, this norm will be approximately zero after a unit step.
(See Sections 10.2 and 10.4.) |
The final printout includes a listing of the status of every variable and constraint.
The following describes the printout for each variable. A full stop (.) is printed for any numerical value that is zero.
| Varbl |
gives the name (V) and index j, for j=1,2,…,n, of the variable.
|
| State |
gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the Feasibility Tolerance, State will be ++ or -- respectively.
A key is sometimes printed before State.
| A |
Alternative optimum possible. The variable is active at one of its bounds, but its Lagrange multiplier is essentially zero. This means that if the variable were allowed to start moving away from its bound then there would be no change to the objective function. The values of the other free variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case the values of the Lagrange multipliers might also change.
|
| D |
Degenerate. The variable is free, but it is equal to (or very close to) one of its bounds.
|
| I |
Infeasible. The variable is currently violating one of its bounds by more than the Feasibility Tolerance.
|
|
| Value |
is the value of the variable at the final iteration.
|
| Lower Bound |
is the lower bound specified for the variable. None indicates that BLj≤-bigbnd.
|
| Upper Bound |
is the upper bound specified for the variable. None indicates that BUj≥bigbnd.
|
| Lagr Mult |
is the Lagrange multiplier for the associated bound. This will be zero if State is FR unless BLj≤-bigbnd and BUj≥bigbnd, in which case the entry will be blank. If x is optimal, the multiplier should be non-negative if State is LL and nonpositive if State is UL.
|
| Slack |
is the difference between the variable Value and the nearer of its (finite) bounds BLj and BUj. A blank entry indicates that the associated variable is not bounded (i.e., BLj≤-bigbnd and BUj≥bigbnd).
|
The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’,
BLj and
BUj are replaced by
BLn+j and
BUn+j respectively, and with the following change in the heading:
| L Con |
gives the name (L) and index j, for j=1,2,…,nL, of the linear constraint.
|
Note that movement off a constraint (as opposed to a variable moving away from its bound) can be interpreted as allowing the entry in the Slack column to become positive.
Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.
9 Example
This example minimizes the function
subject to the bounds
and the general constraints
The initial point, which is infeasible, is
The optimal solution (to five figures) is
Four bound constraints and three general constraints are active at the solution.
The document for
E04MGF/E04MGA includes an example program to solve the same problem using some of the optional parameters described in
Section 11.
9.1 Program Text
Note: the following programs illustrate the use of E04MFF and E04MFA.
Program Text (e04mffe.f)
Program Text (e04mfae.f)
9.2 Program Data
Program Data (e04mffe.d)
Program Data (e04mfae.d)
9.3 Program Results
Program Results (e04mffe.r)
Program Results (e04mfae.r)
Note: the remainder of this document is intended for more advanced users. Section 10 contains a detailed description of the algorithm which may be needed in order to understand Sections 11 and 12. Section 11 describes the optional parameters which may be set by calls to E04MGF/E04MGA and/or E04MHF/E04MHA. Section 12 describes the quantities which can be requested to monitor the course of the computation.
10 Algorithmic Details
This section contains a detailed description of the method used by E04MFF/E04MFA.
10.1 Overview
E04MFF/E04MFA is based on an inertia-controlling method due to
Gill and Murray (1978), and is described in detail by
Gill et al. (1991). Here we briefly summarize the main features of the method. Where possible, explicit reference is made to the names of variables that are parameters of E04MFF/E04MFA or appear in the printed output. E04MFF/E04MFA has two phases: finding an initial feasible point by minimizing the sum of infeasibilities (the
feasibility phase), and minimizing the linear objective function within the feasible region (the
optimality phase). The computations in both phases are performed by the same subroutines. The two-phase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities to the linear objective function. The feasibility phase does
not perform the standard simplex method (i.e., it does not necessarily find a vertex), except in the case when
mL≤n. Once any iterate is feasible, all subsequent iterates remain feasible.
In general, an iterative process is required to solve a linear program. (For simplicity, we shall always consider a typical iteration and avoid reference to the index of the iteration.) Each new iterate
x- is defined by
where the
step length
α is a non-negative scalar, and
p is called the
search direction.
At each point x, a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied ‘exactly’ (to within the tolerance defined by the optional parameter Feasibility Tolerance). The working set is the current prediction of the constraints that hold with equality at a solution of an LP problem. The search direction is constructed so that the constraints in the working set remain unaltered for any value of the step length. For a bound constraint in the working set, this property is achieved by setting the corresponding element of the search direction to zero. Thus, the associated variable is fixed, and specification of the working set induces a partition of x into fixed and free variables. During a given iteration, the fixed variables are effectively removed from the problem; since the relevant elements of the search direction are zero, the columns of A corresponding to fixed variables may be ignored.
Let
mW denote the number of general constraints in the working set and let
nFX denote the number of variables fixed at one of their bounds (
mW and
nFX are the quantities
Lin and
Bnd in the monitoring file output from E04MFF/E04MFA; see
Section 12). Similarly, let
nFR (
nFR=n-nFX) denote the number of free variables. At every iteration,
the variables are reordered so that the last
nFX variables are fixed, with all other relevant vectors and matrices ordered accordingly.
10.2 Definition of Search Direction
Let
AFR denote the
mW by
nFR sub-matrix of general constraints in the working set corresponding to the free variables, and let
pFR denote the search direction with respect to the free variables only. The general constraints in the working set will be unaltered by any move along
p if
In order to compute
pFR, the
TQ factorization of
AFR is used:
where
T is a nonsingular
mW by
mW upper triangular matrix (i.e.,
tij=0 if
i>j), and the nonsingular
nFR by
nFR matrix
QFR is the product of orthogonal transformations (see
Gill et al. (1984)). If the columns of
QFR are partitioned so that
where
Y is
nFR by
mW, then the
nZ (
nZ=nFR-mW) columns of
Z form a basis for the null space of
AFR. Let
nR be an integer such that
0≤nR≤nZ, and let
ZR denote a matrix whose
nR columns are a subset of the columns of
Z. (The integer
nR is the quantity
Zr in the monitoring file output from E04MFF/E04MFA. In many cases,
ZR will include
all the columns of
Z.) The direction
pFR will satisfy
(2) if
where
pR is any
nR-vector.
10.3 Main Iteration
Let
Q denote the
n by
n matrix
where
IFX is the identity matrix of order
nFX. Let
gQ denote the
transformed gradient
and let the vector of the first
nR elements of
gQ be denoted by
gR. The quantity
gR is known as the
reduced gradient of
cTx. If the reduced gradient is zero,
x is a constrained stationary point in the subspace defined by
Z. During the feasibility phase, the reduced gradient will usually be zero only at a vertex (although it may be zero at non-vertices in the presence of constraint dependencies). During the optimality phase, a zero reduced gradient implies that
x minimizes the linear objective when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers
λC and
λB for the general and bound constraints are defined from the equations
Given a positive constant
δ of the order of the
machine precision, a Lagrange multiplier
λj corresponding to an inequality constraint in the working set is said to be
optimal if
λj≤δ when the associated constraint is at its
upper bound, or if
λj≥-δ when the associated constraint is at its
lower bound. If a multiplier is nonoptimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint (with index
Jdel; see
Section 12) from the working set.
If optimal multipliers occur during the feasibility phase and the sum of infeasibilities is nonzero, there is no feasible point, and you can force E04MFF/E04MFA to continue until the minimum value of the sum of infeasibilities has been found; see the discussion of the optional parameter Minimum Sum of Infeasibilities. At such a point, the Lagrange multiplier λj corresponding to an inequality constraint in the working set will be such that -1+δ≤λj≤δ when the associated constraint is at its upper bound, and -δ≤λj≤1+δ when the associated constraint is at its lower bound. Lagrange multipliers for equality constraints will satisfy λj≤1+δ.
If the reduced gradient is not zero, Lagrange multipliers need not be computed and the nonzero elements of the search direction p are given by ZRpR. The choice of step length is influenced by the need to maintain feasibility with respect to the satisfied constraints.
Each change in the working set leads to a simple change to AFR: if the status of a general constraint changes, a row of AFR is altered; if a bound constraint enters or leaves the working set, a column of AFR changes. Explicit representations are recurred of the matrices T and QFR; and of vectors QTg, and QTc.
One of the most important features of E04MFF/E04MFA is its control of the conditioning of the working set, whose nearness to linear dependence is estimated by the ratio of the largest to smallest diagonal elements of the
TQ factor
T (the printed value
Cond T; see
Section 12). In constructing the initial working set, constraints are excluded that would result in a large value of
Cond T.
E04MFF/E04MFA includes a rigorous procedure that prevents the possibility of cycling at a point where the active constraints are nearly linearly dependent (see
Gill et al. (1989)). The main feature of the anti-cycling procedure is that the feasibility tolerance is increased slightly at the start of every iteration. This not only allows a positive step to be taken at every iteration, but also provides, whenever possible, a
choice of constraints to be added to the working set. Let
αM denote the maximum step at which
x+αMp does not violate any constraint by more than its feasibility tolerance. All constraints at a distance
α (
α≤αM) along
p from the current point are then viewed as acceptable candidates for inclusion in the working set. The constraint whose normal makes the largest angle with the search direction is added to the working set.
10.4 Choosing the Initial Working Set
Let
Z be partitioned as
Z=ZRZA. A working set for which
ZR defines the null space can be obtained by including
the rows of
ZAT
as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by
ZR, as described in
Section 10.2.
The artificially augmented working set is given by
so that
pFR will satisfy
AFRpFR=0 and
ZAT
pFR
=
0
. By definition of the
TQ factorization,
A-FR automatically satisfies the following:
where
and hence the
TQ factorization of
(6) is available trivially from
T and
QFR without additional expense.
The matrix
ZA is not kept fixed, since its role is purely to define an appropriate null space; the
TQ factorization can therefore be updated in the normal fashion as the iterations proceed. No work is required to ‘delete’ the artificial constraints associated with
ZA
when
ZRT
gFR
=
0
, since this simply involves repartitioning
QFR. The ‘artificial’ multiplier vector associated with the rows of
ZAT
is equal to
ZAT
gFR
, and the multipliers corresponding to the rows of the ‘true’ working set are the multipliers that would be obtained if the artificial constraints were not present. If an artificial constraint is ‘deleted’ from the working set, an
A appears alongside the entry in the
Jdel column of the monitoring file output (see
Section 12).
The number of columns in
ZA and
ZR and the Euclidean norm of
ZRT
gFR
, appear in the monitoring file output as
Art,
Zr and
Norm Gz respectively (see
Section 12).
Under some circumstances, a different type of artificial constraint is used when solving a linear program. Although the algorithm of E04MFF/E04MFA does not usually perform simplex steps (in the traditional sense), there is one exception: a linear program with fewer general constraints than variables (i.e.,
mL≤n). Use of the simplex method in this situation leads to savings in storage. At the starting point, the ‘natural’ working set (the set of constraints exactly or nearly satisfied at the starting point) is augmented with a suitable number of ‘temporary’ bounds, each of which has the effect of temporarily fixing a variable at its current value. In subsequent iterations, a temporary bound is treated as a standard constraint until it is deleted from the working set, in which case it is never added again. If a temporary bound is ‘deleted’ from the working set, an
F (for ‘Fixed’) appears alongside the entry in the
Jdel column of the monitoring file output (see
Section 12).
11 Optional Parameters
Several optional parameters in E04MFF/E04MFA define choices in the problem specification or the algorithm logic. In order to reduce the number of formal parameters of E04MFF/E04MFA these optional parameters have associated default values that are appropriate for most problems. Therefore, you need only specify those optional parameters whose values are to be different from their default values.
The remainder of this section can be skipped if you wish to use the default values for
all optional parameters. A complete list of optional parameters and their default values is given in
Section 11.1.
Optional parameters may be specified by calling one, or both, of the routines
E04MGF/E04MGA and
E04MHF/E04MHA before a call to E04MFF/E04MFA.
E04MGF/E04MGA reads options from an external options file, with
Begin and
End as the first and last lines respectively and each intermediate line defining a single optional parameter. For example,
Begin
Print Level = 5
End
The call
CALL E04MGF (IOPTNS, INFORM)
can then be used to read the file on unit
IOPTNS.
INFORM will be zero on successful exit.
E04MGF/E04MGA should be consulted for a full description of this method of supplying optional parameters.
E04MHF/E04MHA can be called to supply options directly, one call being necessary for each optional parameter. For example,
CALL E04MHF ('Print Level = 5')
E04MHF/E04MHA should be consulted for a full description of this method of supplying optional parameters.
All optional parameters not specified by you are set to their default values. Optional parameters specified by you are unaltered by E04MFF/E04MFA (unless they define invalid values) and so remain in effect for subsequent calls unless altered by you.
11.1 Optional Parameter Checklist and Default Values
The following list gives the valid options. For each option, we give the keyword, any essential optional qualifiers and the default value. A description for each option can be found in
Section 11.2, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted), the letter
a denotes a phrase (character string) that qualifies an option, the letters
i and
r denote INTEGER and
double precision values required with certain options, and the number
ε is a generic notation for
machine precision (see
X02AJF).
11.2 Description of the Optional Parameters
| Check Frequency | i | Default =50 |
Every ith iteration, a numerical test is made to see if the current solution x satisfies the constraints in the working set. If the largest residual of the constraints in the working set is judged to be too large, the current working set is refactorized and the variables are recomputed to satisfy the constraints more accurately. If i≤0, the default value is used.
This option specifies how the initial working set is chosen. With a Cold Start, E04MFF/E04MFA chooses the initial working set based on the values of the variables and constraints at the initial point. Broadly speaking, the initial working set will include equality constraints and bounds or inequality constraints that violate or ‘nearly’ satisfy their bounds (to within Crash Tolerance).
With a
Warm Start, you must provide a valid definition of every element of the array
ISTATE. E04MFF/E04MFA will override your specification of
ISTATE if necessary, so that a poor choice of the working set will not cause a fatal error. For instance, any elements of
ISTATE which are set to
-2,
-1 or 4 will be reset to zero, as will any elements which are set to
3 when the corresponding elements of
BL and
BU are not equal. A warm start will be advantageous if a good estimate of the initial working set is available – for example, when E04MFF/E04MFA is called repeatedly to solve related problems.
| Crash Tolerance | r | Default =0.01 |
This value is used in conjunction with the optional parameter Cold Start (the default value) when E04MFF/E04MFA selects an initial working set. If 0≤r≤1, the initial working set will include (if possible) bounds or general inequality constraints that lie within r of their bounds. In particular, a constraint of the form
ajT
x≥l
will be included in the initial working set if
ajT
x-l
≤
r
1+l
. If r<0 or r>1, the default value is used.
This special keyword may be used to reset all optional parameters to their default values.
| Expand Frequency | i | Default =5 |
This option is part of an anti-cycling procedure designed to guarantee progress even on highly degenerate problems.
The strategy is to force a positive step at every iteration, at the expense of violating the constraints by a small amount. Suppose that the value of the optional parameter Feasibility Tolerance is δ. Over a period of i iterations, the feasibility tolerance actually used by E04MFF/E04MFA (i.e., the working feasibility tolerance) increases from 0.5δ to δ (in steps of 0.5δ/i).
At certain stages the following ‘resetting procedure’ is used to remove constraint infeasibilities. First, all variables whose upper or lower bounds are in the working set are moved exactly onto their bounds. A count is kept of the number of nontrivial adjustments made. If the count is positive, iterative refinement is used to give variables that satisfy the working set to (essentially) machine precision. Finally, the working feasibility tolerance is reinitialized to 0.5δ.
If a problem requires more than i iterations, the resetting procedure is invoked and a new cycle of i iterations is started with i incremented by 10. (The decision to resume the feasibility phase or optimality phase is based on comparing any constraint infeasibilities with δ.)
The resetting procedure is also invoked when E04MFF/E04MFA reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any nontrivial adjustments are made, iterations are continued.
If i≤0, the default value is used. If i≥9999999, no anti-cycling procedure is invoked.
| Feasibility Tolerance | r | Default =ε |
If r≥ε, r defines the maximum acceptable absolute violation in each constraint at a ‘feasible’ point. For example, if the variables and the coefficients in the general constraints are of order unity, and the latter are correct to about 6 decimal digits, it would be appropriate to specify r as 10-6. If 0≤r<ε, the default value is used.
E04MFF/E04MFA attempts to find a feasible solution before optimizing the objective function. If the sum of infeasibilities cannot be reduced to zero, the optional parameter Minimum Sum of Infeasibilities can be used to find the minimum value of the sum. Let Sinf be the corresponding sum of infeasibilities. If Sinf is quite small, it may be appropriate to raise r by a factor of 10 or 100. Otherwise, some error in the data should be suspected.
Note that a ‘feasible solution’ is a solution that satisfies the current constraints to within the tolerance r.
| Infinite Bound Size | r | Default =1020 |
If r>0, r defines the ‘infinite’ bound infbnd in the definition of the problem constraints. Any upper bound greater than or equal to infbnd will be regarded as +∞ (and similarly any lower bound less than or equal to -infbnd will be regarded as -∞). If r<0, the default value is used.
| Infinite Step Size | r | Default =maxbigbnd,1020 |
If r>0, r specifies the magnitude of the change in variables that will be considered a step to an unbounded solution. (Note that an unbounded solution can occur only when the problem is of type LP.) If the change in x during an iteration would exceed the value of r, the objective function is considered to be unbounded below in the feasible region. If r≤0, the default value is used.
| Iteration Limit | i | Default =max50,5n+mL |
The value of i specifies the maximum number of iterations allowed before termination. With i=0 and Print Level>0, the workspace needed will be computed and printed, but no iterations will be performed. If i<0, the default value is used.
| List | | Default for E04MFF=List |
| Nolist | | Default for E04MFA=Nolist |
Normally each optional parameter specification is printed as it is supplied. Optional parameter Nolist may be used to suppress the printing and optional parameter List may be used to restore printing.
| Minimum Sum of Infeasibilities | No | Default =No |
If no feasible point exists for the constraints, this option is used to control whether or not E04MFF/E04MFA will calculate a point that minimizes the constraint violations. If Minimum Sum of Infeasibilities=No, E04MFF/E04MFA will terminate as soon as it is evident that no feasible point exists for the constraints. The final point will generally not be the point at which the sum of infeasibilities is minimized. If Minimum Sum of Infeasibilities=Yes, E04MFF/E04MFA will continue until the sum of infeasibilities is minimized.
| Monitoring File | i | Default =-1 |
If i≥0 and Print Level≥5, monitoring information produced by E04MFF/E04MFA at every iteration is sent to a file with logical unit number i. If i<0 and/or Print Level<5, no monitoring information is produced.
| Optimality Tolerance | r | Default =ε0.8 |
If r≥ε, r defines the tolerance used to determine if the bounds and general constraints have the right ‘sign’ for the solution to be judged to be optimal.
If 0≤r<ε, the default value is used.
| Print Level | i | Default for E04MFF
=10
Default for E04MFA
=0 |
The value of
i controls the amount of printout produced by E04MFF/E04MFA, as indicated below. A detailed description of the printed output is given in
Section 8.2 (summary output at each iteration and the final solution) and
Section 12 (monitoring information at each iteration).
The following printout is sent to the current advisory message unit (as defined by
X04ABF):
| i |
Output |
| ≥00 |
No output. |
| ≥01 |
The final solution only. |
| ≥05 |
One line of summary output (<80 characters; see Section 8.2) for each iteration (no printout of the final solution). |
| ≥10 |
The final solution and one line of summary output for each iteration. |
The following printout is sent to the logical unit number defined by the optional parameter
Monitoring File:
| i |
Output |
| <05 |
No output. |
| ≥05 |
One long line of output (>80 characters; see Section 12) for each iteration (no printout of the final solution). |
| ≥20 |
At each iteration, the Lagrange multipliers, the variables x, the constraint values Ax and the constraint status. |
| ≥30 |
At each iteration, the diagonal elements of the upper triangular matrix T associated with the TQ factorization (3) (see Section 10.2) of the working set. |
If
Print Level≥5 and the unit number defined by the optional parameter
Monitoring File is the same as that defined by
X04ABF, then the summary output is suppressed.
| Problem Type | a | Default = LP
|
This option specifies the type of objective function to be minimized during the optimality phase. The following is the optional keyword and the dimensions of the array that must be specified in order to define the objective function:
For problems of type FP, the objective function is omitted and
CVEC is not referenced. The following keywords are also acceptable. The minimum abbreviation of each keyword is underlined.
| a |
Option |
| Linear |
LP |
| Feasible |
FP |
12 Description of Monitoring Information
This section describes the long line of output (>80 characters) which forms part of the monitoring information produced by E04MFF/E04MFA. (See also the description of the optional parameters Monitoring File and Print Level.) You can control the level of printed output.
To aid interpretation of the printed results, the following convention is used for numbering the constraints: indices 1 through n refer to the bounds on the variables, and indices n+1 through n+mL refer to the general constraints. When the status of a constraint changes, the index of the constraint is printed, along with the designation L (lower bound), U (upper bound), E (equality), F (temporarily fixed variable) or A (artificial constraint).
When
Print Level≥5 and
Monitoring File≥0, the following line of output is produced at every iteration on the unit number specified by optional parameter
Monitoring File. In all cases, the values of the quantities printed are those in effect
on completion of the given iteration.
| Itn |
is the iteration count.
|
| Jdel |
is the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted.
|
| Jadd |
is the index of the constraint added to the working set. If Jadd is zero, no constraint was added.
|
| Step |
is the step taken along the computed search direction. If a constraint is added during the current iteration, Step will be the step to the nearest constraint. When the problem is of type LP, the step can be greater than one during the optimality phase.
|
| Ninf |
is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
|
| Sinf/Objective |
is the value of the current objective function. If x is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If x is feasible, Objective is the value of the objective function of (1). The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point. During the optimality phase the value of the objective function will be nonincreasing. During the feasibility phase the number of constraint infeasibilities will not increase until either a feasible point is found or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.
|
| Bnd |
is the number of simple bound constraints in the current working set.
|
| Lin |
is the number of general linear constraints in the current working set.
|
| Art |
is the number of artificial constraints in the working set, i.e., the number of columns of ZA (see Section 10.4).
|
| Zr |
is the number of columns of ZR (see Section 10.2). Zr is the dimension of the subspace in which the objective function is currently being minimized. The value
of Zr is the number of variables minus the number of constraints in the working set; i.e., Zr=n-Bnd+Lin+Art.The value of nZ, the number of columns of Z (see Section 10.2) can be calculated as nZ=n-Bnd+Lin. A zero value of nZ implies that x lies at a vertex of the feasible region. |
| Norm Gz |
is
ZRT
gFR
, the Euclidean norm of the reduced gradient with respect to ZR. During the optimality phase, this norm will be approximately zero after a unit step.
|
| NOpt |
is the number of nonoptimal Lagrange multipliers at the current point. NOpt is not printed if the current x is infeasible or no multipliers have been calculated. At a minimizer, NOpt will be zero.
|
| Min Lm |
is the value of the Lagrange multiplier associated with the deleted constraint. If Min Lm is negative, a lower bound constraint has been deleted, if Min Lm is positive, an upper bound constraint has been deleted. If no multipliers are calculated during a given iteration Min Lm will be zero.
|
| Cond T |
is a lower bound on the condition number of the working set.
|