SLPRS

Solves a sparse linear programming problem via the revised simplex algorithm.

Required Arguments

A — Vector of length NZ containing the coefficients of the M constraints. (Input)

IROW — Vector of length NZ containing the row numbers of the corresponding element in A. (Input)

JCOL — Vector of length NZ containing the column numbers of the corresponding elements in A. (Input)

BL — Vector of length M containing the lower limit of the general constraints; if there is no lower limit on the I-th constraint, then BL(I) is not referenced. (Input)

BU — Vector of length M containing the upper limit of the general constraints; if there is no upper limit on the I-th constraint, then BU(I) is not referenced. (Input)

C — Vector of length NVAR containing the coefficients of the objective function. (Input)

IRTYPE — Vector of length M indicating the types of general constraints in the matrix A. (Input)

Let R(I) = A(I, 1)*XSOL(1) + … + A(I, NVAR)*XSOL(NVAR)

Let R(I) = A(I, 1)*XSOL(1) + … + A(I, NVAR)*XSOL(NVAR)

IRTYPE(I) | I-th Constraint |
---|---|

0 | BL(I) = R(I) = BU(I) |

1 | R(I) ≤ BU(I) |

2 | R(I) ≥ BL(I) |

3 | BL(I) ≤ R(I) ≤ BU(I) |

OBJ — Value of the objective function. (Output)

XSOL — Vector of length NVAR containing the primal solution. (Output)

DSOL — Vector of length M containing the dual solution. (Output)

Optional Arguments

M — Number of constraints. (Input)

Default: M = SIZE (IRTYPE,1).

Default: M = SIZE (IRTYPE,1).

NVAR — Number of variables. (Input)

Default: NVAR = SIZE (C,1).

Default: NVAR = SIZE (C,1).

NZ — Number of nonzero coefficients in the matrix A. (Input)

Default: NZ = SIZE (A,1).

Default: NZ = SIZE (A,1).

XLB — Vector of length NVAR containing the lower bound on the variables; if there is no lower bound on a variable, then 1.0E30 should be set as the lower bound. (Input)

Default: XLB = 0.0.

Default: XLB = 0.0.

XUB — Vector of length NVAR containing the upper bound on the variables; if there is no upper bound on a variable, then -1.0E30 should be set as the upper bound. (Input)

Default: XUB = 3.4e38 for single precision and 1.79d + 308 for double precision.

Default: XUB = 3.4e38 for single precision and 1.79d + 308 for double precision.

FORTRAN 90 Interface

Generic: CALL SLPRS (A, IROW, JCOL, BL, BU, C, IRTYPE, OBJ, XSOL, DSOL [, …])

Specific: The specific interface names are S_SLPRS and D_SLPRS.

FORTRAN 77 Interface

Single: CALL SLPRS (M, NVAR, NZ, A, IROW, JCOL, BL, BU, C, IRTYPE, XLB, XUB, OBJ, XSOL, DSOL)

Double: The double precision name is DSLPRS.

Description

This subroutine solves problems of the form

min cTx

subject to

where c is the objective coefficient vector, A is the coefficient matrix, and the vectors bl, bu, xl, and xu are the lower and upper bounds on the constraints and the variables, respectively. SLPRS is designed to take advantage of sparsity in A. The routine is based on DPLO by Hanson and Hiebert.

Comments

Workspace may be explicitly provided, if desired, by use of S2PRS/DS2PRS. The reference is:

CALL S2PRS (M, NVAR, NZ, A, IROW, JCOL, BL, BU, C, IRTYPE, XLB, XUB, OBJ, XSOL, DSOL, IPARAM, RPARAM, COLSCL, ROWSCL, WORK, LW, IWORK, LIW)

The additional arguments are as follows:

IPARAM — Integer parameter vector of length 12. If the default parameters are desired for SLPRS, then set IPARAM(1) to zero and call the routine SLPRS. Otherwise, if any nondefault parameters are desired for IPARAM or RPARAM, then the following steps should be taken before calling SLPRS:

CALL S5PRS (IPARAM, RPARAM)

Set nondefault values for IPARAM and RPARAM.

Note that the call to S5PRS will set IPARAM and RPARAM to their default values so only nondefault values need to be set above. |

IPARAM(1) = 0 indicates that a minimization problem is solved. If set to 1, a maximization problem is solved.

Default: 0

Default: 0

IPARAM(2) = switch indicating the maximum number of iterations to be taken before returning to the user. If set to zero, the maximum number of iterations taken is set to 3*(NVARS+M). If positive, that value is used as the iteration limit.

Default: IPARAM(2) = 0

Default: IPARAM(2) = 0

IPARAM(3) = indicator for choosing how columns are selected to enter the basis. If set to zero, the routine uses the steepest edge pricing strategy which is the best local move. If set to one, the minimum reduced cost pricing strategy is used. The steepest edge pricing strategy generally uses fewer iterations than the minimum reduced cost pricing, but each iteration costs more in terms of the amount of calculation performed. However, this is very problem-dependent.

Default: IPARAM(3) = 0

Default: IPARAM(3) = 0

IPARAM(4) = MXITBR, the number of iterations between recalculating the error in the primal solution is used to monitor the error in solving the linear system. This is an expensive calculation and every tenth iteration is generally enough.

Default: IPARAM(4) = 10

Default: IPARAM(4) = 10

IPARAM(5) = NPP, the number of negative reduced costs (at most) to be found at each iteration of choosing a variable to enter the basis. If set to zero,

NPP = NVARS will be used, implying that all of the reduced costs are computed at each such step. This “Partial pricing” may increase the total number of iterations required. However, it decreases the number of calculation required at each iteration. The effect on overall efficiency is very problem-dependent. If set to some positive number, that value is used as NPP.

Default: IPARAM(5) = 0

NPP = NVARS will be used, implying that all of the reduced costs are computed at each such step. This “Partial pricing” may increase the total number of iterations required. However, it decreases the number of calculation required at each iteration. The effect on overall efficiency is very problem-dependent. If set to some positive number, that value is used as NPP.

Default: IPARAM(5) = 0

IPARAM(6) = IREDFQ, the number of steps between basis matrix redecompositions. Redecompositions also occur whenever the linear systems for the primal and dual systems have lost half their working precision.

Default: IPARAM(6) = 50

Default: IPARAM(6) = 50

IPARAM(7) = LAMAT, the length of the portion of WORK that is allocated to sparse matrix storage and decomposition. LAMAT must be greater than NZ + NVAR + 7.

Default: LAMAT = MAX (NZ + NVAR + 8, 4*NVAR + 7)

Default: LAMAT = MAX (NZ + NVAR + 8, 4*NVAR + 7)

IPARAM(8) = LBM, the length of the portion of IWORK that is allocated to sparse matrix storage and decomposition. LBM must be positive.

Default: LBM = 14 * M

Default: LBM = 14 * M

IPARAM(9) = switch indicating that partial results should be saved after the maximum number of iterations, IPARAM(2), or at the optimum. If IPARAM(9) is not zero, data essential to continuing the calculation is saved to a file, attached to unit number IPARAM(9). The data saved includes all the information about the sparse matrix A and information about the current basis. If IPARAM(9) is set to zero, partial results are not saved. It is the responsibility of the calling program to open the output file.

IPARAM(10) = switch indicating that partial results have been computed and stored on unit number IPARAM(10), if greater than zero. If IPARAM(10) is zero, a new problem is started.

Default: IPARAM(10) = 0

Default: IPARAM(10) = 0

IPARAM(11) = switch indicating that the user supplies scale factors for the columns of the matrix A. If IPARAM(11) = 0, SLPRS computes the scale factors as the reciprocals of the max norm of each column. If IPARAM(11) is set to one, element I of the vector COLSCL is used as the scale factor for column I of the matrix A. The scaling is implicit, so no input data is actually changed.

Default: IPARAM(11) = 0

Default: IPARAM(11) = 0

IPARAM(12) = switch indicating that the user supplied scale factors for the rows of the matrix A. If IPARAM(12) is set to zero, no row scaling is one. If IPARAM(12) is set to 1, element I of the vector ROWSCL is used as the scale factor for row I of the matrix A. The scaling is implicit, so no input data is actually changed.

Default: IPARAM(12) = 0

Default: IPARAM(12) = 0

RPARAM — Real parameter vector of length 7.

RPARAM(1) = COSTSC, a scale factor for the vector of costs. Normally SLPRS computes this scale factor to be the reciprocal of the max norm if the vector costs after the column scaling has been applied. If RPARAM(1) is zero, SLPRS compute COSTSC.

Default: RPARAM(1) = 0.0

Default: RPARAM(1) = 0.0

RPARAM(2) = ASMALL, the smallest magnitude of nonzero entries in the matrix A. If RPARAM(2) is nonzero, checking is done to ensure that all elements of A are at least as large as RPARAM(2). Otherwise, no checking is done.

Default: RPARAM(2) = 0.0

Default: RPARAM(2) = 0.0

RPARAM(3) = ABIG, the largest magnitude of nonzero entries in the matrix A. If RPARAM(3) is nonzero, checking is done to ensure that all elements of A are no larger than RPARAM(3). Otherwise, no checking is done.

Default: RPARAM(3) = 0.0

Default: RPARAM(3) = 0.0

RPARAM(4) = TOLLS, the relative tolerance used in checking if the residuals are feasible. RPARAM(4) is nonzero, that value is used as TOLLS, otherwise the default value is used.

Default: TOLLS = 1000.0*amach(4)

Default: TOLLS = 1000.0*amach(4)

RPARAM(5) = PHI, the scaling factor used to scale the reduced cost error estimates. In some environments, it may be necessary to reset PHI to the range [0.01, 0.1], particularly on machines with short word length and working precision when solving a large problem. If RPARAM(5) is nonzero, that value is used as PHI, otherwise the default value is used.

Default: PHI = 1.0

Default: PHI = 1.0

RPARAM(6) = TOLABS, an absolute error test on feasibility. Normally a relative test is used with TOLLS (see RPARAM(4)). If this test fails, an absolute test will be applied using the value TOLABS.

Default: TOLABS = 0.0

Default: TOLABS = 0.0

RPARAM(7) = pivot tolerance of the underlying sparse factorization routine. If RPARAM(7) is set to zero, the default pivot tolerance is used, otherwise, the RPARAM(7) is used.

Default: RPARAM(7) = 0.1

Default: RPARAM(7) = 0.1

COLSCL — Array of length NVARS containing column scale factors for the matrix A. (Input).

COLSCL is not used if IPARAM(11) is set to zero.

COLSCL is not used if IPARAM(11) is set to zero.

ROWSCL — Array of length M containing row scale factors for the matrix A. (Input)

ROWSCL is not used if IPARAM(12) is set to zero.

ROWSCL is not used if IPARAM(12) is set to zero.

WORK — Work array of length LW.

LW — Length of real work array. LW must be at least

4*NVAR + 9*M + LAMAT + LBM + 4*(M+NVAR) + 2*NZ + NVAR + 1, where LAMAT = IPARAM(7) and LBM = IPARAM(8).

4*NVAR + 9*M + LAMAT + LBM + 4*(M+NVAR) + 2*NZ + NVAR + 1, where LAMAT = IPARAM(7) and LBM = IPARAM(8).

IWORK — Integer work array of length LIW.

LIW — Length of integer work array. LIW must be at least

NVAR + 11*M + LAMAT + 2*LBM + 2*(M + NVAR), where LAMAT = IPARAM(7) and LBM = IPARAM(8).

NVAR + 11*M + LAMAT + 2*LBM + 2*(M + NVAR), where LAMAT = IPARAM(7) and LBM = IPARAM(8).

Example

Solve a linear programming problem, with

defined in sparse coordinate format.

USE SLPRS_INT

USE UMACH_INT

IMPLICIT NONE

INTEGER M, NVAR

PARAMETER (M=200, NVAR=200)

! Specifications for local variables

INTEGER INDEX, IROW(3*M), J, JCOL(3*M), NOUT, NZ

REAL A(3*M), DSOL(M), OBJ, XSOL(NVAR)

INTEGER IRTYPE(M)

REAL B(M), C(NVAR), XL(NVAR), XU(NVAR)

! Specifications for subroutines

DATA B/199*1.7, 1.0/

DATA C/-1.0, -2.0, -3.0, -4.0, -5.0, -6.0, -7.0, -8.0, -9.0, &

-10.0, 190*-1.0/

DATA XL/200*0.1/

DATA XU/200*2.0/

DATA IRTYPE/200*1/

!

CALL UMACH (2, NOUT)

! Define A

INDEX = 1

DO 10 J=2, M

! Superdiagonal element

IROW(INDEX) = J - 1

JCOL(INDEX) = J

A(INDEX) = 0.5

! Diagonal element

IROW(INDEX+1) = J

JCOL(INDEX+1) = J

A(INDEX+1) = 1.0

INDEX = INDEX + 2

10 CONTINUE

NZ = INDEX - 1

!

!

XL(4) = 0.2

CALL SLPRS (A, IROW, JCOL, B, B, C, IRTYPE, OBJ, XSOL, DSOL, &

NZ=NZ, XLB=XL, XUB=XU)

!

WRITE (NOUT,99999) OBJ

!

99999 FORMAT (/, 'The value of the objective function is ', E12.6)

!

END

Output

The value of the objective function is -.280971E+03