lbfgsb_module Module

the dimension of the problem (the number of variables).

the maximum number of variable metric corrections used to define the limited memory matrix. Values of m < 3 are not recommended, and large values of m can result in excessive computing time. The range 3 <= m <= 20 is recommended.

On initial entry it must be set by the user to the values of the initial estimate of the solution vector. Upon successful exit, it contains the values of the variables at the best point found (usually an approximate solution).

the lower bound on x. If the i-th variable has no lower bound, l(i) need not be defined.

the upper bound on x. If the i-th variable has no upper bound, u(i) need not be defined.

nbd represents the type of bounds imposed on the variables, and must be specified as follows:

On first entry f is unspecified. On final exit f is the value of the function at x. If the setulb returns with task(1:2)= 'FG', then f must be set by the user to contain the value of the function at the point x.

On first entry g is unspecified. On final exit g is the value of the gradient at x. If the setulb returns with task(1:2)= 'FG', then g must be set by the user to contain the components of the gradient at the point x.

A tolerance in the termination test for the algorithm. The iteration will stop when:

The iteration will stop when:

working array of length (2mmax + 5)nmax + 11mmax^2 + 8mmax. This array must not be altered by the user.

an integer working array of length 3nmax. This array must not be altered by the user.

Indicates the current job when entering and quitting this subroutine:

Controls the frequency and type of output generated:

working string

A logical working array of dimension 4. On exit with task = 'NEW_X', the following information is available:

An integer working array of dimension 44. On exit with 'task' = NEW_X, the following information is available:

A real(wp) working array of dimension 29. On exit with 'task' = NEW_X, the following information is available:

The name of the iteration file if used (Iprint>=1). If not specified, then 'iterate.dat' is used.

the number of variables.

the maximum number of variable metric corrections allowed in the limited memory matrix.

On entry x is an approximation to the solution. On exit x is the current approximation.

the lower bound of x.

the upper bound of x.

the type of bounds imposed on the variables, and must be specified as follows:

On first entry f is unspecified. On final exit f is the value of the function at x.

On first entry g is unspecified. On final exit g is the value of the gradient at x.

On entry factr >= 0 is specified by the user. The iteration will stop when

On entry pgtol >= 0 is specified by the user. The iteration will stop when

working array used to store information defining the limited memory BFGS matrix: stores S, the matrix of s-vectors;

working array used to store information defining the limited memory BFGS matrix: stores Y, the matrix of y-vectors;

working array used to store information defining the limited memory BFGS matrix: stores S'Y;

working array used to store information defining the limited memory BFGS matrix: stores S'S;

working array used to store information defining the limited memory BFGS matrix: stores the Cholesky factorization of (theta*S'S+LD^(-1)L'); see eq. (2.26) in [3].

working array used to store the LEL^T factorization of the indefinite matrix

working array used to store the lower triangular part of

working array. used at different times to store the Cauchy point and the Newton point.

working array.

working array.

working array.

working array. used to safeguard the projected Newton direction

working array.

working array. In subroutine freev, index is used to store the free and fixed variables at the Generalized Cauchy Point (GCP).

working array used to record the status of the vector x for GCP computation:

integer working array. Within subroutine cauchy, indx2 corresponds to the array iorder. In subroutine freev, a list of variables entering and leaving the free set is stored in indx2, and it is passed on to subroutine formk with this information.

indicates the current job when entering and leaving this subroutine.

Controls the frequency and type of output generated:

working string

working array

working array

working array

The name of the iteration file if used (Iprint>=1). If not specified, then 'iterate.dat' is used.

the dimension of the problem (the number of variables).

the lower bound on x.

the upper bound on x.

In cauchy, iwhere is given finer gradations.

Controls the frequency and type of output generated

the maximum number of variable metric corrections used to define the limited memory matrix.

specifies the matrix S'Y.

specifies the upper triangular matrix J' which is the Cholesky factor of (thetaS'S+LD^(-1)L').

specifies the number of s-vectors (or y-vectors) stored in the compact L-BFGS formula.

specifies vector v.

the product Mv

On exit:

the dimension of the problem.

the starting point for the GCP computation.

the lower bound of x.

the upper bound of x.

On entry nbd represents the type of bounds imposed on the variables, and must be specified as follows:

the gradient of f(x). g must be a nonzero vector.

working array used to store the breakpoints in the piecewise linear path and free variables encountered. On exit:

On entry iwhere indicates only the permanently fixed (iwhere=3) or free (iwhere= -1) components of x.

used to store the break points.

used to store the Cauchy direction P(x-tg)-x.

used to return the GCP on exit.

the maximum number of variable metric corrections used to define the limited memory matrix.

stores information that defines the limited memory BFGS matrix: wy(n,m) stores Y, a set of y-vectors;

stores information that defines the limited memory BFGS matrix: ws(n,m) stores S, a set of s-vectors;

stores information that defines the limited memory BFGS matrix: sy(m,m) stores S'Y;

stores information that defines the limited memory BFGS matrix: wt(m,m) stores the Cholesky factorization of (theta*S'S+LD^(-1)L').

the scaling factor specifying B_0 = theta I.

the actual number of variable metric corrections stored so far.

the location of the first s-vector (or y-vector) in S (or Y).

working array used to store the vector p = W^(T)d.

working array used to store the vector c = W^(T)(xcp-x).

working array used to store the row of W corresponding to a breakpoint.

working array

records the number of quadratic segments explored in searching for the GCP.

controls the frequency and type of output generated:

the norm of the projected gradient at x.

On entry info is 0. On exit:

machine precision

the lower bound on x.

the upper bound on x.

the dimension of the problem.

the number of subspace variables in free set.

specifies the indices of subspace variables.

the number of variables entering the free set.

indx2(ileave),...,indx2(n) are the variables leaving the free set.

On entry indx2(1),...,indx2(nenter) are the variables entering the free set, while indx2(ileave),...,indx2(n) are the variables leaving the free set.

the total number of BFGS updates made so far.

true if the L-BFGS matrix is updatd.

On exit the upper triangle of wn stores the LEL^T factorization of the 2*col x 2*col indefinite matrix

On entry wn1 stores the lower triangular part of

the maximum number of variable metric corrections used to define the limited memory matrix.

information defining the limited memory BFGS matrix: ws(n,m) stores S, a set of s-vectors

information defining the limited memory BFGS matrix: wy(n,m) stores Y, a set of y-vectors

information defining the limited memory BFGS matrix: sy(m,m) stores S'Y

information defining the limited memory BFGS matrix: the scaling factor specifying B_0 = theta I

information defining the limited memory BFGS matrix: the number of variable metric corrections stored

information defining the limited memory BFGS matrix: the location of the 1st s- (or y-) vector in S (or Y)

On exit:

for i=nfree+1,...,n, index(i) are the indices of bound variables

On exit with iter>0, indx2 indicates which variables have changed status since the previous iteration.

indicating whether bounds are present

Controls the frequency and type of output generated

the dimension of the arrays t and iorder.

On entry t stores the elements to be sorted, On exit t(n) stores the least elements of t, and t(1) to t(n-1) stores the remaining elements in the form of a heap.

On entry iorder(i) is the index of t(i). On exit iorder(i) is still the index of t(i), but iorder may be permuted in accordance with t.

iheap == 0 if t(1) to t(n) is not in the form of a heap, iheap /= 0 if otherwise.

the dimension of the problem

the maximum number of variable metric corrections allowed in the limited memory matrix.

the lower bound on x.

the upper bound on x.

initial solution point

Controls the frequency and type of output generated

iteration print file unit

machine precision

records the status of subspace solutions.

the dimension of the problem (the number of variables).

the lower bound on x.

the upper bound on x.

infinity norm of the projected gradient

the dimension of the problem

the maximum number of variable metric corrections used to define the limited memory matrix.

the number of free variables

specifies the coordinate indices of free variables.

the lower bound of x.

the upper bound of x

represents the type of bounds imposed on the variables, and must be specified as follows:

On entry x specifies the Cauchy point xcp. On exit x(i) is the minimizer of Q over the subspace of free variables.

On entry d is the reduced gradient of Q at xcp. On exit d is the Newton direction of Q.

used to safeguard the projected Newton direction

variable that stores the information defining the limited memory BFGS matrix: ws(n,m) stores S, a set of s-vectors

variable that stores the information defining the limited memory BFGS matrix:

variable that stores the information defining the limited memory BFGS matrix: theta is the scaling factor specifying B_0 = theta I

the current iterate

the gradient at the current iterate

variable that stores the information defining the limited memory BFGS matrix: col is the number of variable metric corrections stored

variable that stores the information defining the limited memory BFGS matrix: head is the location of the 1st s- (or y-) vector in S (or Y)

On exit iword specifies the status of the subspace solution:

a real(wp) working array of dimension 2m

The upper triangle of wn stores the LEL^T factorization of the indefinite matrix

If iprint >= 99, some minimum debug printing is done.

On exit:

ftol specifies a nonnegative tolerance for the sufficient decrease condition.

gtol specifies a nonnegative tolerance for the curvature condition.

xtol specifies a nonnegative relative tolerance for an acceptable step. The subroutine exits with a warning if the relative difference between sty and stx is less than xtol.

a nonnegative lower bound for the step.

a nonnegative upper bound for the step.

task is a character variable of length at least 60:

integer work array

real work array

On entry stx is the best step obtained so far and is an endpoint of the interval that contains the minimizer. On exit `stx is the updated best step.

On entry fx is the function at stx. On exit fx is the function at stx.

On entry dx is the derivative of the function at stx. The derivative must be negative in the direction of the step, that is, dx and stp - stx must have opposite signs. On exit dx is the derivative of the function at stx.

On entry sty is the second endpoint of the interval that contains the minimizer. On exit sty is the updated endpoint of the interval that contains the minimizer.

On entry fy is the function at sty. On exit fy is the function at sty.

On entry dy is the derivative of the function at sty. On exit dy is the derivative of the function at the exit sty.

On entry stp is the current step. If brackt is set to .true. then on input stp must be between stx and sty. On exit stp is a new trial step.

the function at stp.

the derivative of the function at stp.

On entry brackt specifies if a minimizer has been bracketed. Initially brackt must be set to .false. On exit brackt specifies if a minimizer has been bracketed. When a minimizer is bracketed brackt is set to .true.

a lower bound for the step.

an upper bound for the step.