Newopt : A Modular Optimisation Package

Introduction

Newopt is a set of general purpose optimisation routines. The design of the package is based on the idea that most popular optimisation algorithms can be broken into discreet steps (e.g. step calculation, hessian update, function evaluation etc.). If these steps are implemented as commands within an interpreted language it is possible to use this language to describe a number of geometry optimisation algorithms in terms of the same set of component steps. Once the algorithms are coded in this way, modifications to the algorithms (and new algorithms) can be generated and tested without recompilation.

Internally, the package is based on an object oriented structure, an optimiser object is generated by optimiser command, which allocates memory for all the matrices and vectors needed for the optimisation. Methods associated with this object can then be invoked to build up the geometry optimisation algorithm.

In addition to this low-level programming approach, a high level driver command (newopt) is provided which manages the optimiser object, allowing selection of one of a set of standard "black box" optimisation algorithms.

The target function to be minimised is provided as a set of Tcl procedures. In this way, the user is able to write Tcl wrappers to allow newopt to optimise any continuous function. A number of target functions are provided for molecular geometry optimisation, the most useful are zopt for Z-matrix (and mixed cartesian/zmatrix) optimisation and copt for pure cartesian optimisation.

The Newopt command

Command line arguments

Argument	Argument type	Mandatory	Default	To specify
function=	name of the function	yes	none	type of problem to be optimised
method=	name of optimisation method	no	bfgs	a standard algorithm for solving an optimisation problem
list_option=	Output keyword	no	medium	Level of output
maxstep=	integer	no	200	the maximum number of points to be allowed
toler=	real	no	n/a	convergence criterion for the trial step length; sets other convergence criteria as well
toler_rmsstep=	real	no	0.0002	convergence criterion for the rms step length
toler_maxstep=	real	no	0.001	convergence criterion for the step length
toler_rmsgrad=	real	no	0.0002	convergence criterion for the rms gradient
toler_maxgrad=	real	no	0.001	convergence criterion for the gradient
toler_energy=	real	no	1.0e-10	forced convergence criterion for the energy
list_variables=	Boolean	no	no	printflag for variables information during optimisation
step=	real	no	0.3	maximum trial step length
punchfile=	filename	no	-	name of a punchfile for output
input_hessian=	matrix	no	-	initial internal coordinate hessian
hessian_update=	keyword	no	powell	hessian update algorithm, see below
final_hessian=	Matrix tag	no	-	final internal coordinate hessian
restart=	Boolean	no	-	restart of previous optimisation run
hard_fail=	Boolean	no	-	return failure code to Tcl
commands=	Tcl list	no	-	Tcl commands to be executed (only with method=custom)
var_list=	Tcl list of numbers	no	-	list of variables to be operated on in conjunction with selected hessian updates
eig_list=	Tcl list of numbers	no	-	list of variables to be operated on in conjunction with searches along hessian eigenmodes
del=	real	no	-	displacement for finite-differencing

Some of these arguments may only required with certain algorithms. If not required, they need not be specified. If they are omitted, default options and values will be put into operation.

function=

newopt function=zopt : {zmatrix=z theory=gamess: \
{basis=dzp listing=gamess.out} }

Optimisation Methods

Method	To do what
descent	steepest-descent minimization. The actual step size is determined through the Shanno line-search algorithm.
bfgs	pseudo-Newton minimisation with BFGS update of the inverse hessian. The actual step size is determined through the Shanno line-search algorithm. Further arguments
bfgs2	pseudo-Newton minimisation with BFGS update of the inverse hessian. The actual step size is determined through a function-only line-search algorithm. Further arguments
conmin	conjugate gradient minimisation. The actual step size is determined through the Shanno line-search algorithm, with at least two trial point evaluations along the line. A conjugate gradient restart step is performed every ndim steps, with ndim the dimension of the optimisation problem. A restart step is also performed under certain conditions of line-search failure.
mineig	line-search minimisation along selected hessian eigenmodes. The mode-indices to be minimised along are input through the eig_list argument.
scaneig	potential energy scan along selected hessian eigenmodes with fixed step-lengths. The mode-indices to be scanned are input through the eig_list= argument. The step-length is input through the step= argument. The scan is stopped if the energy goes up if follow_mode=0
baker	Baker's (partial) rational function optimisation (P)RFO algorithm, with the Powell update of the hessian. The actual step size is bounded by step. For a TS search, the follow_mode argument must be specified as non-zero. Mode-following is achieved through finding maximum overlap of the hessian eigenvectors with the mode to be followed. The mode to be followed may be kept fixed to the starting mode or updated each step to be the actual step direction. This is controlled through the update_mode argument (see below). The argument input_hessian argument is required with this algorithm. Further arguments
diis	direct inversion in the iterative subspace algorithm. Further arguments
custom	user-defined search. The search is performed through the Tcl-commands supplied in the commands= argument. See the newopt programming guide for information on custom programming.

The default of the method argument is bfgs.

The verbose argument may be used to set the level of output during an optimisation. Switching it on causes the eigenvectors of the hessian and the step-vector to be printed, amongst others. Valid options are boolean equivalents; the default value is off.

The list_variables argument specifies the printing of the current variables at each search-step. The default value is yes.

The maxstep argument specifies the maximum number of actual search steps in a search. Note that only the result of a line search counts as a step. The default value is 200.

The toler argument specifies the convergence criteria on the maximum step component. From it, the three other convergence criteria are derived as shown in the table below. If toler is left unspecified, the default values given in the same table are set if they are not specified separately.

Convergence tester	Convergence criterion
	From toler	default
maximum step component	toler	0.001
rms of the step vector	toler * 2/3	0.0002
maximum gradient component	toler * 1/4	0.001
rms of the gradient	toler * 1/6	0.0002

The toler_rmsstep, toler_maxstep, toler_rmsgrad, and toler_maxgrad arguments specify the convergence criteria for the rms-value of the step, maximum value of the step, rms-value of the gradient, and maximum value of the gradient, respectively. If any of these are specified, they override the values set by the toler argument, or the default values tabulated above.

All methods except bfgs, baker and diis are converged on all four criteria listed above.

The toler_energy argument specifies the convergence criterion on the energy. If in two consecutive steps the energy change is less than this criterion, convergence is forced, even if the criteria on gradient and step are not met. This option is currently valid only with method=bfgs, baker and diis. The default value is 1.e-10.

The bfgs and Baker algorithms are tested for convergence in the following manner: the toler_maxgrad criterion must always be satisfied and either the toler_maxstep or toler_energy criteria.

The step argument specifies the maximum allowed step length in the Baker algorithm. The default value is 0.3. It is also used to restrict step-lengths in other algorithms or set the step-length in the scanning along hessian eigenmodes.

The punchfile argument specifies the output of information on the structures during the optimisation to a punchfile. The output may be used to check the progress of an optimisation. The default is not to write to a punchfile.

The input_hessian argument specifies the name of the starting hessian, which may be calculated beforehand through the hessian command. It is required with the method=baker option.

The hessian_update argument specifies the method by which the hessian is updated. Currently valid options are listed in the table below.

Update options	To do what
powell	Powell update of the hessian
bfgs	BFGS update of the hessian
mursar	Murtagh-Sargent update of the hessian
hybrid	hybrid ( i.e. weighted) combination of Powell and Murtagh-Sargent update
fde	finite differencing of selected hessian eigenmodes. The eigenmodes are specified through the var_list argument (default "1 2 3")
fde2	two-point finite differencing of selected hessian eigenmodes. See also fde.
none	hessian is not updated

This option is active only with method=mineig, scaneig, baker and diis. The default value is powell. With all other method= arguments, the hessian, if used, is updated with the BFGS scheme.

The final_hessian argument specifies the name of the final hessian, i.e. the hessian at the end of the optimisation. This argument is active with method=bfgs, bfgs2, and baker options.

The restart argument causes the optimisation to start from saved files (ending _restart), containing information about the point at which the previous run left off. Geometrical, function, gradient and hessian information is taken off those files. Options are boolean equivalents. The default is no.

The hard_fail argument routes any Tcl-error message to output for diagnostics. The default is yes.

The commands argument provides the Tcl-commands to be executed with method=custom. The default is an empty string.

The var_list argument is a string listing the variables or modes to be operated on by the eigenmode finite-difference hessian update (hessian_update=fde) or recomputation options (see below). The default is "1 2 3".

The eig_list argument is a string listing the variables or modes to be search along with method=mineig and scaneig. The default is undefined. The del argument specifies the displacement in finite-differencing steps, be it for hessian or gradient. The default is 0.001.

Target Functions

• zopt Molecular geometry optimisation in Z-matrix and mixed cartesian coordinates
• copt Molecular geometry optimisation in cartesian coordinates
• Custom target functions It is possible to optimise any function that can be evaluated by a ChemShell command, provided the correct interface is adopted.