The command file typically consists of three parts: the preparation part, the analysis block and the final action part. In the preparation part variables are typically allocated, data initialized and functions defined for use at a later time. The analysis block defines how direct analysis is performed. This block is interpreted every time the direct analysis is performed, either run from within some algorithm or as a consequence of user request. In the action part the optimization algorithms that lead to problem solution are run. Test analyses at different parameter sets or some other tests (e.g. tabulating of the objective function) can also be run in this part.

The preparation part and analysis block can usually be swapped. Individual allocations and definitions can be performed right before they are used, although the command file usually looks clearer if this is done in one place. The user must be careful about putting definitions and allocations in the analysis block because this block is iteratively interpreted. What concerns tasks that do not need to be performed in every analysis, it is better if they are invoked outside the analysis block so that they are performed only once.

1.1.1 Defining the Direct Analysis

The term “direct analysis” refers to the evaluation of the objective and constraint functions and possibly their gradients at a given set of optimization parameters. User defines how the direct analysis is performed in the analysis block of the shell command file. This is the block of code in the argument block of the analysis command, i.e. within the curly brackets that follow this command.

The analysis block is interpreted by the shell interpreter every time the direct analysis is performed. Direct analysis can be called by an optimization algorithm or by some other function invoked by the interpreter. Typical examples are tabulating functions or the analyse function for performing test direct analyses.

Data transfer between the direct analyses and the functions that invoke them is implemented through global shell variables with a pre-defined meaning. The shell takes care that the current set of optimization parameters is always in the vector variable parammom when the direct analysis is invoked. In the analysis block the user can therefore obtain parameter values from this variable using the interpreter and expression evaluator functions for accessing variables. In the similar way it is expected that after the direct analysis is performed its results will appear in the appropriate global shell variables. User must take care of that in the analysis block by storing results in these variables. For example, value of the objective function must appear in scalar variable objectivemom, values of constraint functions must appear in scalar variable constraintmom, objective function gradient in vector variable gradobjectivemom, gradients of constraint functions in vector variable gradconstraintmom, simulated measurements (in the case of inverse analyses) in vector variable measmom, etc. These variables with a pre-defined meaning are treated just like other user-defined variables and the same functions can be used for their manipulation. There are however some particularities in behaviour of variable manipulation functions in the case of variables with a pre-defined meaning. Rules are more or less the same, there is only some additional intelligence incorporated, which enables user not to specify dimensions that are already known to the shell. For details, see the “Shell Variables with a Pre-defined Meaning” chapter of the “User Defined Variables in the Optimization Shell Inverse” manual.

Within the analysis block the user is expected to run a numerical simulation with parameters found in vector parammom, combine its results to evaluate the requested function values (objective and constraint functions and their derivatives) and store these results in the appropriate variables with a pre-defined meaning. This can include a number of sub-tasks, for example parameter dependent domain transformation in the case of shape optimization problems (this is reduced to finite element mesh transformation in some cases). Interfacing the simulation programme, i.e. changing input data according to parameter values, running the programme and obtaining results, is usually an important issue, as well as combining of these partial results according to problem definition in order to derive final results. Several modules of the shell provide tools for performing such sub-task, and the user can combine these tools using the file interpreter according to the character of problems that are being solved.

All tools and algorithms of the shell are accessed through the shell file interpreter. This, together with the expression evaluator (the “calculator”) and interpreter flow control functions, gives the user a great flexibility at defining different optimization problems and also the solution procedures. The shell is in the first place designed for use with simulation programmes. For test purposes, however, the user can define optimization problems in such way that evaluation of objective and other functions do not include numerical simulation. The functions are in this case defined analytically using shell variables and expression evaluator. Such examples can be found in the directory of training examples (subdirectory “opt”).

1.2 Optimization algorithms

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1.2.1 optfsqp1 { numob numnonineq numlinineq numnoneq numlineq eps epseqn maxit grad { initial } { lowbound } { upbound } }

Performs the fsqp (feasible sequential quadratic programming) optimization algorithm, which is the basic and most powerful optimization algorithm of the shell.

numob is the number of objective functions (usually one), numnonineq the number of non-linear inequality constraints, numlinineq the number of linear inequality constraints, numnoneq the number of non-linear equality constraints and numlineq the number of linear equality constraints. eps is the final norm requirement for the Newton direction and epseqn maximum violation of nonlinear equality constraints at an optimal point. Both criteria must be satisfied to stop the algorithm (the second one is in effect only if there are equality constraints). maxit is the maximum number of iterations. grad specifies if gradients are provided by direct analyses (1) or calculated numerically (0). initial is the initial guess and lowbound and upbound are vectors of lower and upper bounds on parameters. All three vectors must be in curly brackets. The components which are not specified in the lowbound or upbound vectors are not bounded below or above, respectively. Dimensions must be specified for all three vectors, and all components must be specified for initial.

Warning:

In variables which hold values or derivatives of constraint functions, these must appear in the appropriate order, the same as in the argument block of the function. First must be non-linear inequality constraints, then linear inequality constraints, then non-linear equality constraints and finally linear equality constraints (if any of these are specified, of course).

A detailed description of the fsqp algorithm can be found at

http://www.isr.umd.edu/Labs/CACSE/FSQP/fsqp.html .

1.2.2 inverse { methodspec params }

This function performs different types of optimization algorithm. methodspec determines which optimization algorithm is used. It is followed by parameter specifications params, which are dependent on the type of algorithm used.

methodspec begins either with string 1d or nd, indicating whether we will solve one-dimensional (one parameter) or multi-dimensional problems, respectively. The second part of methodspec is a string that specifies the method more precisely. Method and parameter specifications for different methods are described below.

1.2.2.1 inverse { 1d parabolic x0 step0 tol maxitbrac maxit }

Performs minimization of the objective function of one parameter. Successive three points quadratic approximations of the objective function are used where possible. The minimization is performed in two steps.

In the first step, the interval containing a local minimum is searched for. This is achieved by searching for combination of three points such that the middle point has the lowest value of the objective function. The first point is given by the user (x0), and the second two points are obtained by adding the initial bracketing step (step0) to that point once and twice, respectively. Then the three points are moved, if necessary, until the bracketing condition is reached (i.e. the middle point has the lowest value of the objective function).

In the second step, the bracketing interval that contains the three bracketing points is narrowed in such a way that the bracketing condition remains satisfied. In each iteration a new point is added in the larger of the two intervals defined by the three bracketing points. Among four points we obtain this way, those three which satisfy the bracketing condition and define the smallest interval are kept for the next iteration. The point that is added is usually chosen by finding the minimum of quadratic parabola that through the current bracketing points. This is not done if one of the two intervals becomes much smaller, since in such cases successive quadratic approximations can converge slowly.

x0 is the initial point, and step0 is the initial step of the bracketing stage. The second and the third point of the initial bracketing triple are obtained by adding step0 to x0 once and twice, respectively. tol is the tolerance for function minimum. The algorithm terminates when the difference between the highest and the lowest value of the objective function in the current three bracketing points is below tol. maxitbrac is the maximal allowed number of iterations at searching for bracketing triple. If the algorithm fails to find the three points satisfying the bracketing condition in maxitbrac iterations, it terminates and reports an error. maxit is the maximal allowed number of iterations in the second stage.

1.2.2.2 inverse { nd simplex tol maxit startguess }

Performs minimization of the objective function by simplex method. Apices of a simplex is successively moved in such a way that the simplex moves and shrinks toward function minimum. Simplex is a geometrical body in an n-dimensional space that has n+1 dimensions.

tol is tolerance for function minimum. The algorithm terminates when the difference between the greatest and the least value of the objective function in simplex apices becomes less than tol. maxit is the maximal allowed number of iterations. If the minimum is no reached in maxit iterations, the algorithm terminates and reports an error. startguess is the starting guess, containing the initial simplex. This must be a matrix of dimensions numparam+1 x numparam. Rows of this matrix represent apices of the initial simplex.

1.3 Auxiliary tools

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1.3.1 Testing the Direct Analysis

1.3.1.1 analyse { }

Performs the direct analysis at parameters parammom. The pre-defined vector parammom must therefore be set before this function is called. The values of the pre-defined global variables that hold analysis results are printed to the programme’s standard output and output file.

1.3.2 Tabulating Functions

1.3.2.1 tab1d0 { which val1 val2 numpt pobjective pmeas }

Runs a set of numpt direct analyses so that parameter which is varied between val1 and val2 with a constant step. pobjective and pmeas specify if the objective function and measurements should be printed, respectively (values different than zero indicate that the appropriate quantities should be printed).

At the sampling points, parameters other than which are taken from the pre-defined vector parammom.

1.3.2.2 tab2d0 { whichx x1 x2 numptx whichy y1 y2 numpty pobjective pmeas }

Runs a set of numptx*numpty direct analyses organised in a two-dimensional table so that parameters whichx and whichy are changed. Parameter whichx is varied between x1 and x2 with numptx equidistant sampling values, and parameter whichy is varied between y1 and y2 with numpty equidistant sampling values. pobjective and pmeas specify if the objective function and measurements should be printed, respectively (values different than zero indicate that the appropriate quantities should be printed).

At the sampling points, parameters other than which are taken from the pre-defined vector parammom.

1.3.2.3 tabline0 { kindspec args }

Runs a set of direct analyses along a line in the parameter space and prints the requested results to the programme's standard output and output file. kindspec is a string that specifies what kind of table of direct analyses should be made. The remaining arguments args specify the line along which the table is made, number of points, etc. kindspec can be either lin or exp. The meaning of the remaining arguments for both possibilities is explained below.

1.3.2.3.1 tabline0 { lin numpt pobjective pmeas point1 point21 }

Runs numpt direct analyses with sampling points equidistantly distributed along the straight line between point1 and point2. pobjective and pmeas specify if the objective function and measurements should be printed, respectively (values different than zero indicate that the appropriate quantities should be printed).

1.3.2.3.2 tabline0 { exp numpt factor pobjective pmeas point1 point2 }

Runs numpt direct analyses with sampling points non-equidistantly distributed along the straight line between point1 and point2. pobjective and pmeas specify if the objective function and measurements should be printed, respectively (values different than zero indicate that the appropriate quantities should be printed). factor is the factor for which the distance between the length of the following sampling interval is extended.