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2  Overall Design

The scope of TOMVIEW is large and broad, hence there is a need for a well-designed system. It is natural to use the power of the LabVIEWenvironment, to make the system flexible and easy to use and maintain.

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Figure 1: LabVIEW 7.1 or above is required for TOMVIEW

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2.1  Input and Output

Normally, when solving an optimization problem, a direct call to a solver is made with a long list of parameters in the call. The parameter list is solver-dependent and makes it difficult to make additions and changes to the system.

TOMVIEW solves the problem in two steps. First the problem is defined and stored in a LabVIEW object. The solver is then called with a single string argument and the problem structure. This is handled by the driver VI tomRun which can call any available solver, hiding the details of the call from the user. The solver output is made available in a standardized result structure.

2.2  Problem Types

TOMVIEW solves a number of different optimization problems. The currently defined types are listed in Table 1.

The Prob variable probType contains the current type of optimization problem to be solved. An optimization solver is defined to be of type solvType, where solvType is any of the probType entries in Table 1. A certain solvType might be able to solve a problem of several other types. For example, a constrained nonlinear programming solver (for example SNOPT, MINOS) can solve unconstrained problems, linear and quadratic programs and constrained nonlinear least squares problems.

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Table 1: The different types of optimization problems defined in TOMVIEW.

probType		Type of optimization problem
uc	1	Unconstrained optimization (incl. bound constraints).
qp	2	Quadratic programming.
con	3	Constrained nonlinear optimization.
ls	4	Nonlinear least squares problems (incl. bound constraints).
lls	5	Linear least squares problems.
cls	6	Constrained nonlinear least squares problems.
mip	7	Mixed-Integer programming.
lp	8	Linear programming.
glb	9	Box-bounded global optimization.
glc	10	Global mixed-integer nonlinear programming.
miqp	11	Constrained mixed-integer quadratic programming.
minlp	12	Constrained mixed-integer nonlinear optimization.

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Each probType corresponds to an assign routine (exceptions applies).

There are a number of test problems included with TOMVIEW. The folder testprob contains a large set of test problems, primarily for benchmark testing and development. The problems are implemented in LabVIEW and the user functions are coded in C. The folder quickguide contains smaller problems on the standard format. It is recommended to look at the quick guide problems when implementing custom problems.

The following table defines the test problems included with the TOMVIEW Base Module.

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Table 2: Defined test problem sets in TOMVIEW.

probSet	probType	Description of test problem set
chsProb	3	Hock-Schittkowski constrained test problems.
clsProb	6	Constrained nonlinear least squares problems.
conProb	3	Constrained test problems.
glbProb	9	Box-bounded global optimization test problems.
glcProb	10	Global MINLP test problems.
lgoProb	9	Box-bounded global optimization test problems (LGO).
llsProb	5	Linear least squares problems.
lpProb	8	Linear programming problems.
mghProb	4	More, Garbow, Hillstrom nonlinear least squares problems.
minlpProb	12	Mixed-integer nonlinear programming problems.
mipProb	7	Mixed-integer programming problems.
miqpProb	11	Mixed-integer quadratic programming problems.
qpProb	2	Quadratic programming test problems.
ucProb	1	Unconstrained test problems.
chsProb	1	Hock-Schittkowski unconstrained test problems.

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The problem may be accessed in two different ways, either the probInit.vi is used with a subsequent call to tomRun, or *Examples may be opened for each problem set respectively.

2.3  Solution Process

The process of optimization in TOMVIEW is quite straight forward, even to the beginner. It is efficient and easy to use the interactive system. In general, the following process may be followed:

• Identify the problem type to be solved.
  
  
• Select and use the appropriate assign routine to create a problem instance.
  
  
• Modify solver settings as needed.
  
  
• Select an appropriate solver and call the driver routine tomRun.

If solving one of the pre-defined problems:

• Use probInit to create a problem instance. Select the problem set and number.
  
  
• Modify solver settings if needed.
  
  
• Use tomRun to call the solver.

Depending on the type of problem, the user needs to supply the low-level routines that calculate the objective function, constraint functions for constrained problems, and also if possible, derivatives. To simplify the coding process, TOMVIEW provides gateway routines that ensure that all solvers obtain the correct format.

For example, when working with a least squares problem, it is natural to code the function that computes the vector of residual functions r_i(x₁,x₂,…), since a dedicated least squares solver probably operates on the residual while a general nonlinear solver needs a scalar function, in this case f(x) = 1/2 r^T(x)r(x). Such issues are automatically handled by the gateway functions included in the system.

If derivatives are not given and the selected solver requires this information, the system will automatically ensure that it is provided by numerical differentiation.

2.3.1  Examples in TOMVIEW

This section is to illustrate how to use some routines in TOMVIEW. For example one can try to solve one of the chsProb examples with SNOPT. The following figure illustrates the solution process in LabVIEW.

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Figure 2: Block Diagram - assigning and solving a chs test problem.

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chsProb problem number 8 is solved. It is assigned with the probInit block. It is subsequently solved in the tomRun node with the solver SNOPT. The outputs are the optimal solution x_k, the optimal functional value f_k and the time consumed to solve the problem.

It is trivial to solve a qpProb problem use SQOPT instead. The problem is created and solved in the same manner.

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Figure 3: Block Diagram - assigning and solving a qp test problem.

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The qpProb test problem number 15 is created and the output variables are the same as before. The output cluster may be expanded to include further information.

2.4  Low Level Routines and Gateway Routines

Low level routines are the routines that compute:

• The objective function value
• The gradient vector
• The Hessian matrix (second derivative matrix)
• The residual vector (for nonlinear least squares problems)
• The Jacobian matrix (for nonlinear least squares problems)
• The vector of constraint functions
• The matrix of constraint derivatives (the constraint Jacobian)
• The second part of the second derivative of the Lagrangian function.

The last three routines are only needed for constrained problems. Observe that only the KNITRO solver can make use of second order information.

The names of these routines are defined in the structure fields Prob.FUNCS.f, Prob.FUNCS.g, Prob.FUNCS.H etc. It is a task for the assign routine to set the names of the low level VI's. This is done by using, for example, the routine assign_con with the references as inputs.

Only the low level routines relevant for a certain type of optimization problem need to be coded. Numerical differentiation is automatically used for gradient, Jacobian and constraint gradient if the corresponding user routine is non present or left out when calling assign_con. However, the solver always needs more time to estimate the derivatives compared to if the user supplies code for them. Also the numerical accuracy is lower for estimated derivatives, so it is recommended that the user always tries to code the derivatives, if it is possible.

TOMVIEW uses a variety of gateway routines to handle the special solver needs.

To get a picture of how the low-level routines are used in the system, consider Figure 4 and 5. Figure 4 illustrates the chain of calls when computing the objective function value in SNOPT for a nonlinear least squares problem defined in mghProb_Examples, mghProb_R and mghProb_J. Figure 5 illustrates the chain of calls when computing the numerical approximation of the gradient (by use of the VI callback_fdng) in SNOPT for an unconstrained problem defined in ucProb_Examples and ucProb_F.

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