MathematicalProgram Declaration and Attributes

Attributes

The attributes of mathematical programs are listed in this table.

Attribute	Value-type	See also page
Objective	variable-identifier
Direction	minimize, maximize
Variables	variable-set
Constraints	constraint-set
Type	model-type
ViolationPenalty	reference	Infeasibility Analysis
Text	string	The Text and Comment attributes
Comment	comment string	The Text and Comment attributes
Convention	convention	Globally Overriding Units Through Conventions

Example

The following example illustrates a typical mathematical program.

MathematicalProgram TransportModel {
    Objective   : TransportCost;
    Direction   : minimize;
    Constraints : AllConstraints;
    Variables   : AllVariables;
    Type        : lp;
}

It defines the linear program TransportModel, which is built up from all constraints and variables in the model text. The variable TransportCost serves as the objective function to be minimized.

The Objective attribute

With the Objective attribute you can specify the objective of your mathematical program. Its value must be a reference to a (defined) variable or any other variable expression. When you want to use the objective value in the end-user interface of your model, the Objective attribute must be a variable reference.

Omitting the objective

If you do not specify an objective, your mathematical program will be solved to find a feasible solution and it will then terminate.

The Direction attribute

In conjunction with an objective you must use the Direction attribute to indicate whether the solver should minimize or maximize the objective. During a SOLVE statement you can override this direction by using a WHERE clause for the direction option.

The Variables attribute

With the Variables attribute you can specify which set of variables are to be included in your mathematical program. Its must be either the predefined set AllVariables or a subset thereof. The set AllVariables is predefined by AIMMS, and it contains the names of all the variables declared in your model. Its contents cannot be changed. If you mathematical program contains an objective, AIMMS will automatically add this to set of generated variables during generation.

Variables as parameters

If the Variables attribute is assigned a subset of the set AllVariables, AIMMS will treat all the variables outside this set as if they were parameters. That is, all occurrences of such variables will not result in the generation of individual variables for the solver, but will be accounted for in the right-hand side of the constraint according to their value during generation.

Compare to NonvarStatus

The Variables attribute performs a similar function as the NonvarStatus attribute or the .NonVar suffix of a variable (see also Variable Declaration and Attributes). The Variables attribute in a mathematical program allows you to quickly change the status of an entire class of variables, while the NonvarStatus (in a variable declaration) gives much finer control at the individual level. As shown below, the latter is very useful to perform model algebra.

The Constraints attribute

With the Constraints attribute you can specify which constraints are part of your mathematical program. Its value must be either the predefined set AllConstraints or a subset thereof. The set AllConstraints contains the names of all declared constraints plus the names of all variables which have a definition attribute. Its contents is computed at compile time, and cannot be changed.

• If you specify the set AllConstraints, AIMMS will generate individual constraints for all declared constraints and variables with a definition.

• If you specify a subset of the set AllConstraints, AIMMS will only generate individual constraints for the declared constraints and defined variables in that subset.

If you mathematical program has an objective which is a defined variable, its definition is automatically added to the set of generated constraints during generation.

Defined variables

Variables with a nonempty definition attribute have a somewhat special status. Namely, for every defined variable AIMMS will not only generate this variable, but will also generate a constraint containing its definition. Therefore, defined variables are contained in both the predefined sets AllVariables and AllConstraints. You can add a defined variable to the variable and constraint set of a mathematical program independently.

• If you omit a defined variable from the variable set of a mathematical program, all occurrences of the variable will be fixed to its current value and accounted for in the right-hand side of all constraints.

• If you omit a defined variable from the constraint set of a mathematical program, the defining constraint will not be generated.

Performing model algebra

By changing the contents of the identifier sets that you have entered at the Variables and Constraints attributes of a mathematical program you can perform a simple form of model algebra. That is, you can investigate the effects of adding or removing constraints from within the graphical interface. Furthermore, it allows you to reconfigure your model based on the value of your model data.

Synchronizing variable and constraint sets

When changing the contents of either the variable or the constraint set of a mathematical program, you may find that the contents of the other set also needs some adjustment. For instance, adding a variable to a mathematical program makes no sense if there are no constraints that refer to it. AIMMS offers two special set-valued functions to help you to accomplish this task.

The function VariableConstraints

The function VariableConstraints takes a subset of the predefined set AllVariables as its argument, and returns a subset of the predefined set AllConstraints. The resulting constraint set contains all constraints which use one or more of the variables in the argument set.

The function ConstraintVariables

The function ConstraintVariables performs the opposite task. It takes a subset of the set AllConstraints as its arguments, and returns a subset of the set AllVariables. The resulting variable set contains all variables which are referred to in one or more constraints in the argument set. Also included are all variables referred to in the definitions of other variables inside the set.

Example

Consider the use of the functions VariableConstraints and ConstraintVariables in conjunction with the following declaration of a mathematical program.

MathematicalProgram PartialTransportModel {
    Objective   : TransportCost;
    Direction   : minimize;
    Constraints : PartialConstraintSet;
    Variables   : PartialVariableSet;
}

Assume that the set PartialVariableSet contains a subset of the variables declared in the model. Further assume that you would like to build up the contents of the set PartialConstraintSet together with the required additions to PartialVariableSet so that the contents of both sets are maximal. This is referred to as their transitive closure. By successively calling the functions VariableConstraints and ConstraintVariables, the following loop computes the transitive closure of the variable and constraint sets.

repeat
   PreviousCardinality  := Card( PartialVariableSet );
   PartialConstraintSet := VariableConstraints( PartialVariableSet   );
   PartialVariableSet   := ConstraintVariables( PartialConstraintSet );

   break when Card( PartialVariableSet ) = PreviousCardinality;
endrepeat ;

The break occurs when the set PartialVariableSet has not increased in size.

The Type attribute

With the Type attribute of a mathematical program you can prescribe a solution type. When the specified type is not compatible with the generated mathematical program, AIMMS will return an error message. You can override the type during a SOLVE statement using a WHERE clause for the type option. You can use this, for instance, to easily switch between the mip and rmip types.

Available types

A complete list of the mathematical program types available within AIMMS is given in this table. Most are self-explanatory. When the type rmip is specified, all integer variables are treated as continuous within their bounds. The rmip type is the global version of the Relax attribute associated with individual variables (see also Variable Declaration and Attributes). The types ls and nls can only be selected in the absence of the Objective attribute.

Table 28 Available model types with AIMMS

Type	Description
lp	linear program
ls	linear system
qp	quadratic program
nlp	nonlinear program
nls	nonlinear system
mip	mixed integer program
rmip	relaxed mixed integer program
minlp	mixed integer nonlinear program
rminlp	relaxed mixed integer nonlinear program
qp	quadratic program
miqp	mixed integer quadratic program
qcp	quadratic constraint program
miqcp	mixed integer quadratic constraint program
network	pure network program
mcp	mixed complementarity program
mpcc	mathematical program with complementarity constraint

The Convention attribute

You can use the Convention attribute to specify the unit convention that you want to be used for scaling the variables and constraints in your mathematical program. For further details on this issue you are referred to Globally Overriding Units Through Conventions.

The ViolationPenalty attribute

With the ViolationPenalty attribute you can instruct AIMMS to automatically add artificial terms to the constraints of your mathematical program to help resolve and/or track infeasibilities in your mathematical program. Infeasibility analysis and the use of the ViolationPenalty attribute is discussed in full detail in Infeasibility Analysis.