# Constraint Declaration and Attributes

Definitions

Constraints form the major mechanism for specifying a mathematical program in AIMMS. They are used to restrict the values of variables with interlocking relationships. Constraints are numerical relations containing expressions in terms of variables, parameters and constants.

Constraint attributes

The possible attributes of constraints are given in this table.

Attribute

Value-type

IndexDomain

index-domain

The IndexDomain attribute

Unit

unit-valued expression

Text

string

The Text and Comment attributes

Comment

comment string

The Text and Comment attributes

Definition

expression

Property

NoSave, Sos1, Sos2, IndicatorConstraint

Properties of variables

IsDiversificationFilter, IsRangeFilter,

IncludeInLazyConstraintPool,

IncludeInCutPool, Chance

SosWeight

sos-weights

ActivatingCondition

expression

Probability

expression

Aproximation

element-expression

Domain restriction for constraints

Restricting the domain of constraints through the IndexDomain attribute influences the matrix generation process. Constraints are generated only for those tuples in the index domain that satisfy the domain restriction.

The Definition attribute

With the Definition attribute of a constraint you specify a numerical relationship between variables in your model. Without a definition a constraint is indeterminate. Constraint definitions consist of two or three expressions separated by one of the relational operators =, >= or <=.

Example

The following constraints express the simultaneous requirements that the sum of all transports from a city i must not exceed Supply(i), and that for each city j the Demand(j) must be met.

Constraint SupplyConstraint {
IndexDomain  : i;
Unit         : kton;
Definition   : sum( j, Transport(i,j) ) <= Supply(i);
}
Constraint DemandConstraint {
IndexDomain  : j;
Unit         : kton;
Definition   : sum( i, Transport(i,j) ) >= Demand(j);
}

Allowed relationships

If $$a$$ and $$b$$ are expressions consisting of only parameters and $$f(x,\dots)$$ and $$g(x,\dots)$$ are expressions containing parameters and variables, the following two kinds of relationships are allowed.

$a \leq f(x,\dots) \leq b \qquad \mbox{or} \qquad f(x,\dots) \gtrless g(x,\dots)$

where $$\gtrless$$ denotes any of the relational operators =, >= or <=. Either $$a$$ or $$b$$ can be omitted if there is no lower or upper bound on the expression $$f(x,\dots)$$, respectively. When both $$a$$ and $$b$$ are present, the constraint is referred to as a ranged constraint. The expressions may have linear and nonlinear terms, and may utilize the full range of intrinsic functions of AIMMS except for the random number functions.

Conditional expressions in constraints

You must take extreme care to ensure continuity when the constraints in your model contain logical conditions that include references to variables. Such constraints are viewed by AIMMS as nonlinear constraints, and thus can only be passed to a solver that can handle nonlinearities. It is possible that the outcome of a logical condition, and thus the form of the constraint, changes each time the underlying solver asks AIMMS for function values and gradients. For example, if x(i) is a decision variable, and a constraint contains the expression

sum[ i, if ( x(i) > 0 ) then  x(i)^2 endif ]

it may or may not contain the term x(i)^2, depending on the current value of x(i). In this example, both the expression and its gradient are continuous functions at x(i) = 0.

## Constraint Properties

The Property attribute

With the Property attribute you can specify further characteristics of the constraint at hand. The possible properties of a constraint are NoSave, Sos1, Sos2, Level, Bound, Basic, ShadowPrice, RightHandSideRange, and ShadowPriceRange.

The NoSave property

When you specify the NoSave property you indicate that you do not want AIMMS to store data associated with the constraint in a case, regardless of the specified case identifier selection.

## SOS Properties

The SOS properties

The constraint types Sos1 and Sos2 are used in mixed integer programming, and mutually exclusive. In the context of mathematical programming SOS is an acronym for Special Ordered Sets. A SOS set is associated with every (individual) constraint of type Sos1 or Sos2.

When you specify that a constraint is of type Sos1 or Sos2, an additional SOS-specific attributes becomes available, namely the SosWeight attributes. With this attributes, you can provide further information to the solver about the contents and ordering of the SOS set to be associated with the constraint.

Sos1 constraints

A type Sos1 constraint specifies to the solver that at most one of the variables within the SOS set associated with the constraint is allowed to be nonzero, while all other variables in the SOS set must be zero. Inside a Sos1 constraint all variables in the SOS set must have a lower bound of zero and an upper bound greater than zero.

Sos2 constraints

A type Sos2 constraint specifies to the solver that at most two consecutive variables within the SOS set associated with the constraint are allowed to be nonzero, while all other variables within the SOS set must be zero. All individual variables within the SOS set must have a lower bound of zero and an upper bound greater than zero. The order of the individual variables within the SOS set is determined by their weights (as specified in the SosWeight attribute), where the ordering is from low to high weight.

The SosWeight attribute

With the SosWeight attribute you must specify the contents of the SOS set to be associated with a Sos1 or Sos2 constraint, as well the ordering of its elements. Section 7.5 of the AIMMS Modeling Guide describes how these weights are used during the branch-and- bound process. The syntax of the SosWeight attribute is as follows.

Syntax

sos-weights:

Within the SosWeight attribute you can (but need not) specify a weight for every variable occurring in the constraint. Each weight must be an expression using all the indices in the index domain of the variable plus some or all of the indices in the index domain of the constraint. All weights specified for a particular constraint must be unique, i.e. you cannot specify the same weight for two (individual) variables. The SOS set to be associated with the constraint will be constructed from all variables-within the domain of both the constraint and variable-for which a nonzero weight has been specified in the SosWeight attribute, i.e. if the value of the specified weight is 0.0 for a particular tuple, the corresponding individual variable will not be included in the SOS set. The ordering of variables within the SOS set is from low to high weight.

Consistency

If you do not specify SOS weights, AIMMS will make sure that ordering of variables in each SOS set is consistent over all SOS sets. If you specify SOS weights yourself, you have to make sure that the variable orderings of all SOS sets of type Sos2 are consistent, or your model might become infeasible if feasibility requires that two adjacent variables in one SOS set become nonzero, which are ordered inconsistently in another SOS set. Therefore, AIMMS requires that you specify the SosWeight attributes for all SOS constraints in your model, whenever you specify it for one SOS constraint.

Example

The following is specification of Sos2 constraint to determine the variable y piece-wise linearly from a variable x(i).

Constraint DetermineY {
Property     : Sos2;
Definition   : y = sum[ i, x(i)*c(i) ];
SosWeight    : x(i) : XWeight(i);
}

## Solution Pool Filtering

Solution pool

During the solution process of a MIP problem, the solvers CPLEX, Gurobi and COPT are capable of storing multiple feasible integer solutions in a solution pool, for instance, to capture solutions with attractive properties that are hard to express in a linear fashion.

Filtering

While populating the solution pool, CPLEX offers advanced filtering capabilities, allowing you to control which solutions end up in the solution pool. CPLEX provides two predefined ways to filter solutions:

• if you want to filter solutions based on their difference as compared to a reference solution, use a diversity filter, or

• if you want to filter solutions based on their validity in an additional linear constraint, use a range filter.

To enable filters the CPLEX option Do_Populate need to be on.

Diversity filters

A diversity filter allows you to generate solutions that are similar to (or different from) a set of reference values that you specify for a set of binary variables. In particular, you can use a diversity filter to generate more solutions that are similar to an existing solution or to an existing partial solution. Several diversity filters can be used simultaneously, for example, to generate solutions that share the characteristics of several different solutions.

The IsDiversificationFilter property

In AIMMS, a constraint is used as a diversity filter if the constraint property IsDiversificationFilter has been set. In a diversification filter, the Abs function is used to measure the distance from a given binary variable, and all variables should only occur as the argument of an Abs function.

Example

This following diversification filter forces the solutions to have a distance of at least 1 from variable x.

Constraint filter1 {
Property     :  IsDiversificationFilter;
Definition   :  Abs(x - 1) >= 1;
}

Range filters

A range filter allows you to generate solutions that obey a new constraint, specified as a linear expression within a range. Range filters can be used to express diversity constraints that are more complex than the standard form implemented by diversity filters. In particular, range filters also apply to general integer variables, semi-integer variables, continuous variables, and semi-continuous variables, not just to binary variables.

The IsRangeFilter property

In AIMMS, a constraint is used as a range filter if the constraint property IsRangeFilter has been set for the constraint.

Example

The following range filter specifies that any solution to be added to the solution pool should satisfy the following constraint.

Contraint filter2 {
Property     :  IsRangeFilter;
Definition   :  x + y + z >= 2;
}

## Indicator Constraints, Lazy Constraints and Cut Pools

Indicator constraints

An indicator constraint is a new way of controlling whether or not a constraint takes effect, based on the value of a binary variable. Traditionally, such relationships are expressed by so-called big-$$M$$ formulations. Big-$$M$$ formulations, however, can introduce unwanted side-effects and numerical instabilities into a mathematical program. Using indicator constraints, such relationships between a constraint and a variable can be directly expressed in the constraint declaration. Indicator constraints are supported by the solvers CPLEX, Gurobi and ODH-CPLEX.

The IndicatorConstraint property

You can specify that a constraint is an indicator constraint by settings it IndicatorConstraint property. For indicator constraints, a new attribute called ActivatingCondition will become available in the constraint declaration.

The ActivatingCondition attribute

Through the ActivatingCondition attribute you can specify under which condition the constraint definition should become active during the solution process. Its value should be an expression of the form

binary-variable = expression

where the expression must take one of the values 0 or 1. Note: stochastic variables and parameters are not allowed inside an activation condition.

Example

Consider the following big-$$M$$ constraint

Constraint BigMConstraint {
Definition : x1 + x2 <= M*y;
}

where y is a binary variable. Using the IndicatorConstraint property, the constraint can be reformulated as an indicator constraint as follows

Constraint NonBigMConstraint {
Property             : IndicatorConstraint;
ActivatingCondition  : y = 0;
Definition           : x1 + x2 = 0;
}

The constraint only becomes effective, whenever the binary variable y takes the value 0. To solve the model with the indicator constraint, you need the CPLEX, Gurobi or ODH-CPLEX solver.

Lazy constraints

Sometimes, for a MIP formulation, a user can already identify a group of constraints that are unlikely to be violated (lazy constraints). Simply including these constraints in the original formulation could make the LP subproblem of a MIP optimization very large or too expensive to solve. CPLEX, Gurobi and ODH-CPLEX can handle problems with lazy constraints more efficiently, and therefore AIMMS allows you to identify lazy constraints in your model.

The IncludeInLazyConstraintPool property

You can specify that a constraint should be added to the pool of lazy constraints considered by CPLEX, Gurobi or ODH-CPLEX by setting the property IncludeInLazyConstraintPool. You need the CPLEX, Gurobi or ODH-CPLEX solver to use this constraint property. When solving your MIP model, CPLEX, Gurobi and ODH-CPLEX will only consider these constraints when they are violated.

User cut pools

As discussed in Suffices and Callbacks, AIMMS allows you to add cuts to your mathematical program on the fly during the solution process by using the CallbackAddCut callback. However, when the set of cuts you want to generate is fixed and known upfront, using the CallbackAddCut may add significant overhead to the solution process of your model while you don’t need its flexibility. For those situations, CPLEX allows you to specify a fixed pool of user cuts during the generation of your mathematical program.

The IncludeInCutPool property

By setting the constraint property IncludeInCutPool you can indicate that this constraint should be included in the pool of user cuts associated with your mathematical program. You need the CPLEX solver to use this constraint property. When solving your MIP model, CPLEX will consider the user cuts added in this manner when appropriate.

## Constraint Levels, Bounds and Marginals

Constraint levels and bounds

A constraint in AIMMS can conceptually be divided such that one side consists of all variable terms, whereas the other side consists of all remaining constant terms. The level value of a constraint is the accumulated value of the variable terms, while the constant terms make up the bound of the constraint.

The Level, Bound, Basic and ShadowPrice properties

With the Level, Bound, Basic and ShadowPrice properties you indicate whether you want to store (and have access to) particular parametric data associated with the constraint.

• When you specify the Level property AIMMS will retain the level values of the constraint as provided by the solver. You can access the level values of a constraint by using the constraint name as if it were a parameter.

• By specifying the Bound property, AIMMS will store the upper and lower bound of the constraint as employed by the solver. You get access to the bounds by using the .Lower and .Upper suffices with the constraint identifier.

• If the Basic property has been specified, AIMMS stores basic information is available through the .Basic suffix as an element in of the predefined AIMMS set AllBasicValues. A constraint is said to be basic (nonbasic or superbasic) if its associated slack variable is basic (nonbasic or superbasic).

• With the ShadowPrice property you indicate that you want to store the shadow prices as computed by the solver. You can access these shadow prices by means of the .ShadowPrice attribute.

The shadow price (or dual value) of a constraint is the marginal change in the objective value with respect to a change in the right-hand side (i.e. the constant part) of the constraint. This value is determined by the solver after a SOLVE statement has been executed. The precise mathematical interpretation of the shadow price is discussed in detail in many text books on mathematical programming. Note: if a basic or superbasic constraint has a shadow price of zero then it will be displayed as 0.0, but if a nonbasic constraint has a shadow price of zero then it will be displayed as ZERO.

When the variables and constraints in your model have an associated unit (see Units of Measurement), special care is required in interpreting the values returned through the .ShadowPrice suffix. To obtain the shadow price in terms of the units specified in the model, the values of the .ShadowPrice suffix must be scaled as explained in Unit-based Scaling of Mathematical Programs.

The property RightHandSideRange

By specifying the RightHandSideRange property you request AIMMS to conduct a first type of sensitivity analysis on this constraint during a SOLVE statement of a linear program. The result of this sensitivity analysis are three parameters defined over the domain of the constraint. Two of these parameters represent the smallest and largest values of an interval over which an individual right-hand side (or left-hand side) value can be varied such that the basis remains constant. Consequently, the shadow prices and the reduced costs remain unchanged for variations of a single value within the interval. The third parameter specifies the nominal value for the right-hand side (or left-hand side) of the constraint.

Single sided or ranged constraint

There are three cases we have to consider for the RightHandSideRange property:

• if the constraint is single sided (i.e. $$f(x) \leq a$$) then the smallest, nominal, and largest value for the constraint side are reported (both when constraint is binding and not binding)

• if the constraint is of range type (i.e. $$a \leq f(x) \leq b$$) and it is binding at one side, then the smallest, nominal, and largest value for the binding side of the constraint are reported

• if the constraint is of range type (i.e. $$a \leq f(x) \leq b$$) and it is not binding at neither side, then the lowest upper bound and the highest lower bound are reported.

The values are accessible through the suffices .SmallestRightHandSide, .NominalRightHandSide, and .LargestRightHandSide.

With the ShadowPriceRange property you request AIMMS to conduct a second type of sensitivity analysis on this constraint during a SOLVE statement of a linear program. The result of the sensitivity analysis are two parameters defined over the domain of the variable. The values assigned to the parameters will be the smallest and largest values that the shadow price of the constraint can take while holding the objective value constant. The smallest and largest values of the constraint marginals are accessible through the suffices .SmallestShadowPrice and .LargestShadowPrice.

Linear programs only

As with the advanced sensitivity properties of variables (see Variable Properties), AIMMS also supports the advanced sensitivity analysis conducted through the properties RightHandSideRange and ShadowPriceRange for linear mathematical programs only. Again, if you want to apply these types of analysis to the final solution of a mixed-integer program, you should fix all integer variables to their final solution (using the .NonVar suffix) and re-solve the resulting mathematical program as a linear program.

Storage and computational costs

Setting any of the properties ShadowPrice, ShadowPriceRange or RightHandSideRange may result in an increase of the memory usage. In addition, the computations required to compute the ShadowPriceRange may considerably increase the total solution time of your mathematical program.

## Constraint Suffices for Global Optimization

Suffices for global optimization

AIMMS provides a number of constraint suffices especially for the global optimization solver BARON. They are:

By providing additional knowledge, that cannot be determined automatically by BARON itself, about the constraints in your model through these suffices, the BARON solver may be able to optimize your global optimization model in a more efficient manner. For more detailed information about the specific capabilities of the BARON solver, you are referred to the BARON website http://www.theoptimizationfirm.com/.

The .Convex suffix

The algorithm of the BARON solver exploits convexity-either identified automatically by BARON itself or explicitly supplied in the model formulation-in order to generate polyhedral cutting planes and relaxations for multivariate non-convex problems. Through the .Convex suffix you can explicitly indicate that a particular constraint is convex if BARON is unable to determine its convexity automatically.

The .RelaxationOnly suffix

Using the .RelaxationOnly suffix, you can considerably enhance the convexification capabilities of BARON. Some nonlinear problem reformulations can often tighten the relaxation process of BARON’s branch-and-bound algorithm while making local search considerably more difficult. By assigning a nonzero value to the .RelaxationOnly suffix, you indicate to BARON that the constraint at hand should only be included as a relaxation to the branch-and-bound algorithm, while it should be excluded from the local search.

## Chance Constraints

Chance constraints

The AIMMS modeling language offers facilities for robust optimization models, including support for chance constraints (see also Chance Constraints). By setting the Chance property of a constraint, the constraint will become a chance constraint when solving a mathematical program using robust optimization, using the distributions specified for the random parameters contained in its definition. When setting the Chance property, two new attributes will become available, the Probability attribute and the Approximation attribute.

Only for robust optimization

Note that setting the Chance property does not influence the availability and use of the constraint outside the context of robust optimization. In that case, AIMMS will just use the original, deterministic, constraint definition, completely disregarding the uncertainty of the parameters used in the constraint.

The Probability attribute

Through the Probability attribute, you can specify the probability with which you want the constraint to be satisfied for any feasible solution to the robust counterpart of a robust optimization model. Its value must be a numerical expression in the range $$[0,1]$$.

The Approximation attribute

When constructing the robust counterpart, AIMMS can use several types of approximations to approximate the chance constraint at hand. You can use the Approximation attribute to specify the type of approximation you want to be applied. The chosen type of approximation may lead to a robust counterpart which is easier or harder to solve (see also Chance Constraints). The value of the attribute must be an element expression into the predefined set AllChanceApproximationTypes.