Mathematical modeling features¶

Decision variables¶

Hexaly Optimizer makes a clear distinction between decision variables and intermediate expressions. A decision variable is a variable that cannot be deduced or computed from other variables or expressions. The question to ask when you use Hexaly Optimizer is What are the most atomic decisions I want to take?

This aspect can be a bit disturbing compared to other optimization techniques such as linear programming or constraint programming but it is really important for the performance of the underlying algorithms. For example, in a knapsack problem, the only decisions are the boolean variables x[i] equal to 1 if the object i is in the bag or 0 otherwise. On the opposite, the total weight of elements in the bag, defined as sum[i in 0...nbItems](values[i] * x[i]) is a typical intermediate expression: its value can be deduced from decision variables.

There are six kinds of decisions in Hexaly Optimizer: booleans, floating quantities, integer quantities, intervals, set variables, and list variables.

Constraints¶

A constraint is a kind of tag put on an expression that enforces it to be true (equal to 1). In Hexaly Optimizer, any variable or intermediate expression that has a boolean value (0, 1) can be constrained. Thus, ‘’all’’ expressions involving relational operators (<, <=, >, >=, ==, !=) but also logical and (&&), or (||), xor or immediate if (iif) can be constrained without limitation on the type of the problem. In particular, Hexaly Optimizer is not limited to linear constraints but can also handle highly-nonlinear models.

To tag an expression as a constraint in the modeler, simply prefix it by the keyword constraint.

// These two formulations are equivalent
constraint knapsackWeight <= 102;
weightCst <- knaspackWeight <= 102;
constraint weightCst;

# These two formulations are equivalent
model.constraint(knapsackWeight <= 102)
weightCst = knaspackWeight <= 102
model.constraint(weightCst)

// These two formulations are equivalent
model.constraint(knapsackWeight <= 102);
weightCst = knaspackWeight <= 102;
model.constraint(weightCst);

// These two formulations are equivalent
model.Constraint(knapsackWeight <= 102);
weightCst = knaspackWeight <= 102;
model.Constraint(weightCst);

// These two formulations are equivalent
model.constraint(knapsackWeight <= 102);
weightCst = model.leq(knaspackWeight, 102);
model.constraint(weightCst);

A good practice in operations research is to only model as constraints requirements that are strictly necessary. If a requirement may be violated in some exceptional cases then it is better modeled as a primary objectives in order to be “softly” satisfied (goal programming). Hexaly Optimizer offers a feature making this easy to do: lexicographic objectives.

Objectives¶

At least one objective must be defined using the keyword minimize or maximize. Any expression can be used as objective. If several objectives are defined, they are interpreted as a lexicographic objective function. The lexicographic ordering is induced by the order in which objectives are declared. In this way, expressions frequently encoutered in math programming models like:

maximize 10000 revenues - 100 resources + desiderata;

in order to first maximize revenues, then minimize resources, and ultimately maximize desiderata can be avoided. Indeed, you can directly write

maximize revenues;
minimize resources;
maximize desiderata;

model.maximize(revenues)
model.minimize(resources)
model.maximize(desiderata)

model.maximize(revenues);
model.minimize(resources);
model.maximize(desiderata);

model.Maximize(revenues);
model.Minimize(resources);
model.Maximize(desiderata);

model.maximize(revenues);
model.minimize(resources);
model.maximize(desiderata);

Table of available operators and functions¶

In the table below, each operator is identified with its name in Hexaly Modeler. Note that in Python, C++, C# or Java these names may slightly differ in order to respect coding conventions and reserved keywords of each language:

• In C++ and Java, decisions are suffixed with “Var” (boolVar, floatVar, intVar, setVar and listVar)

• in C# all functions start with a capital letter