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.

Boolean decisions¶

Boolean decisions can take two values 0 or 1. They are declared using the built-in function bool() that returns a new binary decision. Booleans enables you to model any problem where a binary decision is involved (such as the knapsack problem).

To have an idea of what it is possible with boolean decisions, have a look at our code templates.

Floating-point decisions¶

Floating-point decisions are used to model continuous quantitative decisions taking values in a given range. They are declared using the built-in function float(a,b) that returns a floating-point decision with range [a,b]. The largest range of a floating-point decision is defined by the IEEE 754 double-precision floating-point format, which is roughly [-10^307, 10^307].

Integer decisions¶

In a similar way, integer decisions are used to model integer quantitative decisions taking values in a given range. They are declared using the built-in function int(a,b) that returns an integer decision with range [a,b]. The largest range of an integer decision is [-2^63+1, 2^63-1], which is roughly [-10^18, 10^18].

Set and list decisions¶

Set and list decisions allow defining decision variables whose value is a collection of integers within a domain [0, n-1] where n is the unique operand of the operator. See our documentation on collection variables for details.

Interval decisions¶

Interval decisions are used to model the time range of an event. They are declared using the built-in function interval(minStart, maxEnd), where the bounds are integers representing the minimum start and the maximum end of the decision. The start is inclusive and the end is exclusive. Optional intervals can be void: the value of an optional interval decision variable can either be absent (void) or included in [minStart, maxEnd).

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.

Hexaly Modeler
Python
C++
C#
Java

// 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

Hexaly Modeler
Python
C++
C#
Java

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

	Function	Description	Arguments type	Result type	Arity	Symb
Decisional	bool	Boolean decision variable with domain {0,1}	none	bool	0
	float	Float decision variable with domain [a, b]	2 doubles	double	2
	int	Integer decision variable with domain [a, b]	2 integers	int	2
	interval	Interval decision variable with domain [minStart, maxEnd)	2 integers	interval	2
	optionalInterval	Optional Interval decision variable with domain absent or [minStart, maxEnd)	2 integers	interval	2
	list	Ordered collection of integers within a range [0, n - 1]	1 integer	collection	1
	set	Unordered collection of integers within a range [0, n - 1]	1 integer	collection	1
Arithmetic	sum	Sum of all operands	bool, int, double	int, double	n >= 0	+
	sub	Substraction of the first operand by the second one	bool, int, double	int, double	2	-
	prod	Product of all operands	bool, int, double	int, double	n >= 0	*
	min	Minimum of all operands	bool, int, double	int, double	n > 0
	max	Maximum of all operands	bool, int, double	int, double	n > 0
	div	Division of the first operand by the second one	bool, int, double	double	2	/
	mod	Modulo: mod(a, b) = r such that a = q * b + r with q, r integers and r < b.	bool, int	int	2	%
	abs	Absolute value: abs(e) = e if e >= 0, and -e otherwise	bool, int, double	int, double	1
	dist	Distance: dist(a, b) = abs(a - b)	bool, int, double	int, double	2
	sqrt	Square root	bool, int, double	double	1
	cos	Cosine	bool, int, double	double	1
	sin	Sine	bool, int, double	double	1
	tan	Tangent	bool, int, double	double	1
	log	Natural logarithm	bool, int, double	double	1
	exp	Exponential function	bool, int, double	double	1
	pow	Power: pow(a, b) is equal to the value of a raised to the power of b.	bool, int, double	double	2
	ceil	Ceil: round to the smallest following integer	bool, int, double	int	1
	floor	Floor: round to the largest previous integer	bool, int, double	int	1
	round	Round to the nearest integer: round(x) = floor(x + 0.5).	bool, int, double	int	1
	scalar	Scalar product between 2 arrays.	array	int, double	2
	piecewise	Piecewise linear function described by its breakpoints	array, int, double	double	3
Logical	not	Not: not(e) = 1 - e.	bool	bool	1	!
	and	And: equal to 1 if all operands are 1, and 0 otherwise. Takes value 1 when applied to an empty collection.	bool	bool	n >= 0	&&
	or	Or: equal to 0 if all operands are 0, and 1 otherwise. Takes value 0 when applied to an empty collection.	bool	bool	n >= 0	\|\|
	xor	Exclusive or: equal to 0 if the number of operands with value 1 is even, and 1 otherwise. Takes value 0 when applied to an empty collection.	bool	bool	n >= 0
Relational	eq	Equal to: eq(a, b) = 1 if a = b, and 0 otherwise	bool, int, double	bool	2	==
	neq	Not equal to: neq(a, b) = 1 if a != b, and 0 otherwise	bool, int, double	bool	2	!=
	geq	Greater than or equal to: geq(a, b) = 1 if a >= b, 0 otherwise	bool, int, double	bool	2	>=
	leq	Lower than or equal to leq(a, b) = 1 if a <= b, 0 otherwise	bool, int, double	bool	2	<=
	gt	Strictly greater than: gt(a, b) = 1 if a > b, and 0 otherwise. In case of intervals: gt(a, b) = 1 if start(a) >= end(b), and 0 otherwise.	bool, int, double, interval	bool	2	>
	lt	Strictly lower than: lt(a, b) = 1 if a < b, and 0 otherwise. In case of intervals: lt(a, b) = 1 if end(a) <= start(b), and 0 otherwise.	bool, int, double, interval	bool	2	<
Conditional	iif	Ternary operator: iif(a, b, c) = b if a is equal to 1, and c otherwise	bool, int, double	bool, int, double	3	?:
Set related	count	Returns the number of elements in a collection.	collection, interval, array	int	1
	indexOf	Returns the index of a value in a collection or -1 if the value is not present.	collection, int	int	2
	contains	Returns 1 if the collection, array or interval contains the given value or interval and 0 otherwise.	collection, constant array or interval, int or interval	bool	2
	partition	Returns true if all the operands form a partition of their common domain.	collection	bool	n > 0
	disjoint	Returns true if all the operands are pairwise disjoint.	collection	bool	n > 0
	cover	Returns true if all the operands form a cover of their common domain.	collection	bool	n > 0
	array	Creates an array of fixed or variadic size.	bool, int, double, array, list, set	array	n >= 0
	stepArray	Creates a stepArray of fixed size.	bool, int, double	array	n >= 1
	constArray	Creates a constArray of fixed size.	bool, int, double	array	n >= 1
	at	Returns the value in an array or a list at a specified position.	array, list, int	bool, int, double	n >= 2	[]
	find	Returns the position of the first collection containing the given element in the array, or -1 if the value is not present.	array, int	int	2
	sort	Returns the input sorted in ascending order. When used with two arguments, the input is sorted based on the values returned by the lambda.	array, list, set, lambda	array	1 or 2
	distinct	Returns the unordered set of distinct values in an array. When used with two arguments, the distinct values are based on the values returned by the lambda applied to the iterable.	array, list, set, interval, lambda	set	1 or 2
	intersection	Returns the unordered set of values present in both iterables. Returns an interval if operands are intervals.	array, list, set interval	set, interval	n >= 1
	union	Returns the unordered set of values present in at least one iterable.	array, list, set	set	n >= 0
Interval related	start	Returns the start of a non-void interval.	interval	int	1
	end	Returns the end of a non-void interval.	interval	int	1
	length	Returns the length of a non-void interval, equivalent to end(interval) - start(interval).	interval	int	1
	hull	Returns the smallest interval including all the intervals given in operands. When used with a lambda function, the hull is based on the intervals returned by the lambda applied to the iterable.	interval, lambda	interval	n >= 0
	presence	Returns true if the interval is not absent.	interval	bool	1
Other	call	Call a function. It can be used to implement your own operator.	bool, int, double, array	int, double, array, interval	n > 0