A linear programming problem or simply linear program
(LP) consists of:
- a set of linear constraints
- a set of variables
- a linear objective function.
The goal is to assign values to the variables so as to maximize
(or minimize) the value of the objective function while satisfying all
constraints.
Many optimization problems can be modeled in this way. As one basic
example, consider a knapsack with fixed capacity C, and a number of
items with sizes s(i) and values v(i). The
goal is to put as many items as possible in the knapsack (not exceeding
its capacity) while maximizing the sum of their values.
As another example, suppose you are given a set of coins with
certain values, and you are to find the minimum number of coins such
that their values sum up to a fixed amount. Instances of these problems
are solved below.
Solving an LP or integer linear program (ILP) with this library
typically comprises 4 stages:
- an initial state is generated with gen_state/1
- all relevant constraints are added with constraint/3
- maximize/3 or minimize/3
are used to obtain a solved state that represents an optimum
solution
- variable_value/3
and objective/2 are
used on the solved state to obtain variable values and the objective
function at the optimum.
The most frequently used predicates are thus:
- gen_state(-State)
- Generates an initial state corresponding to an empty linear program.
- constraint(+Constraint,
+S0, -S)
- Adds a linear or integrality constraint to the linear program
corresponding to state S0. A linear constraint is of the form
Left Op C,
where Left is a list of Coefficient*Variable
terms (variables in the context of linear programs can be atoms or
compound terms) and C is a non-negative numeric constant. The
list represents the sum of its elements. Op can be =, =<
or >=. The coefficient 1 can be omitted. An
integrality constraint is of the form integral(Variable)
and constrains Variable to an integral value.
- maximize(+Objective,
+S0, -S)
- Maximizes the objective function, stated as a list of
Coefficient*Variable terms that represents the sum of its
elements, with respect to the linear program corresponding to state
S0. \arg{S} is unified with an
internal representation of the solved instance.
- minimize(+Objective,
+S0, -S)
- Analogous to maximize/3.
- variable_value(+State,
+Variable, -Value)
- Value is unified with the value obtained for Variable. State
must correspond to a solved instance.
- objective(+State,
-Objective)
- Unifies Objective with the result of the objective function
at the obtained extremum. State must correspond to a solved
instance.
All numeric quantities are converted to rationals via rationalize/1,
and rational arithmetic is used throughout solving linear programs. In
the current implementation, all variables are implicitly constrained to
be non-negative. This may change in future versions, and
non-negativity constraints should therefore be stated explicitly.
