A.33.1 Introduction

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:

1. an initial state is generated with gen_state/1
2. all relevant constraints are added with constraint/3
3. maximize/3 or minimize/3 are used to obtain a solved state that represents an optimum solution
4. 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.