neil-lindquist/linear-programming

A Common Lisp library for solving linear programming problems

Common Lisp Linear Programming

This is a Common Lisp library for solving linear programming problems. It's designed to provide a high-level and ergonomic API for specifying linear programming problems as lisp expressions.

The core library is implemented purely in Common Lisp with only a few community-standard libraries as dependencies (ASDF, Alexandria, Iterate). However, the solver is designed to support alternative backends without any change to the user's code. Currently, there is a backend for the GNU Linear Programming Kit (GLPK).

Installation

The linear-programming library is avalible in both the main Quicklisp distribution and Ultralisp, so it can loaded with with (ql:quickload :linear-programming). You can check that it works by running (asdf:test-system :linear-programming).

If you are not using Quicklisp, place this repository, Alexandria, and Iterate somewhere where ASDF can find them. Then, it can be loaded with (asdf:load-system :linear-programming) and tested as above.

Usage

See neil-lindquist.github.io/linear-programming/ for further documentation.

Consider the following linear programming problem.

maximize x + 4y + 3z
such that

• 2x + y ≤ 8
• y + z ≤ 7
• x, y, z ≥ 0

First, the problem needs to be specified. Problems are specified with a simple DSL, as described in the syntax reference.

(use-package :linear-programming)

(defvar problem (parse-linear-problem '(max (= w (+ x (* 4 y) (* 3 z))))
                                      '((<= (+ (* 2 x) y) 8)
                                        (<= (+ y z) 7))))

Once the problem is created, it can be solved with the simplex method.

(defvar solution (solve-problem problem))

Finally, the optimal tableau can be inspected to get the resulting objective function, decision variables, and reduced-costs (i.e. the shadow prices for the variable's lower bounds).

(format t "Objective value solution: ~A~%" (solution-variable solution 'w))
(format t "x = ~A (reduced cost: ~A)~%" (solution-variable solution 'x) (solution-reduced-cost solution 'x))
(format t "y = ~A (reduced cost: ~A)~%" (solution-variable solution 'y) (solution-reduced-cost solution 'y))
(format t "z = ~A (reduced cost: ~A)~%" (solution-variable solution 'z) (solution-reduced-cost solution 'z))

;; ==>
;; Objective value solution: 57/2
;; x = 1/2 (reduced cost: 0)
;; y = 7 (reduced cost: 0)
;; z = 0 (reduced cost: 1/2)

Alternatively, the with-solution-variables and with-solved-problem macros simplify some steps and binds the solution variables in their bodies.

(with-solution-variables (w x y z) solution
  (format t "Objective value solution: ~A~%" w)
  (format t "x = ~A (reduced cost: ~A)~%" x (reduced-cost x))
  (format t "y = ~A (reduced cost: ~A)~%" y (reduced-cost y))
  (format t "z = ~A (reduced cost: ~A)~%" z (reduced-cost z)))

;; ==>
;; Objective value solution: 57/2
;; x = 1/2 (reduced cost: 0)
;; y = 7 (reduced cost: 0)
;; z = 0 (reduced cost: 1/2)


(with-solved-problem ((max (= w (+ x (* 4 y) (* 3 z))))
                      (<= (+ (* 2 x) y) 8)
                      (<= (+ y z) 7))
  (format t "Objective value solution: ~A~%" w)
  (format t "x = ~A (reduced cost: ~A)~%" x (reduced-cost x))
  (format t "y = ~A (reduced cost: ~A)~%" y (reduced-cost y))
  (format t "z = ~A (reduced cost: ~A)~%" z (reduced-cost z)))

;; ==>
;; Objective value solution: 57/2
;; x = 1/2 (reduced cost: 0)
;; y = 7 (reduced cost: 0)
;; z = 0 (reduced cost: 1/2)