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 β€ 8
- y + z β€ 7
- x, y, z β₯ 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)