Solving a mixed system of equality and inequality - r

Question

Intro: I sucessfully use the rSymPy library to symbolically solve following example system of equalities.
x + y = 20; x + 2y = 10
library(rSymPy)
sympy("var('x')")
sympy("var('y')")
sympy("solve([
Eq(x+y, 20),
Eq(x+2*y, 10)
],
[x,y])")
# output
#[1] "{x: 30, y: -10}"
Use case: In my use case I want to symbolically solve a system of a mixed system of equality and inequality. Here's a reproduceable example:
x + y = 20; x + 2y > 10
The inequality can be sucessfully coded in rSymPy with Gt:
sympy("Gt(x+2*y, 10)")
# output
# [1] "10 < x + 2*y"
Problem: The code of the mixed system throughs an error:
sympy("solve([
Eq(x + y, 20),
Gt(x+2*y, 10)
],
[x,y])")
# output
# Error in .jcall("RJavaTools", "Ljava/lang/Object;", "invokeMethod", cl, :
# Traceback (most recent call last):
# File "<string>", line 1, in <module>
# File "/Users/.../R/3.0/library/rSymPy/Lib/sympy/solvers/solvers.py", line 308, in solve
# raise NotImplementedError()
# NotImplementedError
Question: How can I refactor the code successfully to solve the mixed system?

Answer 1

1) Define a positive variable z. Then the system can be recast as a system of equalities in terms of z:
x <- Var('x')
y <- Var('y')
z <- Var('z')
sympy("solve( [ Eq(x+y, 20), Eq(x + 2*y - z, 10) ], [x, y] )")
giving:
[1] "{x: 30 - z, y: -10 + z}"
2) This is a linear programming problem so if you are just looking for any feasible solution to the constraints then the lpSolve package can provide such. In this case it gives the solution corresponding to z=10 in (1):
library(lpSolve)
out <- lp(, c(0, 0), matrix(c(1, 1, 1, 2), 2), c("=", ">"), c(20, 10))
out$solution
## [1] 20 0
ADDED first solution in response to comment from poster. Added some further discussion.

Answer 2

It looks like NotImplmentedError is coming form SymPy itself. It looks like it can't solve multivariate inequalities. It can only reduce them (which is what your example did). It doesn't appear that the library supports that type of system.
def _solve_inequality(ie, s, assume=True):
""" A hacky replacement for solve, since the latter only works for
univariate inequalities. """
if not ie.rel_op in ('>', '>=', '<', '<='):
raise NotImplementedError
expr = ie.lhs - ie.rhs
try:
p = Poly(expr, s)
if p.degree() != 1:
raise NotImplementedError
except (PolynomialError, NotImplementedError):
try:
n, d = expr.as_numer_denom()
return reduce_rational_inequalities([[ie]], s, assume=assume)
except PolynomialError:
return solve_univariate_inequality(ie, s, assume=assume)
a, b = p.all_coeffs()
if a.is_positive:
return ie.func(s, -b/a)
elif a.is_negative:
return ie.func(-b/a, s)
else:
raise NotImplementedError

Answer 3

In the meantime I discovered the excellent LIM package which nicely allows symbolical solution to various kinds of linear inverse problems:
The ASCII file linprog.lim contains the symbolical problem in a human readable form (it's located in the same directory as the R code):
## UNKNOWNS
X
Y
## END UNKNOWNS
## EQUALITIES
X + Y = 20
## END EQUALITIES
## INEQUALITIES
X + 2 * Y > 10
## END INEQUALITIES
## PROFIT
X + Y
## END PROFIT
Following R code provides the solution to the linear programming problem:
require(LIM)
model= Setup("linprog.lim")
model.solved= Linp(model, ispos=F, verbose=T)
model.solved
Output:
$residualNorm
[1] 3.552714e-15
$solutionNorm
[1] 20
$X
X Y
[1,] 20 0

Answer 4

Here's another approach using the Rglpk library which can read in and solve GNU's MathProg scripted problems in R:
# in case CRAN install does not work
install.packages("Rglpk", repos="http://cran.us.r-project.org")
library(Rglpk)
## read file
x= Rglpk_read_file("mathprog1.mod", type = "MathProg", verbose=T)
## optimize
Rglpk_solve_LP(obj= x$objective,
mat= x$constraints[[1]],
dir= x$constraints[[2]],
rhs= x$constraints[[3]],
bounds= x$bounds,
types= x$types,
max= x$maximum)
File mathprog1.mod:
# Define Variables
var x;
var y;
# Define Constraints
s.t. A: x + y = 20;
s.t. B: x + 2*y >= 10;
# Define Objective
maximize z: x + y;
# Solve
solve;
end;
R console output:
# $optimum
# [1] 20
# $solution
# [1] 0 20
# $status
# [1] 0