FICO
FICO Xpress Optimization Examples Repository
FICO Optimization Community FICO Xpress Optimization Home
Back to examples browserPrevious exampleNext example

Finding an LP subsystem with as many constraints as possible

Description
Given an infeasible LP, find the feasible subsystem of constraints of maximum cardinality.

Further explanation of this example: 'Xpress Python Reference Manual'

MaxFS_python.zip[download all files]

Source Files
By clicking on a file name, a preview is opened at the bottom of this page.
example_phase1.py[download]





example_phase1.py

# Example: given an infeasible LP, find an (infeasible) solution that minimize
# the total distance from the constraints.
#
# Then solve the obtained MaxFS problem.
#
# (C) Fair Isaac Corp., 1983-2024

import xpress as xp

p = xp.problem()

x = p.addVariable()
y = p.addVariable()

# build a very simple problem with pairs of incompatible constraints

lhs1 = 2*x + 3*y
lhs2 = 3*x + 2*y
lhs3 = 4*x + 5*y

p.addConstraint(lhs1 >= 6, lhs1 <= 5)
p.addConstraint(lhs2 >= 5, lhs2 <= 4)
p.addConstraint(lhs3 >= 8, lhs3 <= 7)

p.optimize()

assert(p.attributes.solstatus == xp.SolStatus.INFEASIBLE)

# We verified the problem is infeasible. Add one binary for each
# constraint to selectively relax them.

m = p.attributes.rows

# get the signs of all constraints: 'E', 'L', or 'G'. Note that this example
# only works with inequality constraints only
sign = []
p.getrowtype(sign, 0, m - 1)

# big-M, large-enough constant to relax all constraints (quite conservative
# here)
M = 1e3

matval = [M]*m
for i in range(m):
    if sign[i] == 'L':
        matval[i] = -M

# Add m new binary columns

p.addcols([1]*m,  # obj. coefficients (as many 1s as there are constraints)
          range(m + 1),  # cumulative number of terms in each column:
                         # 0,1,2,...,m as there is one term per column
          range(m), matval,  # pairs (row_index, coefficient) for each column
          [0]*m, [1]*m,  # lower, upper bound (binary variables, so {0,1})
          ['b_{}'.format(i) for i in range(m)],  # names are b_i, with i is the
                                                 # constraint index
          ['B']*m)                               # type: binary

p.optimize()

# Print constraints constituting a Maximum Feasible Subsystem

b = p.getSolution(range(p.attributes.cols - m, p.attributes.cols))

maxfs = [i for i in range(m) if b[i] > 0.5]

print('MaxFS has ', len(maxfs), 'constraints:', maxfs)

Back to examples browserPrevious exampleNext example