CVXOPT Backend#
AUTHORS:
Ingolfur Edvardsson (2014-05) : initial implementation
- class sage.numerical.backends.cvxopt_backend.CVXOPTBackend#
Bases:
GenericBackendMIP Backend that uses the CVXOPT solver.
There is no support for integer variables.
EXAMPLES:
sage: p = MixedIntegerLinearProgram(solver="CVXOPT") # optional - cvxopt
- add_col(indices, coeffs)#
Add a column.
INPUT:
indices(list of integers) – this list contains the indices of the constraints in which the variable’s coefficient is nonzerocoeffs(list of real values) – associates a coefficient to the variable in each of the constraints in which it appears. Namely, the ith entry ofcoeffscorresponds to the coefficient of the variable in the constraint represented by the ith entry inindices.
Note
indicesandcoeffsare expected to be of the same length.EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.ncols() # optional - cvxopt 0 sage: p.nrows() # optional - cvxopt 0 sage: p.add_linear_constraints(5, 0, None) # optional - cvxopt sage: p.add_col(range(5), range(5)) # optional - cvxopt sage: p.nrows() # optional - cvxopt 5
- add_linear_constraint(coefficients, lower_bound, upper_bound, name=None)#
Add a linear constraint.
INPUT:
coefficientsan iterable with(c,v)pairs wherecis a variable index (integer) andvis a value (real value).lower_bound- a lower bound, either a real value orNoneupper_bound- an upper bound, either a real value orNonename- an optional name for this row (default:None)
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.add_variables(5) # optional - cvxopt 4 sage: p.add_linear_constraint(zip(range(5), range(5)), 2.0, 2.0) # optional - cvxopt sage: p.row(0) # optional - cvxopt ([1, 2, 3, 4], [1, 2, 3, 4]) sage: p.row_bounds(0) # optional - cvxopt (2.00000000000000, 2.00000000000000) sage: p.add_linear_constraint(zip(range(5), range(5)), 1.0, 1.0, name='foo') # optional - cvxopt sage: p.row_name(-1) # optional - cvxopt 'foo'
- add_variable(lower_bound=0.0, upper_bound=None, binary=False, continuous=True, integer=False, obj=None, name=None)#
Add a variable.
This amounts to adding a new column to the matrix. By default, the variable is both positive and real. Variable types are always continuous, and thus the parameters
binary,integer, andcontinuoushave no effect.INPUT:
lower_bound- the lower bound of the variable (default: 0)upper_bound- the upper bound of the variable (default:None)binary-Trueif the variable is binary (default:False).continuous-Trueif the variable is continuous (default:True).integer-Trueif the variable is integer (default:False).obj- (optional) coefficient of this variable in the objective function (default: 0.0)name- an optional name for the newly added variable (default:None).
OUTPUT: The index of the newly created variable
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.ncols() # optional - cvxopt 0 sage: p.add_variable() # optional - cvxopt 0 sage: p.ncols() # optional - cvxopt 1 sage: p.add_variable() # optional - cvxopt 1 sage: p.add_variable(lower_bound=-2.0) # optional - cvxopt 2 sage: p.add_variable(continuous=True) # optional - cvxopt 3 sage: p.add_variable(name='x',obj=1.0) # optional - cvxopt 4 sage: p.col_name(3) # optional - cvxopt 'x_3' sage: p.col_name(4) # optional - cvxopt 'x' sage: p.objective_coefficient(4) # optional - cvxopt 1.00000000000000
- col_bounds(index)#
Return the bounds of a specific variable.
INPUT:
index(integer) – the variable’s id.
OUTPUT:
A pair
(lower_bound, upper_bound). Each of them can be set toNoneif the variable is not bounded in the corresponding direction, and is a real value otherwise.EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.add_variable() # optional - cvxopt 0 sage: p.col_bounds(0) # optional - cvxopt (0.0, None) sage: p.variable_upper_bound(0, 5) # optional - cvxopt sage: p.col_bounds(0) # optional - cvxopt (0.0, 5)
- col_name(index)#
Return the
indexth col nameINPUT:
index(integer) – the col’s idname(char *) – its name. When set toNULL(default), the method returns the current name.
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.add_variable(name="I am a variable") # optional - cvxopt 0 sage: p.col_name(0) # optional - cvxopt 'I am a variable'
- get_objective_value()#
Return the value of the objective function.
Note
Behaviour is undefined unless
solvehas been called before.EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "cvxopt") # optional - cvxopt sage: p.add_variables(2) # optional - cvxopt 1 sage: p.add_linear_constraint([(0,1), (1,2)], None, 3) # optional - cvxopt sage: p.set_objective([2, 5]) # optional - cvxopt sage: p.solve() # optional - cvxopt 0 sage: N(p.get_objective_value(),4) # optional - cvxopt 7.5 sage: N(p.get_variable_value(0),4) # optional - cvxopt 3.6e-7 sage: N(p.get_variable_value(1),4) # optional - cvxopt 1.5
- get_variable_value(variable)#
Return the value of a variable given by the solver.
Note
Behaviour is undefined unless
solvehas been called before.EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.add_variables(2) # optional - cvxopt 1 sage: p.add_linear_constraint([(0,1), (1, 2)], None, 3) # optional - cvxopt sage: p.set_objective([2, 5]) # optional - cvxopt sage: p.solve() # optional - cvxopt 0 sage: N(p.get_objective_value(),4) # optional - cvxopt 7.5 sage: N(p.get_variable_value(0),4) # optional - cvxopt 3.6e-7 sage: N(p.get_variable_value(1),4) # optional - cvxopt 1.5
- is_maximization()#
Test whether the problem is a maximization
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.is_maximization() # optional - cvxopt True sage: p.set_sense(-1) # optional - cvxopt sage: p.is_maximization() # optional - cvxopt False
- is_variable_binary(index)#
Test whether the given variable is of binary type. CVXOPT does not allow integer variables, so this is a bit moot.
INPUT:
index(integer) – the variable’s id
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.ncols() # optional - cvxopt 0 sage: p.add_variable() # optional - cvxopt 0 sage: p.set_variable_type(0,0) # optional - cvxopt Traceback (most recent call last): ... ValueError: ... sage: p.is_variable_binary(0) # optional - cvxopt False
- is_variable_continuous(index)#
Test whether the given variable is of continuous/real type. CVXOPT does not allow integer variables, so this is a bit moot.
INPUT:
index(integer) – the variable’s id
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.ncols() # optional - cvxopt 0 sage: p.add_variable() # optional - cvxopt 0 sage: p.is_variable_continuous(0) # optional - cvxopt True sage: p.set_variable_type(0,1) # optional - cvxopt Traceback (most recent call last): ... ValueError: ... sage: p.is_variable_continuous(0) # optional - cvxopt True
- is_variable_integer(index)#
Test whether the given variable is of integer type. CVXOPT does not allow integer variables, so this is a bit moot.
INPUT:
index(integer) – the variable’s id
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.ncols() # optional - cvxopt 0 sage: p.add_variable() # optional - cvxopt 0 sage: p.set_variable_type(0,-1) # optional - cvxopt sage: p.set_variable_type(0,1) # optional - cvxopt Traceback (most recent call last): ... ValueError: ... sage: p.is_variable_integer(0) # optional - cvxopt False
- ncols()#
Return the number of columns/variables.
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.ncols() # optional - cvxopt 0 sage: p.add_variables(2) # optional - cvxopt 1 sage: p.ncols() # optional - cvxopt 2
- nrows()#
Return the number of rows/constraints.
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.nrows() # optional - cvxopt 0 sage: p.add_variables(5) # optional - cvxopt 4 sage: p.add_linear_constraints(2, 2.0, None) # optional - cvxopt sage: p.nrows() # optional - cvxopt 2
- objective_coefficient(variable, coeff=None)#
Set or get the coefficient of a variable in the objective function
INPUT:
variable(integer) – the variable’s idcoeff(double) – its coefficient
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.add_variable() # optional - cvxopt 0 sage: p.objective_coefficient(0) # optional - cvxopt 0.0 sage: p.objective_coefficient(0,2) # optional - cvxopt sage: p.objective_coefficient(0) # optional - cvxopt 2.0
- problem_name(name=None)#
Return or define the problem’s name
INPUT:
name(str) – the problem’s name. When set toNone(default), the method returns the problem’s name.
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.problem_name() # optional - cvxopt '' sage: p.problem_name("There once was a french fry") # optional - cvxopt sage: print(p.problem_name()) # optional - cvxopt There once was a french fry
- row(i)#
Return a row
INPUT:
index(integer) – the constraint’s id.
OUTPUT:
A pair
(indices, coeffs)whereindiceslists the entries whose coefficient is nonzero, and to whichcoeffsassociates their coefficient on the model of theadd_linear_constraintmethod.EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.add_variables(5) # optional - cvxopt 4 sage: p.add_linear_constraint(list(zip(range(5), range(5))), 2, 2) # optional - cvxopt sage: p.row(0) # optional - cvxopt ([1, 2, 3, 4], [1, 2, 3, 4]) sage: p.row_bounds(0) # optional - cvxopt (2, 2)
- row_bounds(index)#
Return the bounds of a specific constraint.
INPUT:
index(integer) – the constraint’s id.
OUTPUT:
A pair
(lower_bound, upper_bound). Each of them can be set toNoneif the constraint is not bounded in the corresponding direction, and is a real value otherwise.EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.add_variables(5) # optional - cvxopt 4 sage: p.add_linear_constraint(list(zip(range(5), range(5))), 2, 2) # optional - cvxopt sage: p.row(0) # optional - cvxopt ([1, 2, 3, 4], [1, 2, 3, 4]) sage: p.row_bounds(0) # optional - cvxopt (2, 2)
- row_name(index)#
Return the
indexth row nameINPUT:
index(integer) – the row’s id
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.add_linear_constraints(1, 2, None, names=["Empty constraint 1"]) # optional - cvxopt sage: p.row_name(0) # optional - cvxopt 'Empty constraint 1'
- set_objective(coeff, d=0.0)#
Set the objective function.
INPUT:
coeff– a list of real values, whose ith element is the coefficient of the ith variable in the objective function.d(double) – the constant term in the linear function (set to \(0\) by default)
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver # optional - cvxopt sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.add_variables(5) # optional - cvxopt 4 sage: p.set_objective([1, 1, 2, 1, 3]) # optional - cvxopt sage: [p.objective_coefficient(x) for x in range(5)] # optional - cvxopt [1, 1, 2, 1, 3]
- set_sense(sense)#
Set the direction (maximization/minimization).
INPUT:
sense(integer) :+1 => Maximization
-1 => Minimization
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.is_maximization() # optional - cvxopt True sage: p.set_sense(-1) # optional - cvxopt sage: p.is_maximization() # optional - cvxopt False
- set_variable_type(variable, vtype)#
Set the type of a variable.
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "cvxopt") # optional - cvxopt sage: p.add_variables(5) # optional - cvxopt 4 sage: p.set_variable_type(3, -1) # optional - cvxopt sage: p.set_variable_type(3, -2) # optional - cvxopt Traceback (most recent call last): ... ValueError: ...
- set_verbosity(level)#
Does not apply for the cvxopt solver
- solve()#
Solve the problem.
Note
This method raises
MIPSolverExceptionexceptions when the solution cannot be computed for any reason (none exists, or the LP solver was not able to find it, etc…)EXAMPLES:
sage: p = MixedIntegerLinearProgram(solver = "cvxopt", maximization=False) # optional - cvxopt sage: x=p.new_variable(nonnegative=True) # optional - cvxopt sage: p.set_objective(-4*x[0] - 5*x[1]) # optional - cvxopt sage: p.add_constraint(2*x[0] + x[1] <= 3) # optional - cvxopt sage: p.add_constraint(2*x[1] + x[0] <= 3) # optional - cvxopt sage: N(p.solve(), digits=2) # optional - cvxopt -9.0 sage: p = MixedIntegerLinearProgram(solver = "cvxopt", maximization=False) # optional - cvxopt sage: x=p.new_variable(nonnegative=True) # optional - cvxopt sage: p.set_objective(x[0] + 2*x[1]) # optional - cvxopt sage: p.add_constraint(-5*x[0] + x[1] <= 7) # optional - cvxopt sage: p.add_constraint(-5*x[0] + x[1] >= 7) # optional - cvxopt sage: p.add_constraint(x[0] + x[1] >= 26 ) # optional - cvxopt sage: p.add_constraint( x[0] >= 3) # optional - cvxopt sage: p.add_constraint( x[1] >= 4) # optional - cvxopt sage: N(p.solve(),digits=4) # optional - cvxopt 48.83 sage: p = MixedIntegerLinearProgram(solver = "cvxopt") # optional - cvxopt sage: x=p.new_variable(nonnegative=True) # optional - cvxopt sage: p.set_objective(x[0] + x[1] + 3*x[2]) # optional - cvxopt sage: p.solver_parameter("show_progress",True) # optional - cvxopt sage: p.add_constraint(x[0] + 2*x[1] <= 4) # optional - cvxopt sage: p.add_constraint(5*x[2] - x[1] <= 8) # optional - cvxopt sage: N(p.solve(), digits=2) # optional - cvxopt pcost dcost gap pres dres k/t ... 8.8 sage: #CVXOPT gives different values for variables compared to the other solvers. sage: c = MixedIntegerLinearProgram(solver = "cvxopt") # optional - cvxopt sage: p = MixedIntegerLinearProgram(solver = "ppl") # optional - cvxopt sage: g = MixedIntegerLinearProgram() # optional - cvxopt sage: xc=c.new_variable(nonnegative=True) # optional - cvxopt sage: xp=p.new_variable(nonnegative=True) # optional - cvxopt sage: xg=g.new_variable(nonnegative=True) # optional - cvxopt sage: c.set_objective(xc[2]) # optional - cvxopt sage: p.set_objective(xp[2]) # optional - cvxopt sage: g.set_objective(xg[2]) # optional - cvxopt sage: #we create a cube for all three solvers sage: c.add_constraint(xc[0] <= 100) # optional - cvxopt sage: c.add_constraint(xc[1] <= 100) # optional - cvxopt sage: c.add_constraint(xc[2] <= 100) # optional - cvxopt sage: p.add_constraint(xp[0] <= 100) # optional - cvxopt sage: p.add_constraint(xp[1] <= 100) # optional - cvxopt sage: p.add_constraint(xp[2] <= 100) # optional - cvxopt sage: g.add_constraint(xg[0] <= 100) # optional - cvxopt sage: g.add_constraint(xg[1] <= 100) # optional - cvxopt sage: g.add_constraint(xg[2] <= 100) # optional - cvxopt sage: N(c.solve(),digits=4) # optional - cvxopt 100.0 sage: N(c.get_values(xc[0]),digits=3) # optional - cvxopt 50.0 sage: N(c.get_values(xc[1]),digits=3) # optional - cvxopt 50.0 sage: N(c.get_values(xc[2]),digits=4) # optional - cvxopt 100.0 sage: N(p.solve(),digits=4) # optional - cvxopt 100.0 sage: N(p.get_values(xp[0]),2) # optional - cvxopt 0.00 sage: N(p.get_values(xp[1]),2) # optional - cvxopt 0.00 sage: N(p.get_values(xp[2]),digits=4) # optional - cvxopt 100.0 sage: N(g.solve(),digits=4) # optional - cvxopt 100.0 sage: N(g.get_values(xg[0]),2) # optional - cvxopt 0.00 sage: N(g.get_values(xg[1]),2) # optional - cvxopt 0.00 sage: N(g.get_values(xg[2]),digits=4) # optional - cvxopt 100.0
- solver_parameter(name, value=None)#
Return or define a solver parameter
INPUT:
name(string) – the parametervalue– the parameter’s value if it is to be defined, orNone(default) to obtain its current value.
Note
The list of available parameters is available at
solver_parameter().EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver="CVXOPT") # optional - cvxopt sage: p.solver_parameter("show_progress") # optional - cvxopt False sage: p.solver_parameter("show_progress", True) # optional - cvxopt sage: p.solver_parameter("show_progress") # optional - cvxopt True
- variable_lower_bound(index, value=False)#
Return or define the lower bound on a variable
INPUT:
index(integer) – the variable’s idvalue– real value, orNoneto mean that the variable has not lower bound. When set toFalse(default), the method returns the current value.
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.add_variable() # optional - cvxopt 0 sage: p.col_bounds(0) # optional - cvxopt (0.0, None) sage: p.variable_lower_bound(0, 5) # optional - cvxopt sage: p.col_bounds(0) # optional - cvxopt (5, None)
- variable_upper_bound(index, value=False)#
Return or define the upper bound on a variable
INPUT:
index(integer) – the variable’s idvalue– real value, orNoneto mean that the variable has not upper bound. When set toFalse(default), the method returns the current value.
EXAMPLES:
sage: from sage.numerical.backends.generic_backend import get_solver sage: p = get_solver(solver = "CVXOPT") # optional - cvxopt sage: p.add_variable() # optional - cvxopt 0 sage: p.col_bounds(0) # optional - cvxopt (0.0, None) sage: p.variable_upper_bound(0, 5) # optional - cvxopt sage: p.col_bounds(0) # optional - cvxopt (0.0, 5)