picos.modeling.problem¶
Implementation of Problem
.
Exceptions
- exception picos.modeling.problem.SolutionFailure(code, message)[source]¶
Bases:
RuntimeError
Solving the problem failed.
- __init__(code, message)[source]¶
Construct a
SolutionFailure
.- Parameters
code (int) – Status code, as defined in
Problem.solve
.message (str) – Text description of the failure.
- __new__(**kwargs)¶
Classes
- class picos.modeling.problem.Problem(copyOptions=None, useOptions=None, **extra_options)[source]¶
Bases:
object
PICOS’ representation of an optimization problem.
- Example
>>> from picos import Problem, RealVariable >>> X = RealVariable("X", (2,2), lower = 0) >>> P = Problem() >>> P.set_objective("max", X.tr) >>> C1 = P.add_constraint(X.sum <= 10) >>> C2 = P.add_constraint(X[0,0] == 1) >>> print(P) Linear Program maximize tr(X) over 2×2 real variable X (bounded below) subject to ∑(X) ≤ 10 X[0,0] = 1 >>> # PICOS will select a suitable solver if you don't specify one. >>> solution = P.solve(solver = "cvxopt") >>> solution.claimedStatus 'optimal' >>> solution.searchTime 0.002137422561645508 >>> round(P, 1) 10.0 >>> print(X) [ 1.00e+00 4.89e-10] [ 4.89e-10 9.00e+00] >>> round(C1.dual, 1) 1.0
- __init__(copyOptions=None, useOptions=None, **extra_options)[source]¶
Create an empty problem and optionally set initial solver options.
- Parameters
copyOptions – An
Options
object to copy instead of using the default options.useOptions – An
Options
object to use (without making a copy) instead of using the default options.extra_options – A sequence of additional solver options to apply on top of the default options or those given by
copyOptions
oruseOptions
.
- add_constraint(constraint, key=None)[source]¶
Add a single constraint to the problem and return it.
- Parameters
constraint (
Constraint
) – The constraint to be added.key – DEPRECATED
- Returns
The constraint that was added to the problem.
Note
This method is superseded by the more compact and more flexible
require
method or, at your preference, the+=
operator.
- add_list_of_constraints(lst, it=None, indices=None, key=None)[source]¶
Add constraints from an iterable to the problem.
- Parameters
lst – Iterable of constraints to add.
it – DEPRECATED
indices – DEPRECATED
key – DEPRECATED
- Returns
A list of all constraints that were added.
- Example
>>> import picos as pic >>> import cvxopt as cvx >>> from pprint import pprint >>> prob=pic.Problem() >>> x=[prob.add_variable('x[{0}]'.format(i),2) for i in range(5)] >>> pprint(x) [<2×1 Real Variable: x[0]>, <2×1 Real Variable: x[1]>, <2×1 Real Variable: x[2]>, <2×1 Real Variable: x[3]>, <2×1 Real Variable: x[4]>] >>> y=prob.add_variable('y',5) >>> IJ=[(1,2),(2,0),(4,2)] >>> w={} >>> for ij in IJ: ... w[ij]=prob.add_variable('w[{},{}]'.format(*ij),3) ... >>> u=pic.new_param('u',cvx.matrix([2,5])) >>> C1=prob.add_list_of_constraints([u.T*x[i] < y[i] for i in range(5)]) >>> C2=prob.add_list_of_constraints([abs(w[i,j])<y[j] for (i,j) in IJ]) >>> C3=prob.add_list_of_constraints([y[t] > y[t+1] for t in range(4)]) >>> print(prob) Feasibility Problem find an assignment for 2×1 real variable x[i] ∀ i ∈ [0…4] 3×1 real variable w[i,j] ∀ (i,j) ∈ zip([1,2,4],[2,0,2]) 5×1 real variable y subject to uᵀ·x[i] ≤ y[i] ∀ i ∈ [0…4] ‖w[i,j]‖ ≤ y[j] ∀ (i,j) ∈ zip([1,2,4],[2,0,2]) y[i] ≥ y[i+1] ∀ i ∈ [0…3]
Note
This method is superseded by the more compact and more flexible
require
method or, at your preference, the+=
operator.
- add_variable(name, size=1, vtype='continuous', lower=None, upper=None)[source]¶
Legacy method to create a PICOS variable.
- Parameters
name (str) – The name of the variable.
size (anything recognized by
load_shape
) – The shape of the variable.vtype (str) –
Domain of the variable. Can be any of
'continuous'
– real valued,'binary'
– either zero or one,'integer'
– integer valued,'symmetric'
– symmetric matrix,'antisym'
or'skewsym'
– skew-symmetric matrix,'complex'
– complex matrix,'hermitian'
– complex hermitian matrix.
lower (anything recognized by
load_data
) – A lower bound on the variable.upper (anything recognized by
load_data
) – An upper bound on the variable.
- Returns
A
BaseVariable
instance.- Example
>>> from picos import Problem >>> P = Problem() >>> x = P.add_variable("x", 3) >>> x <3×1 Real Variable: x> >>> # Variable are not stored inside the problem any more: >>> P.variables mappingproxy(OrderedDict()) >>> # They are only part of the problem if they actually appear: >>> P.set_objective("min", abs(x)**2) >>> P.variables mappingproxy(OrderedDict([('x', <3×1 Real Variable: x>)]))
Deprecated since version 2.0: Variables can now be created independent of problems, and do not need to be added to any problem explicitly.
- as_dual()[source]¶
Return the Lagrangian dual problem of the standardized problem.
Deprecated since version 2.0: Use
dual
instead.
- check_current_value_feasibility(tol=1e-05, inttol=None)[source]¶
Check if the problem is feasibly valued.
Checks whether all variables that appear in constraints are valued and satisfy both their bounds and the constraints up to the given tolerance.
- Parameters
tol (float) – Largest tolerated absolute violation of a constraint or variable bound. If
None
, then theabs_prim_fsb_tol
solver option is used.inttol – DEPRECATED
- Returns
A tuple
(feasible, violation)
wherefeasible
is a bool stating whether the solution is feasible andviolation
is eitherNone
, iffeasible == True
, or the amount of violation, otherwise.- Raises
picos.uncertain.IntractableWorstCase – When computing the worst-case (expected) value of the constrained expression is not supported.
- clone(copyOptions=True)[source]¶
Create a semi-deep copy of the problem.
The copy is constrained by the same constraint objects and has the same objective function and thereby references the existing variables and parameters that appear in these objects.
The clone can be modified to describe a new problem but when its variables and parameters are valued, in particular when a solution is applied to the new problem, then the same values are found in the corresponding variables and parameters of the old problem. If this is not a problem to you, then cloning can be much faster than copying.
- Parameters
copyOptions (bool) – Whether to make an independent copy of the problem’s options. Disabling this will apply any option changes to the original problem as well but yields a (very small) reduction in cloning time.
- continuous_relaxation(copy_other_mutables=True)[source]¶
Return a continuous relaxation of the problem.
This is done by replacing integer variables with continuous ones.
- get_constraint(idOrIndOrCon)[source]¶
Return a (list of) constraint(s) of the problem.
- Parameters
idOrIndOrCon (picos.constraints.Constraint or int or tuple or list) –
One of the following:
A constraint object. It will be returned when the constraint is part of the problem, otherwise a KeyError is raised.
The integer ID of the constraint.
The integer offset of the constraint in the list of all constraints that are part of the problem, in the order that they were added.
A list or tuple of length 1. Its only element is the index of a constraint group (of constraints that were added together), where groups are indexed in the order that they were added to the problem. The whole group is returned as a list of constraints. That list has the constraints in the order that they were added.
A list or tuple of length 2. The first element is a constraint group offset as above, the second an offset within that list.
- Returns
A
constraint
or a list thereof.- Example
>>> import picos as pic >>> import cvxopt as cvx >>> from pprint import pprint >>> prob=pic.Problem() >>> x=[prob.add_variable('x[{0}]'.format(i),2) for i in range(5)] >>> y=prob.add_variable('y',5) >>> Cx=prob.add_list_of_constraints([(1|x[i]) < y[i] for i in range(5)]) >>> Cy=prob.add_constraint(y>0) >>> print(prob) Linear Feasibility Problem find an assignment for 2×1 real variable x[i] ∀ i ∈ [0…4] 5×1 real variable y subject to ∑(x[i]) ≤ y[i] ∀ i ∈ [0…4] y ≥ 0 >>> # Retrieve the second constraint, indexed from zero: >>> prob.get_constraint(1) <1×1 Affine Constraint: ∑(x[1]) ≤ y[1]> >>> # Retrieve the fourth consraint from the first group: >>> prob.get_constraint((0,3)) <1×1 Affine Constraint: ∑(x[3]) ≤ y[3]> >>> # Retrieve the whole first group of constraints: >>> pprint(prob.get_constraint((0,))) [<1×1 Affine Constraint: ∑(x[0]) ≤ y[0]>, <1×1 Affine Constraint: ∑(x[1]) ≤ y[1]>, <1×1 Affine Constraint: ∑(x[2]) ≤ y[2]>, <1×1 Affine Constraint: ∑(x[3]) ≤ y[3]>, <1×1 Affine Constraint: ∑(x[4]) ≤ y[4]>] >>> # Retrieve the second "group", containing just one constraint: >>> prob.get_constraint((1,)) [<5×1 Affine Constraint: y ≥ 0>]
- get_valued_variable(name)[source]¶
Retrieve values of variables referenced by the problem.
This method works the same
get_variable
but it returns the variable’svalues
instead of the variable objects.- Raises
NotValued – If any of the selected variables is not valued.
- get_variable(name)[source]¶
Retrieve variables referenced by the problem.
Retrieves either a single variable with the given name or a group of variables all named
name[param]
with different values forparam
. If the values forparam
are the integers from zero to the size of the group minus one, then the group is returned as alist
ordered byparam
, otherwise it is returned as adict
with the values ofparam
as keys.Note
Since PICOS 2.0, variables are independent of problems and only appear in a problem for as long as they are referenced by the problem’s objective function or constraints.
- Parameters
name (str) – The name of a variable, or the base name of a group of variables.
- Returns
- Example
>>> from picos import Problem, RealVariable >>> from pprint import pprint >>> # Create a number of variables with structured names. >>> vars = [RealVariable("x")] >>> for i in range(4): ... vars.append(RealVariable("y[{}]".format(i))) >>> for key in ["alice", "bob", "carol"]: ... vars.append(RealVariable("z[{}]".format(key))) >>> # Make the variables appear in a problem. >>> P = Problem() >>> P.set_objective("min", sum([var for var in vars])) >>> print(P) Linear Program minimize x + y[0] + y[1] + y[2] + y[3] + z[alice] + z[bob] + z[carol] over 1×1 real variable x, y[0], y[1], y[2], y[3], z[alice], z[bob], z[carol] >>> # Retrieve the variables from the problem. >>> P.get_variable("x") <1×1 Real Variable: x> >>> pprint(P.get_variable("y")) [<1×1 Real Variable: y[0]>, <1×1 Real Variable: y[1]>, <1×1 Real Variable: y[2]>, <1×1 Real Variable: y[3]>] >>> pprint(P.get_variable("z")) {'alice': <1×1 Real Variable: z[alice]>, 'bob': <1×1 Real Variable: z[bob]>, 'carol': <1×1 Real Variable: z[carol]>} >>> P.get_variable("z")["alice"] is P.get_variable("z[alice]") True
- is_continuous()[source]¶
Whether all variables are of continuous types.
Deprecated since version 2.0: Use
continuous
instead.
- is_pure_integer()[source]¶
Whether all variables are of integral types.
Deprecated since version 2.0: Use
pure_integer
instead.
- obj_value()[source]¶
Objective function value.
- Raises
AttributeError – If the problem is a feasibility problem or if the objective function is not valued. This is legacy behavior. Note that
value
just returnsNone
while functions that do raise an exception to denote an unvalued expression would raiseNotValued
instead.
Deprecated since version 2.0: Use
value
instead.
- prepared(steps=None, **extra_options)[source]¶
Perform a dry-run returning the reformulated (prepared) problem.
This behaves like
solve
in that it takes a number of additional temporary options, finds a solution strategy matching the problem and options, and performs the strategy’s reformulations in turn to obtain modified problems. However, it stops after the given number of steps and never hands the reformulated problem to a solver. Instead of a solution,prepared
then returns the last reformulated problem.Unless this method returns the problem itself, the special attributes
prepared_strategy
andprepared_steps
are added to the returned problem. They then contain the (partially) executed solution strategy and the number of performed reformulations, respectively.- Parameters
steps (int) – Number of reformulations to perform.
None
means as many as there are. If this parameter is , then the problem itself is returned. If it is , then only the implicit first reformulationExtraOptions
is executed, which may also output the problem itself, depending onextra_options
.extra_options – Additional solver options to use with this dry-run only.
- Returns
The reformulated problem, with
extra_options
set unless they were “consumed” by a reformulation (e.g. option_dualize).- Raises
NoStrategyFound – If no solution strategy was found.
ValueError – If there are not as many reformulation steps as requested.
- Example
>>> from picos import Problem, RealVariable >>> x = RealVariable("x", 2) >>> P = Problem() >>> P.set_objective("min", abs(x)**2) >>> Q = P.prepared(solver = "cvxopt") >>> print(Q.prepared_strategy) # Show prepared reformulation steps. 1. ExtraOptions 2. EpigraphReformulation 3. SquaredNormToConicReformulation 4. CVXOPTSolver >>> Q.prepared_steps # Check how many steps have been performed. 3 >>> print(P) Quadratic Program minimize ‖x‖² over 2×1 real variable x >>> print(Q) Second Order Cone Program minimize __..._t over 1×1 real variable __..._t 2×1 real variable x subject to ‖fullroot(‖x‖²)‖² ≤ __..._t ∧ __..._t ≥ 0
- reformulated(specification, **extra_options)[source]¶
Return the problem reformulated to match a specification.
Internally this creates a dummy solver accepting problems of the desired form and then calls
prepared
with the dummy solver passed via option_ad_hoc_solver. See meth:prepared for more details.- Parameters
specification (Specification) – A problem class that the resulting problem must be a member of.
extra_options – Additional solver options to use with this reformulation only.
- Returns
The reformulated problem, with
extra_options
set unless they were “consumed” by a reformulation (e.g. dualize).- Raises
NoStrategyFound – If no reformulation strategy was found.
- Example
>>> from picos import Problem, RealVariable >>> from picos.modeling import Specification >>> from picos.expressions import AffineExpression >>> from picos.constraints import ( ... AffineConstraint, SOCConstraint, RSOCConstraint) >>> # Define the class/specification of second order conic problems: >>> S = Specification(objectives=[AffineExpression], ... constraints=[AffineConstraint, SOCConstraint, RSOCConstraint]) >>> # Define a quadratic program and reformulate it: >>> x = RealVariable("x", 2) >>> P = Problem() >>> P.set_objective("min", abs(x)**2) >>> Q = P.reformulated(S) >>> print(P) Quadratic Program minimize ‖x‖² over 2×1 real variable x >>> print(Q) Second Order Cone Program minimize __..._t over 1×1 real variable __..._t 2×1 real variable x subject to ‖fullroot(‖x‖²)‖² ≤ __..._t ∧ __..._t ≥ 0
Note
This method is intended for educational purposes. You do not need to use it when solving a problem as PICOS will perform the necessary reformulations automatically.
- remove_all_constraints()[source]¶
Remove all constraints from the problem.
Note
This method does not remove bounds set directly on variables.
- remove_constraint(idOrIndOrCon)[source]¶
Delete a constraint from the problem.
- Parameters
idOrIndOrCon – See
get_constraint
.- Example
>>> import picos >>> from pprint import pprint >>> P = picos.Problem() >>> x = [P.add_variable('x[{0}]'.format(i), 2) for i in range(4)] >>> y = P.add_variable('y', 4) >>> Cxy = P.add_list_of_constraints( ... [(1 | x[i]) <= y[i] for i in range(4)]) >>> Cy = P.add_constraint(y >= 0) >>> Cx0to2 = P.add_list_of_constraints([x[i] <= 2 for i in range(3)]) >>> Cx3 = P.add_constraint(x[3] <= 1) >>> pprint(list(P.constraints.values())) [<1×1 Affine Constraint: ∑(x[0]) ≤ y[0]>, <1×1 Affine Constraint: ∑(x[1]) ≤ y[1]>, <1×1 Affine Constraint: ∑(x[2]) ≤ y[2]>, <1×1 Affine Constraint: ∑(x[3]) ≤ y[3]>, <4×1 Affine Constraint: y ≥ 0>, <2×1 Affine Constraint: x[0] ≤ [2]>, <2×1 Affine Constraint: x[1] ≤ [2]>, <2×1 Affine Constraint: x[2] ≤ [2]>, <2×1 Affine Constraint: x[3] ≤ [1]>] >>> # Delete the 2nd constraint (counted from 0): >>> P.remove_constraint(1) >>> pprint(list(P.constraints.values())) [<1×1 Affine Constraint: ∑(x[0]) ≤ y[0]>, <1×1 Affine Constraint: ∑(x[2]) ≤ y[2]>, <1×1 Affine Constraint: ∑(x[3]) ≤ y[3]>, <4×1 Affine Constraint: y ≥ 0>, <2×1 Affine Constraint: x[0] ≤ [2]>, <2×1 Affine Constraint: x[1] ≤ [2]>, <2×1 Affine Constraint: x[2] ≤ [2]>, <2×1 Affine Constraint: x[3] ≤ [1]>] >>> # Delete the 2nd group of constraints, i.e. the constraint y > 0: >>> P.remove_constraint((1,)) >>> pprint(list(P.constraints.values())) [<1×1 Affine Constraint: ∑(x[0]) ≤ y[0]>, <1×1 Affine Constraint: ∑(x[2]) ≤ y[2]>, <1×1 Affine Constraint: ∑(x[3]) ≤ y[3]>, <2×1 Affine Constraint: x[0] ≤ [2]>, <2×1 Affine Constraint: x[1] ≤ [2]>, <2×1 Affine Constraint: x[2] ≤ [2]>, <2×1 Affine Constraint: x[3] ≤ [1]>] >>> # Delete the 3rd remaining group of constraints, i.e. x[3] < [1]: >>> P.remove_constraint((2,)) >>> pprint(list(P.constraints.values())) [<1×1 Affine Constraint: ∑(x[0]) ≤ y[0]>, <1×1 Affine Constraint: ∑(x[2]) ≤ y[2]>, <1×1 Affine Constraint: ∑(x[3]) ≤ y[3]>, <2×1 Affine Constraint: x[0] ≤ [2]>, <2×1 Affine Constraint: x[1] ≤ [2]>, <2×1 Affine Constraint: x[2] ≤ [2]>] >>> # Delete 2nd constraint of the 2nd remaining group, i.e. x[1] < |2|: >>> P.remove_constraint((1,1)) >>> pprint(list(P.constraints.values())) [<1×1 Affine Constraint: ∑(x[0]) ≤ y[0]>, <1×1 Affine Constraint: ∑(x[2]) ≤ y[2]>, <1×1 Affine Constraint: ∑(x[3]) ≤ y[3]>, <2×1 Affine Constraint: x[0] ≤ [2]>, <2×1 Affine Constraint: x[2] ≤ [2]>]
- remove_variable(name)[source]¶
Does nothing.
Deprecated since version 2.0: Whether a problem references a variable is now determined dynamically, so this method has no effect.
- require(*constraints, ret=False)[source]¶
Add constraints to the problem.
- Parameters
constraints – A sequence of constraints or constraint groups (iterables yielding constraints) or a mix thereof.
ret (bool) – Whether to return the added constraints.
- Returns
When
ret=True
, returns either the single constraint that was added, the single group of constraint that was added in the form of alist
or, when multiple arguments are given, a list of constraints or constraint groups represented as above. Whenret=False
, returns nothing.- Example
>>> from picos import Problem, RealVariable >>> x = RealVariable("x", 5) >>> P = Problem() >>> P.require(x >= -1, x <= 1) # Add individual constraints. >>> P.require([x[i] <= x[i+1] for i in range(4)]) # Add groups. >>> print(P) Linear Feasibility Problem find an assignment for 5×1 real variable x subject to x ≥ [-1] x ≤ [1] x[i] ≤ x[i+1] ∀ i ∈ [0…3]
Note
For a single constraint
C
,P.require(C)
may also be written asP += C
. For multiple constraints,P.require([C1, C2])
can be abbreviatedP += [C1, C2]
whileP.require(C1, C2)
can be written as eitherP += (C1, C2)
or justP += C1, C2
.
- reset(resetOptions=False)[source]¶
Reset the problem instance to its initial empty state.
- Parameters
resetOptions (bool) – Whether also solver options should be reset to their default values.
- set_all_options_to_default()[source]¶
Set all solver options to their default value.
Deprecated since version 2.0: Use
Problem.options
instead.
- set_objective(direction=None, expression=None)[source]¶
Set the optimization direction and objective function of the problem.
- Parameters
direction (str) –
Case insensitive search direction string. One of
"min"
or"minimize"
,"max"
or"maximize"
,"find"
orNone
(for a feasibility problem).
expression (Expression) – The objective function. Must be
None
for a feasibility problem.
- set_option(key, val)[source]¶
Set a single solver option to the given value.
- Parameters
key (str) – String name of the option, see below for a list.
val – New value for the option.
Deprecated since version 2.0: Use
Problem.options
instead.
- set_var_value(name, value)[source]¶
Set the
value
of a variable.For a
Problem
P
, this is the same asP.variables[name] = value
.- Parameters
Deprecated since version 2.0: Use
variables
instead.
- solve(**extra_options)[source]¶
Hand the problem to a solver.
You can select the solver manually with the
solver
option. Otherwise a suitable solver will be selected among those that are available on the platform.The default behavior (options
primals=True
,duals=None
) is to raise aSolutionFailure
when the primal solution is not found optimal by the solver, while the dual solution is allowed to be missing or incomplete.When this method succeeds and unless
apply_solution=False
, you can access the solution as follows:The problem’s
value
denotes the objective function value.The variables’
value
is set according to the primal solution. You can in fact query the value of any expression involving valued variables like this.The constraints’
dual
is set according to the dual solution.The value of any parameter involved in the problem may have changed, depending on the parameter.
- Parameters
extra_options –
A sequence of additional solver options to use with this solution search only. In particular, this lets you
select a solver via the
solver
option,obtain non-optimal primal solutions by setting
primals=None
,require a complete and optimal dual solution with
duals=True
, andskip valuing variables or constraints with
apply_solution=False
.
- Returns ~picos.Solution or list(~picos.Solution)
A solution object or list thereof.
- Raises
In the following cases:
No solution strategy was found.
Multiple solutions were requested but none were returned.
A primal solution was explicitly requested (
primals=True
) but the primal solution is missing/incomplete or not claimed optimal.A dual solution was explicitly requested (
duals=True
) but the dual solution is missing/incomplete or not claimed optimal.
The case number is stored in the
code
attribute of the exception.
- update_options(**options)[source]¶
Set multiple solver options at once.
- Parameters
options – A parameter sequence of options to set.
Deprecated since version 2.0: Use
Problem.options
instead.
- verbosity()[source]¶
Return the problem’s current verbosity level.
Deprecated since version 2.0: Use
Problem.options
instead.
- CONIC_FORM = <Specification: Optimize AffineExpression subject to AffineConstraint, ComplexAffineConstraint, ComplexLMIConstraint, DummyConstraint, LMIConstraint, ProductConeConstraint, RSOCConstraint, SOCConstraint using any variables and any options.>¶
The specification for problems returned by
conic_form
.
- property conic_form¶
The problem in conic form.
Reformulates the problem such that the objective is affine and all constraints are
ConicConstraint
instances.- Raises
NoStrategyFound – If no reformulation strategy was found.
- Example
>>> from picos import Problem, RealVariable >>> x = RealVariable("x", 2) >>> P = Problem() >>> P.set_objective("min", abs(x)**2) >>> print(P) Quadratic Program minimize ‖x‖² over 2×1 real variable x >>> print(P.conic_form) Second Order Cone Program minimize __..._t over 1×1 real variable __..._t 2×1 real variable x subject to ‖fullroot(‖x‖²)‖² ≤ __..._t ∧ __..._t ≥ 0
Note
This property is intended for educational purposes. You do not need to use it when solving a problem as PICOS will perform the necessary reformulations automatically.
- property constraints¶
Maps constraint IDs to constraints that are part of the problem.
- Returns
A read-only view to an
OrderedDict
. The order is that in which constraints were added.
- property continuous¶
Whether all variables are of continuous types.
- property countCons¶
The same as
len
applied toconstraints
.Deprecated since version 2.0: Still used internally by legacy code; will be removed together with that code.
- property countVar¶
The same as
len
applied tovariables
.Deprecated since version 2.0: Still used internally by legacy code; will be removed together with that code.
- property dual¶
The Lagrangian dual problem of the standardized problem.
More precisely, this property invokes the following:
The primal problem is posed as an equivalent conic standard form minimization problem, with variable bounds expressed as additional constraints.
The Lagrangian dual problem of the reposed primal is computed.
The optimization direction and objective function sign of the dual are adjusted such that, given strong duality and primal feasibility, the optimal values of both problems are equal. In particular, if the primal problem is a minimization or a maximization problem, the dual problem returned will be the respective other.
- Raises
NoStrategyFound – If no reformulation strategy was found.
Note
This property is intended for educational purposes. If you want to solve the primal problem via its dual, use the dualize option instead.
- property maximize¶
Maximization objective as an
Expression
.This can be used to set a maximization objective. For querying the objective, it is recommended to use
objective
instead.
- property minimize¶
Minimization objective as an
Expression
.This can be used to set a minimization objective. For querying the objective, it is recommended to use
objective
instead.
- property mutables¶
Maps names to variables and parameters in use by the problem.
- Returns
A read-only view to an
OrderedDict
. The order is deterministic and depends on the order of operations performed on theProblem
instance as well as on the mutables’ names.
- property no¶
Normalized objective as an
Objective
instance.Either a minimization or a maximization objective, with feasibility posed as “minimize 0”.
The same as the
normalized
attribute of theobjective
.
- property numberConeConstraints¶
Number of quadratic conic constraints stored.
Deprecated since version 2.0: Still used internally by legacy code; will be removed together with that code.
- property numberLSEConstraints¶
Number of
LogSumExpConstraint
stored.Deprecated since version 2.0: Still used internally by legacy code; will be removed together with that code.
- property numberOfVars¶
The sum of the dimensions of all referenced variables.
Deprecated since version 2.0: Still used internally by legacy code; will be removed together with that code.
- property numberQuadConstraints¶
Number of quadratic constraints stored.
Deprecated since version 2.0: Still used internally by legacy code; will be removed together with that code.
- property numberSDPConstraints¶
Number of
LMIConstraint
stored.Deprecated since version 2.0: Still used internally by legacy code; will be removed together with that code.
- property pure_integer¶
Whether all variables are of integral types.
- property status¶
The solution status string as claimed by
last_solution
.
- property strategy¶
Solution strategy as a
Strategy
object.A strategy is available once you order the problem to be solved and it will be reused for successive solution attempts (of a modified problem) while it remains valid with respect to the problem’s
footprint
.When a strategy is reused, modifications to the objective and constraints of a problem are passed step by step through the strategy’s reformulation pipeline while existing reformulation work is not repeated. If the solver also supports these kinds of updates, then modifying and re-solving a problem can be much faster than solving the problem from scratch.
- Example
>>> from picos import Problem, RealVariable >>> x = RealVariable("x", 2) >>> P = Problem() >>> P.set_objective("min", abs(x)**2) >>> print(P.strategy) None >>> sol = P.solve(solver = "cvxopt") # Creates a solution strategy. >>> print(P.strategy) 1. ExtraOptions 2. EpigraphReformulation 3. SquaredNormToConicReformulation 4. CVXOPTSolver >>> # Add another constraint handled by SquaredNormToConicReformulation: >>> P.add_constraint(abs(x - 2)**2 <= 1) <Squared Norm Constraint: ‖x - [2]‖² ≤ 1> >>> P.strategy.valid(solver = "cvxopt") True >>> P.strategy.valid(solver = "glpk") False >>> sol = P.solve(solver = "cvxopt") # Reuses the strategy.
It’s also possible to create a startegy from scratch:
>>> from picos.modeling import Strategy >>> from picos.reforms import (EpigraphReformulation, ... ConvexQuadraticToConicReformulation) >>> from picos.solvers import CVXOPTSolver >>> # Mimic what solve() does when no strategy exists: >>> P.strategy = Strategy(P, CVXOPTSolver, EpigraphReformulation, ... ConvexQuadraticToConicReformulation)
- property type¶
The problem type as a string, such as “Linear Program”.
- property value¶
Objective function value.
If all mutables that appear in the objective function are valued, in particular after a successful solution search, this is the numeric value of the objective function. If the objective function is not fully valued or if the problem is a feasiblity problem without an objective function, this is
None
.In the case of an uncertain objective, this is the worst-case (expected) objective value.
- Raises
picos.uncertain.IntractableWorstCase – When computing the worst-case (expected) value of an uncertain objective is not supported.