"""
Class definitions for all of the symmetric cones (and their superclass,
-class:`SymmetricCone`) supported by CVXOPT.
+:class:`SymmetricCone`) supported by CVXOPT.
"""
from cvxopt import matrix
dimension : int
The dimension of this cone.
+ Raises
+ ------
+
+ ValueError
+ If you try to create a cone with dimension zero or less.
+
"""
def __init__(self, dimension):
"""
class SymmetricPSD(SymmetricCone):
- """
+ r"""
The cone of real symmetric positive-semidefinite matrices.
This cone has a dimension ``n`` associated with it, but we let ``n``
refer to the dimension of the domain of our matrices and not the
dimension of the (much larger) space in which the matrices
themselves live. In other words, our ``n`` is the ``n`` that appears
- in the usual notation `S^{n}` for symmetric matrices.
+ in the usual notation :math:`S^{n}` for symmetric matrices.
As a result, the cone ``SymmetricPSD(n)`` lives in a space of dimension
- ``(n**2 + n)/2)``.
+ :math:`\left(n^{2} + n\right)/2)`.
Examples
--------
class SymmetricLinearGame:
- """
+ r"""
A representation of a symmetric linear game.
- The data for a linear game are,
+ The data for a symmetric linear game are,
* A "payoff" operator ``L``.
- * A cone ``K``.
- * A point ``e`` in the interior of ``K``.
- * A point ``f`` in the interior of the dual of ``K``.
-
- In a symmetric game, the cone ``K`` is be self-dual. We therefore
- name the two interior points ``e1`` and ``e2`` to indicate that
- they come from the same cone but are "chosen" by players one and
- two respectively.
+ * A symmetric cone ``K``.
+ * Two points ``e1`` and ``e2`` in the interior of ``K``.
The ambient space is assumed to be the span of ``K``.
+ With those data understood, the game is played as follows. Players
+ one and two choose points :math:`x` and :math:`y` respectively, from
+ their respective strategy sets,
+
+ .. math::
+ \begin{aligned}
+ \Delta_{1}
+ &=
+ \left\{
+ x \in K \ \middle|\ \left\langle x, e_{2} \right\rangle = 1
+ \right\}\\
+ \Delta_{2}
+ &=
+ \left\{
+ y \in K \ \middle|\ \left\langle y, e_{1} \right\rangle = 1
+ \right\}.
+ \end{aligned}
+
+ Afterwards, a "payout" is computed as :math:`\left\langle
+ L\left(x\right), y \right\rangle` and is paid to player one out of
+ player two's pocket. The game is therefore zero sum, and we suppose
+ that player one would like to guarantee himself the largest minimum
+ payout possible. That is, player one wishes to,
+
+ .. math::
+ \begin{aligned}
+ \text{maximize }
+ &\underset{y \in \Delta_{2}}{\min}\left(
+ \left\langle L\left(x\right), y \right\rangle
+ \right)\\
+ \text{subject to } & x \in \Delta_{1}.
+ \end{aligned}
+
+ Player two has the simultaneous goal to,
+
+ .. math::
+ \begin{aligned}
+ \text{minimize }
+ &\underset{x \in \Delta_{1}}{\max}\left(
+ \left\langle L\left(x\right), y \right\rangle
+ \right)\\
+ \text{subject to } & y \in \Delta_{2}.
+ \end{aligned}
+
+ These goals obviously conflict (the game is zero sum), but an
+ existence theorem guarantees at least one optimal min-max solution
+ from which neither player would like to deviate. This class is
+ able to find such a solution.
+
+ Parameters
+ ----------
+
+ L : list of list of float
+ A matrix represented as a list of ROWS. This representation
+ agrees with (for example) SageMath and NumPy, but not with CVXOPT
+ (whose matrix constructor accepts a list of columns).
+
+ K : :class:`SymmetricCone`
+ The symmetric cone instance over which the game is played.
+
+ e1 : iterable float
+ The interior point of ``K`` belonging to player one; it
+ can be of any iterable type having the correct length.
+
+ e2 : iterable float
+ The interior point of ``K`` belonging to player two; it
+ can be of any enumerable type having the correct length.
+
+ Raises
+ ------
+
+ ValueError
+ If either ``e1`` or ``e2`` lie outside of the cone ``K``.
+
Examples
--------
[ 2]
[ 3].
-
Lists can (and probably should) be used for every argument::
>>> from cones import NonnegativeOrthant
def __init__(self, L, K, e1, e2):
"""
Create a new SymmetricLinearGame object.
-
- INPUT:
-
- - ``L`` -- an square matrix represented as a list of lists
- of real numbers. ``L`` itself is interpreted as a list of
- ROWS, which agrees with (for example) SageMath and NumPy,
- but not with CVXOPT (whose matrix constructor accepts a
- list of columns).
-
- - ``K`` -- a SymmetricCone instance.
-
- - ``e1`` -- the interior point of ``K`` belonging to player one;
- it can be of any enumerable type having the correct length.
-
- - ``e2`` -- the interior point of ``K`` belonging to player two;
- it can be of any enumerable type having the correct length.
-
"""
self._K = K
self._e1 = matrix(e1, (K.dimension(), 1))
def solution(self):
"""
- Solve this linear game and return a Solution object.
+ Solve this linear game and return a :class:`Solution`.
- OUTPUT:
+ Returns
+ -------
- If the cone program associated with this game could be
- successfully solved, then a Solution object containing the
- game's value and optimal strategies is returned. If the game
- could *not* be solved -- which should never happen -- then a
- GameUnsolvableException is raised. It can be printed to get the
- raw output from CVXOPT.
+ :class:`Solution`
+ A :class:`Solution` object describing the game's value and
+ the optimal strategies of both players.
+
+ Raises
+ ------
+ GameUnsolvableException
+ If the game could not be solved (if an optimal solution to its
+ associated cone program was not found).
Examples
--------
return Solution(p1_value, p1_optimal, p2_optimal)
def dual(self):
- """
+ r"""
Return the dual game to this game.
+ If :math:`G = \left(L,K,e_{1},e_{2}\right)` is a linear game,
+ then its dual is :math:`G^{*} =
+ \left(L^{*},K^{*},e_{2},e_{1}\right)`. However, since this cone
+ is symmetric, :math:`K^{*} = K`.
+
Examples
--------
projects / dunshire.git / commitdiff