X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=dunshire%2Fgames.py;h=ae1426a2c611f7e315f94fc5fea0e98f1da0905b;hb=cd77ba5250ed98ece623730c26af845366847487;hp=71da5edbc561de3608e324c309c8ea3914213ce1;hpb=019b62bedacc909118317f255725ff2891f161aa;p=dunshire.git diff --git a/dunshire/games.py b/dunshire/games.py index 71da5ed..ae1426a 100644 --- a/dunshire/games.py +++ b/dunshire/games.py @@ -4,12 +4,11 @@ Symmetric linear games and their solutions. This module contains the main :class:`SymmetricLinearGame` class that knows how to solve a linear game. """ - from cvxopt import matrix, printing, solvers from .cones import CartesianProduct from .errors import GameUnsolvableException, PoorScalingException from .matrices import (append_col, append_row, condition_number, identity, - inner_product) + inner_product, norm, specnorm) from . import options printing.options['dformat'] = options.FLOAT_FORMAT @@ -23,7 +22,7 @@ class Solution: -------- >>> print(Solution(10, matrix([1,2]), matrix([3,4]))) - Game value: 10.0000000 + Game value: 10.000... Player 1 optimal: [ 1] [ 2] @@ -323,6 +322,8 @@ class SymmetricLinearGame: if not self._e2 in K: raise ValueError('the point e2 must lie in the interior of K') + # Initial value of cached method. + self._L_specnorm_value = None def __str__(self): @@ -581,7 +582,7 @@ class SymmetricLinearGame: return matrix(0, (self.dimension(), 1), tc='d') - def _A(self): + def A(self): """ Return the matrix ``A`` used in our CVXOPT construction. @@ -609,7 +610,7 @@ class SymmetricLinearGame: >>> e1 = [1,1,1] >>> e2 = [1,2,3] >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> print(SLG._A()) + >>> print(SLG.A()) [0.0000000 1.0000000 2.0000000 3.0000000] @@ -698,7 +699,7 @@ class SymmetricLinearGame: return matrix([-1, self._zero()]) - def _C(self): + def C(self): """ Return the cone ``C`` used in our CVXOPT construction. @@ -720,7 +721,7 @@ class SymmetricLinearGame: >>> e1 = [1,2,3] >>> e2 = [1,1,1] >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> print(SLG._C()) + >>> print(SLG.C()) Cartesian product of dimension 6 with 2 factors: * Nonnegative orthant in the real 3-space * Nonnegative orthant in the real 3-space @@ -770,7 +771,7 @@ class SymmetricLinearGame: @staticmethod - def _b(): + def b(): """ Return the ``b`` vector used in our CVXOPT construction. @@ -801,7 +802,7 @@ class SymmetricLinearGame: >>> e1 = [1,2,3] >>> e2 = [1,1,1] >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> print(SLG._b()) + >>> print(SLG.b()) [1.0000000] @@ -809,6 +810,57 @@ class SymmetricLinearGame: return matrix([1], tc='d') + def player1_start(self): + """ + Return a feasible starting point for player one. + + This starting point is for the CVXOPT formulation and not for + the original game. The basic premise is that if you normalize + :meth:`e2`, then you get a point in :meth:`K` that makes a unit + inner product with :meth:`e2`. We then get to choose the primal + objective function value such that the constraint involving + :meth:`L` is satisfied. + """ + p = self.e2() / (norm(self.e2()) ** 2) + dist = self.K().ball_radius(self.e1()) + nu = - self._L_specnorm()/(dist*norm(self.e2())) + x = matrix([nu, p], (self.dimension() + 1, 1)) + s = - self._G()*x + + return {'x': x, 's': s} + + + def player2_start(self): + """ + Return a feasible starting point for player two. + """ + q = self.e1() / (norm(self.e1()) ** 2) + dist = self.K().ball_radius(self.e2()) + omega = self._L_specnorm()/(dist*norm(self.e1())) + y = matrix([omega]) + z2 = q + z1 = y*self.e2() - self.L().trans()*z2 + z = matrix([z1, z2], (self.dimension()*2, 1)) + + return {'y': y, 'z': z} + + + def _L_specnorm(self): + """ + Compute the spectral norm of ``L`` and cache it. + """ + if self._L_specnorm_value is None: + self._L_specnorm_value = specnorm(self.L()) + return self._L_specnorm_value + + def epsilon_scale(self, solution): + # Don't return anything smaller than 1... we can't go below + # out "minimum tolerance." + norm_p1_opt = norm(solution.player1_optimal()) + norm_p2_opt = norm(solution.player2_optimal()) + scale = self._L_specnorm()*(norm_p1_opt + norm_p2_opt) + return max(1, scale) + def solution(self): """ @@ -844,11 +896,11 @@ class SymmetricLinearGame: >>> e2 = [1,1,1] >>> SLG = SymmetricLinearGame(L, K, e1, e2) >>> print(SLG.solution()) - Game value: -6.1724138 + Game value: -6.172... Player 1 optimal: - [ 0.551...] - [-0.000...] - [ 0.448...] + [0.551...] + [0.000...] + [0.448...] Player 2 optimal: [0.448...] [0.000...] @@ -864,7 +916,7 @@ class SymmetricLinearGame: >>> e2 = [4,5,6] >>> SLG = SymmetricLinearGame(L, K, e1, e2) >>> print(SLG.solution()) - Game value: 0.0312500 + Game value: 0.031... Player 1 optimal: [0.031...] [0.062...] @@ -900,8 +952,8 @@ class SymmetricLinearGame: >>> print(SLG.solution()) Game value: 18.767... Player 1 optimal: - [-0.000...] - [ 9.766...] + [0.000...] + [9.766...] Player 2 optimal: [1.047...] [0.000...] @@ -918,8 +970,8 @@ class SymmetricLinearGame: >>> print(SLG.solution()) Game value: 24.614... Player 1 optimal: - [ 6.371...] - [-0.000...] + [6.371...] + [0.000...] Player 2 optimal: [2.506...] [0.000...] @@ -930,9 +982,10 @@ class SymmetricLinearGame: soln_dict = solvers.conelp(self._c(), self._G(), self._h(), - self._C().cvxopt_dims(), - self._A(), - self._b(), + self.C().cvxopt_dims(), + self.A(), + self.b(), + primalstart=self.player1_start(), options=opts) except ValueError as error: if str(error) == 'math domain error': @@ -1036,7 +1089,7 @@ class SymmetricLinearGame: True """ - return (condition_number(self._G()) + condition_number(self._A()))/2 + return (condition_number(self._G()) + condition_number(self.A()))/2 def dual(self):