X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=dunshire%2Fgames.py;h=0a473915716de7f4be4ce7d99cbc64d87c960795;hb=8371d92c42c7faded26d8ef327129ad6d8c72d73;hp=71da5edbc561de3608e324c309c8ea3914213ce1;hpb=019b62bedacc909118317f255725ff2891f161aa;p=dunshire.git diff --git a/dunshire/games.py b/dunshire/games.py index 71da5ed..0a47391 100644 --- a/dunshire/games.py +++ b/dunshire/games.py @@ -4,12 +4,13 @@ Symmetric linear games and their solutions. This module contains the main :class:`SymmetricLinearGame` class that knows how to solve a linear game. """ +from math import sqrt from cvxopt import matrix, printing, solvers -from .cones import CartesianProduct +from .cones import CartesianProduct, IceCream, NonnegativeOrthant 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 +24,7 @@ class Solution: -------- >>> print(Solution(10, matrix([1,2]), matrix([3,4]))) - Game value: 10.0000000 + Game value: 10.000... Player 1 optimal: [ 1] [ 2] @@ -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 +810,72 @@ 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) + + # Compute the distance from p to the outside of K. + if isinstance(self.K(), NonnegativeOrthant): + # How far is it to a wall? + dist = min(list(self.e1())) + elif isinstance(self.K(), IceCream): + # How far is it to the boundary of the ball that defines + # the ice-cream cone at a given height? Now draw a + # 45-45-90 triangle and the shortest distance to the + # outside of the cone should be 1/sqrt(2) of that. + # It works in R^2, so it works everywhere, right? + height = self.e1()[0] + radius = norm(self.e1()[1:]) + dist = (height - radius) / sqrt(2) + else: + raise NotImplementedError + + nu = - specnorm(self.L())/(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) + + # Compute the distance from p to the outside of K. + if isinstance(self.K(), NonnegativeOrthant): + # How far is it to a wall? + dist = min(list(self.e2())) + elif isinstance(self.K(), IceCream): + # How far is it to the boundary of the ball that defines + # the ice-cream cone at a given height? Now draw a + # 45-45-90 triangle and the shortest distance to the + # outside of the cone should be 1/sqrt(2) of that. + # It works in R^2, so it works everywhere, right? + height = self.e2()[0] + radius = norm(self.e2()[1:]) + dist = (height - radius) / sqrt(2) + else: + raise NotImplementedError + + omega = specnorm(self.L())/(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 solution(self): """ @@ -844,11 +911,11 @@ 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 +931,7 @@ 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 +967,8 @@ 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 +985,8 @@ 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 +997,10 @@ 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 +1104,7 @@ 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):