X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=dunshire%2Fgames.py;h=719198b7007896c0eb68df23ab91d6456888fb12;hb=0f8a3d6c4dc67db51fa8e909339bc61a4cddf635;hp=3ed89bb3f2f70b30d0313cbe5a578e4f53e47421;hpb=e41ad668f4f16d8948181ae307cb98430b37ed1d;p=dunshire.git diff --git a/dunshire/games.py b/dunshire/games.py index 3ed89bb..719198b 100644 --- a/dunshire/games.py +++ b/dunshire/games.py @@ -4,10 +4,8 @@ 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, IceCream, NonnegativeOrthant +from .cones import CartesianProduct from .errors import GameUnsolvableException, PoorScalingException from .matrices import (append_col, append_row, condition_number, identity, inner_product, norm, specnorm) @@ -15,6 +13,7 @@ from . import options printing.options['dformat'] = options.FLOAT_FORMAT + class Solution: """ A representation of the solution of a linear game. It should contain @@ -24,7 +23,7 @@ class Solution: -------- >>> print(Solution(10, matrix([1,2]), matrix([3,4]))) - Game value: 10.0000000 + Game value: 10.000... Player 1 optimal: [ 1] [ 2] @@ -324,6 +323,8 @@ 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): @@ -822,30 +823,46 @@ class SymmetricLinearGame: :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)) + 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): """ Solve this linear game and return a :class:`Solution`. @@ -880,11 +897,11 @@ 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...] @@ -900,7 +917,7 @@ 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...] @@ -936,8 +953,8 @@ 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...] @@ -954,12 +971,50 @@ 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...] + This is another one that was difficult numerically, and caused + trouble even after we fixed the first two:: + + >>> from dunshire import * + >>> L = [[57.22233908627052301199, 41.70631373437460354126], + ... [83.04512571985074487202, 57.82581810406928468637]] + >>> K = NonnegativeOrthant(2) + >>> e1 = [7.31887017043399268346, 0.89744171905822367474] + >>> e2 = [0.11099824781179848388, 6.12564670639315345113] + >>> SLG = SymmetricLinearGame(L,K,e1,e2) + >>> print(SLG.solution()) + Game value: 70.437... + Player 1 optimal: + [9.009...] + [0.000...] + Player 2 optimal: + [0.136...] + [0.000...] + + And finally, here's one that returns an "optimal" solution, but + whose primal/dual objective function values are far apart:: + + >>> from dunshire import * + >>> L = [[ 6.49260076597376212248, -0.60528030227678542019], + ... [ 2.59896077096751731972, -0.97685530240286766457]] + >>> K = IceCream(2) + >>> e1 = [1, 0.43749513972645248661] + >>> e2 = [1, 0.46008379832200291260] + >>> SLG = SymmetricLinearGame(L, K, e1, e2) + >>> print(SLG.solution()) + Game value: 11.596... + Player 1 optimal: + [ 1.852...] + [-1.852...] + Player 2 optimal: + [ 1.777...] + [-1.777...] + """ try: opts = {'show_progress': False} @@ -969,6 +1024,8 @@ class SymmetricLinearGame: self.C().cvxopt_dims(), self.A(), self.b(), + primalstart=self.player1_start(), + dualstart=self.player2_start(), options=opts) except ValueError as error: if str(error) == 'math domain error': @@ -1003,6 +1060,20 @@ class SymmetricLinearGame: printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT raise GameUnsolvableException(self, soln_dict) + # For the game value, we could use any of: + # + # * p1_value + # * p2_value + # * (p1_value + p2_value)/2 + # * the game payoff + # + # We want the game value to be the payoff, however, so it + # makes the most sense to just use that, even if it means we + # can't test the fact that p1_value/p2_value are close to the + # payoff. + payoff = self.payoff(p1_optimal, p2_optimal) + soln = Solution(payoff, p1_optimal, p2_optimal) + # The "optimal" and "unknown" results, we actually treat the # same. Even if CVXOPT bails out due to numerical difficulty, # it will have some candidate points in mind. If those @@ -1013,7 +1084,8 @@ class SymmetricLinearGame: # close enough (one could be low by ABS_TOL, the other high by # it) because otherwise CVXOPT might return "unknown" and give # us two points in the cone that are nowhere near optimal. - if abs(p1_value - p2_value) > 2*options.ABS_TOL: + # + if abs(p1_value - p2_value) > self.epsilon_scale(soln)*options.ABS_TOL: printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT raise GameUnsolvableException(self, soln_dict) @@ -1023,19 +1095,7 @@ class SymmetricLinearGame: printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT raise GameUnsolvableException(self, soln_dict) - # For the game value, we could use any of: - # - # * p1_value - # * p2_value - # * (p1_value + p2_value)/2 - # * the game payoff - # - # We want the game value to be the payoff, however, so it - # makes the most sense to just use that, even if it means we - # can't test the fact that p1_value/p2_value are close to the - # payoff. - payoff = self.payoff(p1_optimal, p2_optimal) - return Solution(payoff, p1_optimal, p2_optimal) + return soln def condition(self):