X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=dunshire%2Fgames.py;h=07a70b0a2b651a40c207d5d3bcdfd807d55f92df;hb=0b6e486f52b6c42f78ba408543be0cc4b66fada7;hp=77c83300a586e7e9f8a98a766d242f98738635f1;hpb=fa3639d3a5cc52e104a81dc75d8688c64c274a71;p=dunshire.git diff --git a/dunshire/games.py b/dunshire/games.py index 77c8330..07a70b0 100644 --- a/dunshire/games.py +++ b/dunshire/games.py @@ -4,16 +4,15 @@ 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) -from . import options +from .options import ABS_TOL, FLOAT_FORMAT, DEBUG_FLOAT_FORMAT + +printing.options['dformat'] = FLOAT_FORMAT -printing.options['dformat'] = options.FLOAT_FORMAT class Solution: """ @@ -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 tolerance_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/2.0) + + def solution(self): """ Solve this linear game and return a :class:`Solution`. @@ -960,6 +977,44 @@ class SymmetricLinearGame: [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} @@ -970,13 +1025,14 @@ class SymmetricLinearGame: self.A(), self.b(), primalstart=self.player1_start(), + dualstart=self.player2_start(), options=opts) except ValueError as error: if str(error) == 'math domain error': # Oops, CVXOPT tried to take the square root of a # negative number. Report some details about the game # rather than just the underlying CVXOPT crash. - printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT + printing.options['dformat'] = DEBUG_FLOAT_FORMAT raise PoorScalingException(self) else: raise error @@ -1001,9 +1057,23 @@ class SymmetricLinearGame: # that CVXOPT is convinced the problem is infeasible (and that # cannot happen). if soln_dict['status'] in ['primal infeasible', 'dual infeasible']: - printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT + printing.options['dformat'] = 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 @@ -1014,29 +1084,18 @@ 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: - printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT + # + if abs(p1_value - p2_value) > self.tolerance_scale(soln)*ABS_TOL: + printing.options['dformat'] = DEBUG_FLOAT_FORMAT raise GameUnsolvableException(self, soln_dict) # And we also check that the points it gave us belong to the # cone, just in case... if (p1_optimal not in self._K) or (p2_optimal not in self._K): - printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT + printing.options['dformat'] = 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): @@ -1067,10 +1126,8 @@ class SymmetricLinearGame: >>> e1 = [1] >>> e2 = e1 >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> actual = SLG.condition() - >>> expected = 1.8090169943749477 - >>> abs(actual - expected) < options.ABS_TOL - True + >>> SLG.condition() + 1.809... """ return (condition_number(self._G()) + condition_number(self.A()))/2