# s. t. I_3 + A0*x0 + A1*x1 >= 0
c = array([[1], [1]])
A0 = array([[1, 0, 0],
[0, -1, 0],
[0, 0, -1]])
A1 = array([[0, 1, 0],
[1, 0, 1],
[0, 1, 0]])
# starting point
startPoint = array([[0], [0]])
# create the solver object
problem = SDPSolver(c, [[eye(3), A0, A1]])
# enable graphs
problem.setDrawPlot(True)
# enable informative output
problem.setPrintOutput(True)
# enable bounding into ball with radius 1
#problem.bound(1)
# solve!
x = problem.solve(startPoint, problem.dampedNewton)
print(x)
# print eigenvalues
#print(problem.eigenvalues())
class POPSolver:
"""
Class providing POP (Polynomial Optimization Problem) Solver.
Solves problem in this form:
min f(x)
s.t. g(x) >= 0
by Pavel Trutman, [email protected]
"""
def __init__(self, f, g, d):
"""
Initialization of the POP problem.
Args:
f (dictionary: tuple => int): representation of the objective function f(x)
g (dictionary: tuple => int): representation of the constraining function g(x)
d (int): degree of the relaxation
"""
# get number of variables
key = list(f.keys())[0]
self.n = len(key)
self.d = d
# disable output
logging.basicConfig(stream = sys.stdout, format = '%(message)s')
self.logStdout = logging.getLogger()
# generate all variables up to degree 2*d
allVar = self.generateVariablesUpDegree(2*self.d)
# collect all variables used
varUsed = allVar
# generate moment matrix and localizing matrix
self.MM = self.momentMatrix(self.d, varUsed)
self.LM = self.localizingMatrix(self.d - 1, varUsed, g)
# generate objective function for SDP
self.c = zeros((len(varUsed) - 1, 1))
for variable in range(1, len(varUsed)):
self.c[variable - 1, 0] = f.get(varUsed[variable], 0)
# initialize SDP Solver
self.SDP = SDPSolver(self.c, [self.MM, self.LM])
def solve(self, startPoint):
"""
Solves a POP problem.
Args:
startPoint (Matrix): some feasible point of the SDP problem
Returns:
Matrix: solution of the problem
"""
# solve the SDP porblem
y = self.SDP.solve(startPoint, self.SDP.dampedNewton)
# extract solutions of the POP problem
x = y[0:self.n, :]
return x
def setPrintOutput(self, printOutput):
"""
Enables or disables printing of the computation state.
Args:
printOuput (bool): True - enables the output, False - disables the output
Returns:
None
"""
self.SDP.setPrintOutput(printOutput)
if printOutput:
self.logStdout.setLevel(logging.INFO)
else:
self.logStdout.setLevel(logging.WARNING)
def momentMatrix(self, d, varUsed):
"""
Constructs moment matrix.
Args:
d (int): degree of the relaxation
varUsed (list of tuples): all variables that are used
Returns:
list: list of moment matrices
"""
varUpD = self.generateVariablesUpDegree(d)
varUsedNum = len(varUsed)
dimM = len(varUpD)
MM = [zeros((dimM, dimM)) for i in range(0, varUsedNum)]
for i in range(0, dimM):
for j in range(i, dimM):
# sum up the degrees
varCur = tuple(sum(t) for t in zip(varUpD[i], varUpD[j]))
# find this variable amongs used vars
index = [k for k in range(0, varUsedNum) if varUsed[k] == varCur]
if len(index) > 0:
pos = index[0]
MM[pos][i, j] = 1
MM[pos][j, i] = 1
return MM
def localizingMatrix(self, d, varUsed, g):
"""
Constructs localizing matrix.
Args:
d (int): degree of the relaxation
varUsed (list of tuples): all variables that are used
g (dictionary: tuple => int): representation of the constraining function g(x)
Returns:
list: list of localizing matrices
"""
varUpD = self.generateVariablesUpDegree(d)
varUsedNum = len(varUsed)
dimM = len(varUpD)
LM = [zeros((dimM, dimM)) for i in range(0, varUsedNum)]
for mon, coef in g.items():
for i in range(0, dimM):
for j in range(i, dimM):
# sum up the degrees
varCur = tuple(sum(t) for t in zip(varUpD[i], varUpD[j], mon))
# find this variable amongs used vars
index = [k for k in range(0, varUsedNum) if varUsed[k] == varCur]
if len(index) > 0:
pos = index[0]
LM[pos][i, j] += coef
if i != j:
LM[pos][j, i] += coef
return LM
def generateVariablesDegree(self, d, n):
"""
Generates whole set of variables of given degree.
Args:
d (int): degree of the variables
n (int): number of unknowns
Returns:
list: list of all variables
"""
# generate zero degree variables
if d == 0:
return [(0,)*n]
# generrate one degree variables
elif d == 1:
variables = []
for i in range(0, n):
t = [0]*n
t[i] = 1
variables.append(tuple(t))
return variables
# there is only one unkown with the degree d
elif n == 1:
return [(d,)]
# generate variables in general case
else:
variables = []
for i in range(0, d + 1):
innerVariables = self.generateVariablesDegree(d - i, n - 1)
variables.extend([v + (i,) for v in innerVariables])
return variables
def generateVariablesUpDegree(self, d):
"""
Generates whole set of variables up to given degree.
Args:
d (int): maximal degree of the variables
Returns:
list: list of variables
"""
variables = []
for i in range(0, d + 1):
variables.extend(self.generateVariablesDegree(i, self.n))
return variables
def getFeasiblePoint(self, R):
"""Finds feasible point for SDP problem arisen from the POP problem.
Args:
R (int): a radius of a ball from which x points are choosen
Returns:
Matrix: feasible point for the SDP problem
"""
N = comb(self.n + self.d, self.n)
N = ceil(N*1.5 + 1)
# generate all variable
usedVars = self.generateVariablesUpDegree(2*self.d)[1:]
y = zeros((len(usedVars), 1))
i = 0
# choose many points x to moment matrix have full rank
while i < N:
# select x from the ball with given radius
x = uniform(-R, R, (self.n, 1))
if norm(x) < R:
# generate points y from it
for alpha in range(0, len(usedVars)):
yTemp = 1
for j in range(0, self.n):
yTemp *= x[j, 0]**usedVars[alpha][j]
y[alpha, 0] += yTemp
i += 1
y = y / N
return y