def _bcd(prob, fix_sets, max_iter, solver, mu, rho, mu_max, ep, lambd, linear,
proximal):
"""
block coordinate descent
:param prob: Problem
:param max_iter: maximum number of iterations
:param solver: solver used to solved each fixed problem
:return: it: number of iterations; max_slack: maximum slack variable
"""
obj_pre = np.inf
for it in range(max_iter):
np.random.shuffle(fix_sets)
#print "======= iteration", it, "======="
for subset in fix_sets:
problem_variables = prob.variables()
problem_variables.sort(key=lambda x: x.id)
fix_var = [prob.variables()[idx] for idx in subset]
# fix variables in fix_set
fixed_p = fix(prob, fix_var)
# linearize
if linear:
fixed_p_obj = cvx.linearize(fixed_p.objective.expr)
fixed_p_constr = fixed_p.constraints
if fixed_p.objective.NAME == 'minimize':
fixed_p = cvx.Problem(cvx.Minimize(fixed_p_obj),
fixed_p_constr)
else:
fixed_p = cvx.Problem(cvx.Maximize(fixed_p_obj),
fixed_p_constr)
# add slack variables
fixed_p, var_slack = add_slack(fixed_p, mu)
# proximal operator
if proximal:
fixed_p = proximal_op(fixed_p, var_slack, lambd)
# solve
fixed_p.solve(solver=solver)
max_slack = 0
if not var_slack == []:
max_slack = np.max(
[np.max(np.abs(var.value)) for var in var_slack])
print("max abs slack =", max_slack, "mu =", mu,
"original objective value =",
prob.objective.args[0].value, "fixed objective value =",
fixed_p.objective.args[0].value, "status=",
fixed_p.status)
else:
print("original objective value =",
prob.objective.args[0].value, "status=", fixed_p.status)
mu = min(mu * rho, mu_max) # adaptive mu
if np.linalg.norm(obj_pre - prob.objective.expr.value
) <= ep and max_slack <= ep: # quit
return it, max_slack
else:
obj_pre = prob.objective.args[0].value
return it, max_slack
def is_dmcp(obj):
"""
:param obj: an obj
:return: a boolean indicating if the obj (Function, Expression, Variable) is DMCP
"""
for var in obj.variables():
fix_var = [avar for avar in obj.variables() if not avar.id == var.id]
fix_var.sort(key=lambda x: x.id)
if not fix(obj, fix_var).is_dcp():
return False
return True
def test_fixExpr(self):
'''
Tests whether or not the fix variable function works for expressions.
'''
# Define expression
expr = cvx.abs(self.x1 + self.x1 + self.x2 + self.x3 + self.x4)
# Fix variables and get list of parameters
new_expr = fix(expr, self.fix_vars)
list_params = new_expr.parameters()
# Assertion test
# 2 variables since fix_vars has 2 elements
self.assertEqual(len(list_params), 2)
def find_MIS(prob, is_all):
"""
find maximal independent sets of a graph until all vertices are included
:param prob: a problem
:param is_all: to find all minimal sets or not
:return: a list of maximal independent sets
"""
if prob.is_dcp():
return [prob.variables()]
# graph of conflict vars
V = prob.variables()
node_num = len(V)
g = np.zeros((node_num, node_num)) # table of edges
varid = [var.id for var in V]
varid.sort()
stack, g = search_conflict_l(prob.objective.expr, [], varid, g)
for con in prob.constraints:
stack, g = search_conflict_l(-con.expr, [], varid, g)
# find all independent sets of the conflict graph
i_subsets = find_all_iset(V, g)
# sort all independent sets
subsets_len = [len(subset) for subset in i_subsets]
sort_idx = np.argsort(subsets_len) # sort the subsets by card
# check all sorted sets
result = []
U = [] # union of all collected vars
# collecting from a subset with the largest cardinality
for count in range(1, len(sort_idx) + 1):
flag = 1
for subs in result:
if is_subset(
i_subsets[sort_idx[-count]], subs
): # the current one is a subset of a previously collected one
flag = 0
break
if flag:
set_id = [var.id for var in i_subsets[sort_idx[-count]]]
fix_set = [var for var in V if var.id not in set_id]
if fix(prob, fix_set).is_dcp():
result.append(i_subsets[sort_idx[-count]])
U = union(U, i_subsets[sort_idx[-count]])
# break if the collected vars cover all vars
if not is_all and is_subset(V, U):
break
return result
def test_fixProb(self):
'''
Tests whether or not the fix variable function works for problems/
'''
# Define problem
obj = cvx.Minimize(cvx.abs(self.x1*self.x2 + self.x3*self.x4))
constr = [self.x1 + self.x2 + self.x3 + self.x4 == 1]
prob = cvx.Problem(obj, constr)
# Fix variables and get list of parameters
new_prob = fix(prob, self.fix_vars)
list_params = new_prob.parameters()
# Assertion test
# 2 variables since x1 and x3 for both objective and constraints are fixed
self.assertEqual(len(list_params), 2)
def test_fixPSD(self):
'''
Tests whether or not the fix variable function works for problems/
'''
# Define problem
x = cvx.Variable((4,4), PSD=True)
obj = cvx.Minimize(cvx.norm(x))
constr = [x >> 0]
prob = cvx.Problem(obj, constr)
# Fix variables and get list of parameters
new_prob = fix(prob, [x])
list_params = new_prob.parameters()
# Assertion test
# 1 variable since x is a single symmetric positive semi-definite matrix
self.assertEqual(len(list_params), 1)
def find_dcp_maxset(expr, vars, current=[]):
"""
find maximal subsets of variables, so that expr is a dcp expression within each subset
:param expr: an expression
:param vars: variables that are not fixed
:param current: current list of subsets
:return: a list of subsets of variables and each subset is a list, or None
"""
if expr.is_dcp():
return [expr.variables()]
result = []
next_level = []
for var in vars:
vars_active = erase(vars, var) # active variables
if vars_active == []: # an empty list indicates that the expression is not multi-dcp
return None
# if the set of active vars is not a subset of the current result
if all([
not is_subset(vars_active, current_set)
for current_set in current
]):
vars_active_id = [var.id for var in vars_active]
fix_vars_temp = [
var for var in expr.variables() if not var.id in vars_active_id
]
if fix(expr, fix_vars_temp).is_dcp() == True:
result.append(vars_active) # find a subset
current.append(vars_active)
else:
next_level.append(
vars_active) # to be decomposed in the next level
for set in next_level:
result_temp = find_dcp_maxset(expr, set, current)
if result_temp is None:
return None
else:
for set in result_temp:
result.append(set)
return result
def find_maxset_prob(prob, vars, current=[]):
"""
Analyze a problem to find maximal subsets of variables,
so that the problem is dcp restricting on each subset
:param prob: Problem
:return: a list of subsets of Variables, or None
"""
if prob.is_dcp():
return [prob.variables()]
result = []
next_level = []
for var in vars:
vars_active = erase(vars, var) # active variables
if vars_active == []: # an empty list indicates that the problem is not multi-convex
return None
# if the set of active vars is not a subset of the current result
if all([
not is_subset(vars_active, current_set)
for current_set in current
]):
vars_active_id = [var.id for var in vars_active]
fix_vars_temp = [
var for var in prob.variables() if not var.id in vars_active_id
]
if fix(prob, fix_vars_temp).is_dcp() == True:
result.append(vars_active) # find a subset
current.append(vars_active)
else:
next_level.append(
vars_active) # to be decomposed in the next level
for set in next_level:
result_temp = find_maxset_prob(prob, set, current)
if result_temp is None:
return None
else:
for set in result_temp:
result.append(set)
return result
def find_dcp_set(expr, vars):
"""
find subsets of variables, so that expr is a dcp expression within each subset
:param expr:
:param vars: variables that are not fixed
:return: a list of lists of variables, or None
"""
if vars == []: # an empty list indicates that the expression is not multi-dcp
return None
vars_id = [var.id for var in vars]
fix_vars = [var for var in expr.variables() if not var.id in vars_id]
if fix(expr, fix_vars).is_dcp() == True:
return [vars]
else:
result = []
for var in vars: # erase each variable from vars
vars_temp = erase(vars, var) # active variables
result_temp = find_dcp_set(expr, vars_temp)
if result_temp is None:
return None
for var_set in result_temp:
result.append(var_set)
return result