def transform_sigma_max(expr):
X = only_arg(expr)
m, n = dims(X)
S = epi_var(expr, "sigma_max_S", size=(m+n, m+n))
t = epi_var(expr, "sigma_max")
t_In = expression.diag_vec(expression.multiply(expression.ones(n, 1), t))
t_Im = expression.diag_vec(expression.multiply(expression.ones(m, 1), t))
return t, [
expression.eq_constraint(expression.index(S, 0, n, 0, n), t_In),
expression.eq_constraint(expression.index(S, n, n+m, 0, n), X),
expression.eq_constraint(expression.index(S, n, n+m, n, n+m), t_Im),
expression.semidefinite(S)]
def transform_sigma_max(expr):
X = only_arg(expr)
m, n = dims(X)
S = epi_var(expr, "sigma_max_S", size=(m + n, m + n))
t = epi_var(expr, "sigma_max")
t_In = expression.diag_vec(expression.multiply(expression.ones(n, 1), t))
t_Im = expression.diag_vec(expression.multiply(expression.ones(m, 1), t))
return t, [
expression.eq_constraint(expression.index(S, 0, n, 0, n), t_In),
expression.eq_constraint(expression.index(S, n, n + m, 0, n), X),
expression.eq_constraint(expression.index(S, n, n + m, n, n + m),
t_Im),
expression.semidefinite(S)
]
def transform_max_entries(expr):
x = only_arg(expr)
m, n = dims(x)
t = epi_var(expr, "max_entries")
if not expr.has_axis:
return t, [expression.leq_constraint(x, t)]
if expr.axis == 0:
return t, [
expression.leq_constraint(
x, expression.multiply(expression.ones(m, 1), t))]
if expr.axis == 1:
return t, [
expression.leq_constraint(
x, expression.multiply(t, expression.ones(1, n)))]
raise TransformError("unknown axis attribute", expr)
def transform_norm_p(expr):
p = expr.p
x = only_arg(expr)
t = epi_var(expr, "norm_p", size=(1,1))
if p == float("inf"):
return t, [expression.leq_constraint(x, t),
expression.leq_constraint(expression.negate(x), t)]
if p == 1:
return transform_expr(expression.sum_entries(expression.abs_val(x)))
if p == 2:
return t, [expression.soc_constraint(t, x)]
r = epi_var(expr, "norm_p_r", size=dims(x))
t1 = expression.multiply(expression.ones(*dims(x)), t)
if p < 0:
p, _ = power_tools.pow_neg(p)
p = Fraction(p)
constrs = gm_constrs(t1, [x, r], (-p/(1-p), 1/(1-p)))
elif 0 < p < 1:
p, _ = power_tools.pow_mid(p)
p = Fraction(p)
constrs = gm_constrs(r, [x, t1], (p, 1-p))
elif p > 1:
abs_x, constrs = transform_expr(expression.abs_val(x))
p, _ = power_tools.pow_high(p)
p = Fraction(p)
constrs += gm_constrs(abs_x, [r, t1], (1/p, 1-1/p))
constrs.append(expression.eq_constraint(expression.sum_entries(r), t))
return t, constrs
def transform_max_entries(expr):
x = only_arg(expr)
m, n = dims(x)
t = epi_var(expr, "max_entries")
if not expr.has_axis:
return t, [expression.leq_constraint(x, t)]
if expr.axis == 0:
return t, [
expression.leq_constraint(
x, expression.multiply(expression.ones(m, 1), t))
]
if expr.axis == 1:
return t, [
expression.leq_constraint(
x, expression.multiply(t, expression.ones(1, n)))
]
raise TransformError("unknown axis attribute", expr)
def transform_norm_nuc(expr):
X = only_arg(expr)
m, n = dims(X)
T = epi_var(expr, "norm_nuc", size=(m + n, m + n))
obj = expression.multiply(expression.scalar_constant(0.5),
expression.trace(T))
return obj, [
expression.semidefinite(T),
expression.eq_constraint(expression.index(T, 0, m, m, m + n), X)
]
def transform_norm_nuc(expr):
X = only_arg(expr)
m, n = dims(X)
T = epi_var(expr, "norm_nuc", size=(m+n, m+n))
obj = expression.multiply(
expression.scalar_constant(0.5),
expression.trace(T))
return obj, [
expression.semidefinite(T),
expression.eq_constraint(expression.index(T, 0, m, m, m+n), X)]
def transform_sum_largest(expr):
x = only_arg(expr)
k = expr.k
q = epi_var(expr, "sum_largest")
t = epi_var(expr, "sum_largest_t", size=dims(x))
obj = expression.add(expression.sum_entries(t),
expression.multiply(expression.scalar_constant(k), q))
constr = [
expression.leq_constraint(x, expression.add(t, q)),
expression.leq_constraint(expression.scalar_constant(0), t)
]
return obj, constr
def transform_sum_largest(expr):
x = only_arg(expr)
k = expr.k
q = epi_var(expr, "sum_largest")
t = epi_var(expr, "sum_largest_t", size=dims(x))
obj = expression.add(
expression.sum_entries(t),
expression.multiply(expression.scalar_constant(k), q))
constr = [
expression.leq_constraint(x, expression.add(t, q)),
expression.leq_constraint(expression.scalar_constant(0), t)]
return obj, constr
def transform_huber(expr):
n = epi_var(expr, "huber_n")
s = epi_var(expr, "huber_s")
# n**2 + 2*M*|s|
t, constr = transform_expr(
expression.add(
expression.power(n, 2),
expression.multiply(expression.scalar_constant(2 * expr.M),
expression.abs_val(s))))
# x == s + n
x = only_arg(expr)
constr.append(expression.eq_constraint(x, expression.add(s, n)))
return t, constr
def transform_quad_over_lin(expr):
assert len(expr.arg) == 2
x, y = expr.arg
assert dim(y) == 1
t = epi_var(expr, "qol", size=(1,1))
return t, [
expression.soc_constraint(
expression.add(y, t),
expression.vstack(
expression.add(y, expression.negate(t)),
expression.reshape(
expression.multiply(expression.scalar_constant(2), x),
dim(x), 1))),
expression.leq_constraint(expression.scalar_constant(0), y)]
def transform_huber(expr):
n = epi_var(expr, "huber_n")
s = epi_var(expr, "huber_s")
# n**2 + 2*M*|s|
t, constr = transform_expr(
expression.add(
expression.power(n, 2),
expression.multiply(
expression.scalar_constant(2*expr.M),
expression.abs_val(s))))
# x == s + n
x = only_arg(expr)
constr.append(expression.eq_constraint(x, expression.add(s, n)))
return t, constr
def transform_quad_over_lin(expr):
assert len(expr.arg) == 2
x, y = expr.arg
assert dim(y) == 1
t = epi_var(expr, "qol", size=(1, 1))
return t, [
expression.soc_constraint(
expression.add(y, t),
expression.hstack(
expression.add(y, expression.negate(t)),
expression.reshape(
expression.multiply(expression.scalar_constant(2), x), 1,
dim(x)))),
expression.leq_constraint(expression.scalar_constant(0), y)
]
def transform_norm_p(expr):
p = expr.p
x = only_arg(expr)
t = epi_var(expr, "norm_p")
if p == float("inf"):
return t, [
expression.leq_constraint(x, t),
expression.leq_constraint(expression.negate(x), t)
]
if p == 1:
return transform_expr(expression.sum_entries(expression.abs_val(x)))
if p == 2:
if not expr.has_axis:
return t, [
expression.soc_constraint(t, expression.reshape(x, 1, dim(x)))
]
if expr.axis == 0:
return t, [
expression.soc_constraint(expression.reshape(t, dim(x, 1), 1),
expression.transpose(x))
]
if expr.axis == 1:
return t, [expression.soc_constraint(t, x)]
r = epi_var(expr, "norm_p_r", size=dims(x))
t1 = expression.multiply(expression.ones(*dims(x)), t)
if p < 0:
p, _ = power_tools.pow_neg(p)
p = Fraction(p)
constrs = gm_constrs(t1, [x, r], (-p / (1 - p), 1 / (1 - p)))
elif 0 < p < 1:
p, _ = power_tools.pow_mid(p)
p = Fraction(p)
constrs = gm_constrs(r, [x, t1], (p, 1 - p))
elif p > 1:
abs_x, constrs = transform_expr(expression.abs_val(x))
p, _ = power_tools.pow_high(p)
p = Fraction(p)
constrs += gm_constrs(abs_x, [r, t1], (1 / p, 1 - 1 / p))
constrs.append(expression.eq_constraint(expression.sum_entries(r), t))
return t, constrs
def transform_lambda_max(expr):
t = epi_var(expr, "lambda_max", size=(1,1))
X = only_arg(expr)
n = dim(X, 0)
tI = expression.diag_vec(expression.multiply(expression.ones(n, 1), t))
return t, [expression.psd_constraint(tI, X)]
def gm(t, x, y):
return expression.soc_elemwise_constraint(
expression.add(x, y), expression.add(x, expression.negate(y)),
expression.multiply(expression.scalar_constant(2), t))
def gm(t, x, y):
return expression.soc_elemwise_constraint(
expression.add(x, y),
expression.add(x, expression.negate(y)),
expression.multiply(expression.scalar_constant(2), t))
def transform_lambda_max(expr):
t = epi_var(expr, "lambda_max", size=(1, 1))
X = only_arg(expr)
n = dim(X, 0)
tI = expression.diag_vec(expression.multiply(expression.ones(n, 1), t))
return t, [expression.psd_constraint(tI, X)]
