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mosek_conif.py
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mosek_conif.py
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"""Implements the GPkit interface to MOSEK (version >= 9)
through python-based Optimizer API"""
from __future__ import print_function
import numpy as np
import mosek
def mskoptimize(c, A, k, p_idxs, *args, **kwargs):
# pylint: disable=unused-argument
# pylint: disable-msg=too-many-locals
# pylint: disable-msg=too-many-statements
# pylint: disable-msg=too-many-branches
"""
Definitions
-----------
"[a,b] array of floats" indicates array-like data with shape [a,b]
n is the number of monomials in the gp
m is the number of variables in the gp
p is the number of posynomial constraints in the gp
Arguments
---------
c : floats array of shape n
Coefficients of each monomial
A : gpkit.small_classes.CootMatrix, of shape (n, m)
Exponents of the various free variables for each monomial.
k : ints array of shape p+1
k[0] is the number of monomials (rows of A) present in the objective
k[1:] is the number of monomials (rows of A) present in each constraint
p_idxs : ints array of shape n.
sel = p_idxs == i selects rows of A
and entries of c of the i-th posynomial
fi(x) = c[sel] @ exp(A[sel,:] @ x).
The 0-th posynomial gives the objective function, and the
remaining posynomials should be constrained to be <= 1.
Returns
-------
dict
Contains the following keys
"status": string
"optimal", "infeasible", "unbounded", or "unknown".
"primal" np.ndarray or None
The values of the ``m`` primal variables.
"la": np.ndarray or None
The dual variables to the ``p`` posynomial constraints, when
those constraints are represented in log-sum-exp ("LSE") form.
"""
#
# Initial transformations of problem data.
#
# separate monomial constraints (call them "lin"),
# from those which require
# an LSE representation (call those "lse").
#
# NOTE: the epigraph of the objective function always gets an "lse"
# representation, even if the objective is a monomial.
#
log_c = np.log(np.array(c))
lse_posys = [0]
lin_posys = []
for i, val in enumerate(k[1:]):
if val > 1:
lse_posys.append(i+1)
else:
lin_posys.append(i+1)
if lin_posys:
A = A.tocsr()
lin_posys_set = frozenset(lin_posys)
lin_idxs = [i for i, p in enumerate(p_idxs) if p in lin_posys_set]
lse_idxs = np.ones(A.shape[0], dtype=bool)
lse_idxs[lin_idxs] = False
A_lse = A[lse_idxs, :].tocoo()
log_c_lse = log_c[lse_idxs]
A_lin = A[lin_idxs, :].tocoo()
log_c_lin = log_c[lin_idxs]
else:
log_c_lin = None # A_lin won't be referenced later,
A_lse = A # so no need to define it.
log_c_lse = log_c
k_lse = [k[i] for i in lse_posys]
n_lse = sum(k_lse)
p_lse = len(k_lse)
lse_p_idx = []
for i, ki in enumerate(k_lse):
lse_p_idx.extend([i] * ki)
lse_p_idx = np.array(lse_p_idx)
#
# Create MOSEK task. Add variables, bound constraints, and conic
# constraints.
#
# Say that MOSEK's optimization variable is a block vector,
# [x;t;z], where ...
# x is the user-defined primal variable (length m),
# t is an aux variable for exponential cones (length 3 * n_lse),
# z is an epigraph variable for LSE terms (length p_lse).
#
# The variable z[0] is special,
# because it's the epigraph of the objective function
# in LSE form. The sign of this variable is not constrained.
#
# The variables z[1:] are epigraph terms for "log",
# in constraints that naturally write as
# LSE(Ai @ x + log_ci) <= 0. These variables need to be <= 0.
#
# The main constraints on (x, t, z) are described
# in next comment block.
#
env = mosek.Env()
task = env.Task(0, 0)
m = A.shape[1]
msk_nvars = m + 3 * n_lse + p_lse
task.appendvars(msk_nvars)
# "x" is free
task.putvarboundlist(np.arange(m), [mosek.boundkey.fr] * m,
np.zeros(m), np.zeros(m))
# t[3 * i + i] == 1, other components are free.
bound_types = [mosek.boundkey.fr, mosek.boundkey.fx, mosek.boundkey.fr]
task.putvarboundlist(np.arange(m, m + 3*n_lse), bound_types * n_lse,
np.ones(3*n_lse), np.ones(3*n_lse))
# z[0] is free; z[1:] <= 0.
bound_types = [mosek.boundkey.fr] + [mosek.boundkey.up] * (p_lse - 1)
task.putvarboundlist(np.arange(m + 3*n_lse, msk_nvars), bound_types,
np.zeros(p_lse), np.zeros(p_lse))
# t[3*i], t[3*i + 1], t[3*i + 2] belongs to the exponential cone
task.appendconesseq([mosek.conetype.pexp] * n_lse, [0.0] * n_lse,
[3] * n_lse, m)
#
# Exponential cone affine constraints (other than t[3*i + 1] == 1).
#
# For each i in {0, ..., n_lse - 1}, we need
# t[3*i + 2] == A_lse[i, :] @ x + log_c_lse[i] - z[lse_p_idx[i]].
#
# For each j from {0, ..., p_lse - 1}, the "t" should also satisfy
# sum(t[3*i] for i where i == lse_p_idx[j]) <= 1.
#
# When combined with bound constraints ``t[3*i + 1] == 1``, the
# above constraints imply
# LSE(A_lse[sel, :] @ x + log_c_lse[sel]) <= z[i]
# for ``sel = lse_p_idx == i``.
#
task.appendcons(n_lse + p_lse)
# Linear equations between (x,t,z).
# start with coefficients on "x"
rows = list(A_lse.row)
cols = list(A_lse.col)
vals = list(A_lse.data)
# add coefficients on "t"
rows += list(range(n_lse))
cols += (m + 3*np.arange(n_lse) + 2).tolist()
vals += [-1.0] * n_lse
# add coefficients on "z"
rows += list(range(n_lse))
cols += [m + 3*n_lse + lse_p_idx[i] for i in range(n_lse)]
vals += [-1.0] * n_lse
task.putaijlist(rows, cols, vals)
cur_con_idx = n_lse
# Linear inequalities on certain sums of "t".
rows, cols, vals = [], [], []
for i in range(p_lse):
sels = np.nonzero(lse_p_idx == i)[0]
rows.extend([cur_con_idx] * sels.size)
cols.extend(m + 3 * sels)
vals.extend([1] * sels.size)
cur_con_idx += 1
task.putaijlist(rows, cols, vals)
# Build the right-hand-sides of the [in]equality constraints
type_constraint = [mosek.boundkey.fx] * n_lse + [mosek.boundkey.up] * p_lse
h = np.concatenate([-log_c_lse, np.ones(p_lse)])
task.putconboundlist(np.arange(h.size, dtype=int), type_constraint, h, h)
#
# Affine constraints, not needing the exponential cone
#
# Require A_lin @ x <= -log_c_lin.
#
if log_c_lin is not None:
task.appendcons(log_c_lin.size)
rows = cur_con_idx + np.array(A_lin.row)
task.putaijlist(rows, A_lin.col, A_lin.data)
type_constraint = [mosek.boundkey.up] * log_c_lin.size
con_indices = np.arange(cur_con_idx, cur_con_idx + log_c_lin.size)
h = -log_c_lin #pylint: disable=invalid-unary-operand-type
task.putconboundlist(con_indices, type_constraint, h, h)
cur_con_idx += log_c_lin.size
#
# Set the objective function
#
task.putclist([int(m + 3*n_lse)], [1])
task.putobjsense(mosek.objsense.minimize)
#
# Set solver parameters, and call .solve().
#
verbose = kwargs.get("verbose", True)
if verbose:
def streamprinter(text):
"Stream printer for output from mosek."
print(text)
env.set_Stream(mosek.streamtype.log, streamprinter)
task.set_Stream(mosek.streamtype.log, streamprinter)
task.putintparam(mosek.iparam.infeas_report_auto, mosek.onoffkey.on)
task.putintparam(mosek.iparam.log_presolve, 0)
task.optimize()
if verbose:
task.solutionsummary(mosek.streamtype.msg)
#
# Recover the solution
#
msk_solsta = task.getsolsta(mosek.soltype.itr)
if msk_solsta == mosek.solsta.optimal:
# recover primal variables
x = [0.] * m
task.getxxslice(mosek.soltype.itr, 0, m, x)
x = np.array(x)
# recover dual variables for log-sum-exp epigraph constraints
# (skip epigraph of the objective function).
z_duals = [0.] * (p_lse - 1)
task.getsuxslice(mosek.soltype.itr, m + 3*n_lse + 1,
msk_nvars, z_duals)
z_duals = np.array(z_duals)
z_duals[z_duals < 0] = 0
# recover dual variables for the remaining user-provided constraints
if log_c_lin is not None:
aff_duals = [0.] * log_c_lin.size
task.getsucslice(mosek.soltype.itr, n_lse + p_lse, cur_con_idx,
aff_duals)
aff_duals = np.array(aff_duals)
aff_duals[aff_duals < 0] = 0
# merge z_duals with aff_duals
merged_duals = np.zeros(len(k))
merged_duals[lse_posys[1:]] = z_duals
merged_duals[lin_posys] = aff_duals
merged_duals = merged_duals[1:]
else:
merged_duals = z_duals
# wrap things up in a dictionary
solution = {'status': 'optimal', 'primal': x, 'la': merged_duals}
elif msk_solsta == mosek.solsta.prim_infeas_cer:
solution = {'status': 'infeasible', 'primal': None, 'la': None}
elif msk_solsta == mosek.solsta.dual_infeas_cer:
solution = {'status': 'unbounded', 'primal': None, 'la': None}
else:
solution = {'status': 'unknown', 'primal': None, 'la': None}
task.__exit__(None, None, None)
env.__exit__(None, None, None)
return solution