def solve_opf(self):
if self.node_type == 1:
gen = False
else:
gen = True
f_fcn = lambda x, return_hessian=False: self.f(x, return_hessian)
gh_fcn = lambda x: self.gh(x, gen)
hess_fcn = lambda x, lmbda, cost_mult=1: self.hess(
x, lmbda, cost_mult, gen)
solution = pips(f_fcn,
self.x0,
xmin=self.lb,
xmax=self.ub,
gh_fcn=gh_fcn,
hess_fcn=hess_fcn)
x, f, s, lam, _ = solution["x"], solution["f"], solution["eflag"], \
solution["lmbda"], solution["output"]
return [x, f, s]
from scipy.sparse import csr_matrix
from pips import pips
def f2(x):
f = -x[0] * x[1] - x[1] * x[2]
df = -r_[x[1], x[0] + x[2], x[1]]
# actually not used since 'hess_fcn' is provided
d2f = -array([[0, 1, 0], [1, 0, 1], [0, 1, 0]], float64)
return f, df, d2f
def gh2(x):
h = dot(array([ [1, -1, 1], [1, 1, 1] ]), x**2) + array([-2.0, -10.0])
dh = 2 * csr_matrix( array([ [x[0], x[0]], [-x[1], x[1]], [x[2], x[2]] ]) )
g = array([])
dg = None
return h, g, dh, dg
def hess2(x, lam):
mu = lam["ineqnonlin"]
Lxx = csr_matrix( array([ r_[dot(2 * array([1, 1]), mu), -1, 0],
r_[-1, dot(2 * array([-1, 1]), mu), -1],
r_[0, -1, dot(2 * array([1, 1]), mu)] ]) )
return Lxx
x0 = array([1, 1, 0], float64)
opt = {"verbose": True}
solution = pips(f2, x0, gh_fcn=gh2, hess_fcn=hess2, opt=opt)
print solution["output"]["iterations"], "F:", solution["x"], solution["f"]
def _solve(self, x0, A, l, u, xmin, xmax):
""" Solves using Python Interior Point Solver (PIPS).
"""
s = pips(self._costfcn, x0, A, l, u, xmin, xmax,
self._consfcn, self._hessfcn, self.opt)
return s
def qps_pips(H, c, A, l, u, xmin=None, xmax=None, x0=None, opt=None):
"""Uses the Python Interior Point Solver (PIPS) to solve the following
QP (quadratic programming) problem::
min 1/2 x'*H*x + C'*x
x
subject to::
l <= A*x <= u (linear constraints)
xmin <= x <= xmax (variable bounds)
Note the calling syntax is almost identical to that of QUADPROG from
MathWorks' Optimization Toolbox. The main difference is that the linear
constraints are specified with C{A}, C{L}, C{U} instead of C{A}, C{B},
C{Aeq}, C{Beq}.
Example from U{http://www.uc.edu/sashtml/iml/chap8/sect12.htm}:
>>> from numpy import array, zeros, Inf
>>> from scipy.sparse import csr_matrix
>>> H = csr_matrix(array([[1003.1, 4.3, 6.3, 5.9],
... [4.3, 2.2, 2.1, 3.9],
... [6.3, 2.1, 3.5, 4.8],
... [5.9, 3.9, 4.8, 10 ]]))
>>> c = zeros(4)
>>> A = csr_matrix(array([[1, 1, 1, 1 ],
... [0.17, 0.11, 0.10, 0.18]]))
>>> l = array([1, 0.10])
>>> u = array([1, Inf])
>>> xmin = zeros(4)
>>> xmax = None
>>> x0 = array([1, 0, 0, 1])
>>> solution = qps_pips(H, c, A, l, u, xmin, xmax, x0)
>>> round(solution["f"], 11) == 1.09666678128
True
>>> solution["converged"]
True
>>> solution["output"]["iterations"]
10
All parameters are optional except C{H}, C{c}, C{A} and C{l} or C{u}.
@param H: Quadratic cost coefficients.
@type H: csr_matrix
@param c: vector of linear cost coefficients
@type c: array
@param A: Optional linear constraints.
@type A: csr_matrix
@param l: Optional linear constraints. Default values are M{-Inf}.
@type l: array
@param u: Optional linear constraints. Default values are M{Inf}.
@type u: array
@param xmin: Optional lower bounds on the M{x} variables, defaults are
M{-Inf}.
@type xmin: array
@param xmax: Optional upper bounds on the M{x} variables, defaults are
M{Inf}.
@type xmax: array
@param x0: Starting value of optimization vector M{x}.
@type x0: array
@param opt: optional options dictionary with the following keys, all of
which are also optional (default values shown in parentheses)
- C{verbose} (False) - Controls level of progress output
displayed
- C{feastol} (1e-6) - termination tolerance for feasibility
condition
- C{gradtol} (1e-6) - termination tolerance for gradient
condition
- C{comptol} (1e-6) - termination tolerance for
complementarity condition
- C{costtol} (1e-6) - termination tolerance for cost
condition
- C{max_it} (150) - maximum number of iterations
- C{step_control} (False) - set to True to enable step-size
control
- C{max_red} (20) - maximum number of step-size reductions if
step-control is on
- C{cost_mult} (1.0) - cost multiplier used to scale the
objective function for improved conditioning. Note: The
same value must also be passed to the Hessian evaluation
function so that it can appropriately scale the objective
function term in the Hessian of the Lagrangian.
@type opt: dict
@rtype: dict
@return: The solution dictionary has the following keys:
- C{x} - solution vector
- C{f} - final objective function value
- C{converged} - exit status
- True = first order optimality conditions satisfied
- False = maximum number of iterations reached
- None = numerically failed
- C{output} - output dictionary with keys:
- C{iterations} - number of iterations performed
- C{hist} - dictionary of arrays with trajectories of the
following: feascond, gradcond, coppcond, costcond, gamma,
stepsize, obj, alphap, alphad
- C{message} - exit message
- C{lmbda} - dictionary containing the Langrange and Kuhn-Tucker
multipliers on the constraints, with keys:
- C{eqnonlin} - nonlinear equality constraints
- C{ineqnonlin} - nonlinear inequality constraints
- C{mu_l} - lower (left-hand) limit on linear constraints
- C{mu_u} - upper (right-hand) limit on linear constraints
- C{lower} - lower bound on optimization variables
- C{upper} - upper bound on optimization variables
@see: L{pips}
@author: Ray Zimmerman (PSERC Cornell)
@author: Richard Lincoln
"""
if isinstance(H, dict):
p = H
else:
p = {'H': H, 'c': c, 'A': A, 'l': l, 'u': u}
if xmin is not None: p['xmin'] = xmin
if xmax is not None: p['xmax'] = xmax
if x0 is not None: p['x0'] = x0
if opt is not None: p['opt'] = opt
if 'H' not in p or p['H'] == None:#p['H'].nnz == 0:
if p['A'] is None or p['A'].nnz == 0 and \
'xmin' not in p and \
'xmax' not in p:
# 'xmin' not in p or len(p['xmin']) == 0 and \
# 'xmax' not in p or len(p['xmax']) == 0:
print 'qps_pips: LP problem must include constraints or variable bounds'
return
else:
if p['A'] is not None and p['A'].nnz >= 0:
nx = p['A'].shape[1]
elif 'xmin' in p and len(p['xmin']) > 0:
nx = p['xmin'].shape[0]
elif 'xmax' in p and len(p['xmax']) > 0:
nx = p['xmax'].shape[0]
p['H'] = sparse((nx, nx))
else:
nx = p['H'].shape[0]
p['xmin'] = -Inf * ones(nx) if 'xmin' not in p else p['xmin']
p['xmax'] = Inf * ones(nx) if 'xmax' not in p else p['xmax']
p['c'] = zeros(nx) if p['c'] is None else p['c']
p['x0'] = zeros(nx) if 'x0' not in p else p['x0']
def qp_f(x, return_hessian=False):
f = 0.5 * dot(x * p['H'], x) + dot(p['c'], x)
df = p['H'] * x + p['c']
if not return_hessian:
return f, df
d2f = p['H']
return f, df, d2f
p['f_fcn'] = qp_f
sol = pips(p)
return sol["x"], sol["f"], sol["eflag"], sol["output"], sol["lmbda"]
def pipsopf_solver(om, ppopt, out_opt=None):
"""Solves AC optimal power flow using PIPS.
Inputs are an OPF model object, a PYPOWER options vector and
a dict containing keys (can be empty) for each of the desired
optional output fields.
outputs are a C{results} dict, C{success} flag and C{raw} output dict.
C{results} is a PYPOWER case dict (ppc) with the usual baseMVA, bus
branch, gen, gencost fields, along with the following additional
fields:
- C{order} see 'help ext2int' for details of this field
- C{x} final value of optimization variables (internal order)
- C{f} final objective function value
- C{mu} shadow prices on ...
- C{var}
- C{l} lower bounds on variables
- C{u} upper bounds on variables
- C{nln}
- C{l} lower bounds on nonlinear constraints
- C{u} upper bounds on nonlinear constraints
- C{lin}
- C{l} lower bounds on linear constraints
- C{u} upper bounds on linear constraints
C{success} is C{True} if solver converged successfully, C{False} otherwise
C{raw} is a raw output dict in form returned by MINOS
- xr final value of optimization variables
- pimul constraint multipliers
- info solver specific termination code
- output solver specific output information
@see: L{opf}, L{pips}
@author: Ray Zimmerman (PSERC Cornell)
@author: Carlos E. Murillo-Sanchez (PSERC Cornell & Universidad
Autonoma de Manizales)
@author: Richard Lincoln
"""
##----- initialization -----
## optional output
if out_opt is None:
out_opt = {}
## options
verbose = ppopt['VERBOSE']
feastol = ppopt['PDIPM_FEASTOL']
gradtol = ppopt['PDIPM_GRADTOL']
comptol = ppopt['PDIPM_COMPTOL']
costtol = ppopt['PDIPM_COSTTOL']
max_it = ppopt['PDIPM_MAX_IT']
max_red = ppopt['SCPDIPM_RED_IT']
step_control = (ppopt['OPF_ALG'] == 565) ## OPF_ALG == 565, PIPS-sc
if feastol == 0:
feastol = ppopt['OPF_VIOLATION']
opt = { 'feastol': feastol,
'gradtol': gradtol,
'comptol': comptol,
'costtol': costtol,
'max_it': max_it,
'max_red': max_red,
'step_control': step_control,
'cost_mult': 1e-4,
'verbose': verbose }
## unpack data
ppc = om.get_ppc()
baseMVA, bus, gen, branch, gencost = \
ppc["baseMVA"], ppc["bus"], ppc["gen"], ppc["branch"], ppc["gencost"]
vv, _, nn, _ = om.get_idx()
## problem dimensions
nb = bus.shape[0] ## number of buses
nl = branch.shape[0] ## number of branches
ny = om.getN('var', 'y') ## number of piece-wise linear costs
## linear constraints
A, l, u = om.linear_constraints()
## bounds on optimization vars
_, xmin, xmax = om.getv()
## build admittance matrices
Ybus, Yf, Yt = makeYbus(baseMVA, bus, branch)
## try to select an interior initial point
ll, uu = xmin.copy(), xmax.copy()
ll[xmin == -Inf] = -1e10 ## replace Inf with numerical proxies
uu[xmax == Inf] = 1e10
x0 = (ll + uu) / 2
Varefs = bus[bus[:, BUS_TYPE] == REF, VA] * (pi / 180)
## angles set to first reference angle
x0[vv["i1"]["Va"]:vv["iN"]["Va"]] = Varefs[0]
if ny > 0:
ipwl = find(gencost[:, MODEL] == PW_LINEAR)
# PQ = r_[gen[:, PMAX], gen[:, QMAX]]
# c = totcost(gencost[ipwl, :], PQ[ipwl])
c = gencost.flatten('F')[sub2ind(gencost.shape, ipwl, NCOST+2*gencost[ipwl, NCOST])] ## largest y-value in CCV data
x0[vv["i1"]["y"]:vv["iN"]["y"]] = max(c) + 0.1 * abs(max(c))
# x0[vv["i1"]["y"]:vv["iN"]["y"]] = c + 0.1 * abs(c)
## find branches with flow limits
il = find((branch[:, RATE_A] != 0) & (branch[:, RATE_A] < 1e10))
nl2 = len(il) ## number of constrained lines
##----- run opf -----
f_fcn = lambda x, return_hessian=False: opf_costfcn(x, om, return_hessian)
gh_fcn = lambda x: opf_consfcn(x, om, Ybus, Yf[il, :], Yt[il,:], ppopt, il)
hess_fcn = lambda x, lmbda, cost_mult: opf_hessfcn(x, lmbda, om, Ybus, Yf[il, :], Yt[il, :], ppopt, il, cost_mult)
solution = pips(f_fcn, x0, A, l, u, xmin, xmax, gh_fcn, hess_fcn, opt)
x, f, info, lmbda, output = solution["x"], solution["f"], \
solution["eflag"], solution["lmbda"], solution["output"]
success = (info > 0)
## update solution data
Va = x[vv["i1"]["Va"]:vv["iN"]["Va"]]
Vm = x[vv["i1"]["Vm"]:vv["iN"]["Vm"]]
Pg = x[vv["i1"]["Pg"]:vv["iN"]["Pg"]]
Qg = x[vv["i1"]["Qg"]:vv["iN"]["Qg"]]
V = Vm * exp(1j * Va)
##----- calculate return values -----
## update voltages & generator outputs
bus[:, VA] = Va * 180 / pi
bus[:, VM] = Vm
gen[:, PG] = Pg * baseMVA
gen[:, QG] = Qg * baseMVA
gen[:, VG] = Vm[ gen[:, GEN_BUS].astype(int) ]
## compute branch flows
Sf = V[ branch[:, F_BUS].astype(int) ] * conj(Yf * V) ## cplx pwr at "from" bus, p["u"].
St = V[ branch[:, T_BUS].astype(int) ] * conj(Yt * V) ## cplx pwr at "to" bus, p["u"].
branch[:, PF] = Sf.real * baseMVA
branch[:, QF] = Sf.imag * baseMVA
branch[:, PT] = St.real * baseMVA
branch[:, QT] = St.imag * baseMVA
## line constraint is actually on square of limit
## so we must fix multipliers
muSf = zeros(nl)
muSt = zeros(nl)
if len(il) > 0:
muSf[il] = \
2 * lmbda["ineqnonlin"][:nl2] * branch[il, RATE_A] / baseMVA
muSt[il] = \
2 * lmbda["ineqnonlin"][nl2:nl2+nl2] * branch[il, RATE_A] / baseMVA
## update Lagrange multipliers
bus[:, MU_VMAX] = lmbda["upper"][vv["i1"]["Vm"]:vv["iN"]["Vm"]]
bus[:, MU_VMIN] = lmbda["lower"][vv["i1"]["Vm"]:vv["iN"]["Vm"]]
gen[:, MU_PMAX] = lmbda["upper"][vv["i1"]["Pg"]:vv["iN"]["Pg"]] / baseMVA
gen[:, MU_PMIN] = lmbda["lower"][vv["i1"]["Pg"]:vv["iN"]["Pg"]] / baseMVA
gen[:, MU_QMAX] = lmbda["upper"][vv["i1"]["Qg"]:vv["iN"]["Qg"]] / baseMVA
gen[:, MU_QMIN] = lmbda["lower"][vv["i1"]["Qg"]:vv["iN"]["Qg"]] / baseMVA
bus[:, LAM_P] = \
lmbda["eqnonlin"][nn["i1"]["Pmis"]:nn["iN"]["Pmis"]] / baseMVA
bus[:, LAM_Q] = \
lmbda["eqnonlin"][nn["i1"]["Qmis"]:nn["iN"]["Qmis"]] / baseMVA
branch[:, MU_SF] = muSf / baseMVA
branch[:, MU_ST] = muSt / baseMVA
## package up results
nlnN = om.getN('nln')
## extract multipliers for nonlinear constraints
kl = find(lmbda["eqnonlin"] < 0)
ku = find(lmbda["eqnonlin"] > 0)
nl_mu_l = zeros(nlnN)
nl_mu_u = r_[zeros(2*nb), muSf, muSt]
nl_mu_l[kl] = -lmbda["eqnonlin"][kl]
nl_mu_u[ku] = lmbda["eqnonlin"][ku]
mu = {
'var': {'l': lmbda["lower"], 'u': lmbda["upper"]},
'nln': {'l': nl_mu_l, 'u': nl_mu_u},
'lin': {'l': lmbda["mu_l"], 'u': lmbda["mu_u"]} }
results = ppc
results["bus"], results["branch"], results["gen"], \
results["om"], results["x"], results["mu"], results["f"] = \
bus, branch, gen, om, x, mu, f
pimul = r_[
results["mu"]["nln"]["l"] - results["mu"]["nln"]["u"],
results["mu"]["lin"]["l"] - results["mu"]["lin"]["u"],
-ones(ny > 0),
results["mu"]["var"]["l"] - results["mu"]["var"]["u"],
]
raw = {'xr': x, 'pimul': pimul, 'info': info, 'output': output}
return results, success, raw
def pipsopf_solver(om, ppopt, z, nu, rho):
"""Solves AC optimal power flow using PIPS.
Inputs are an OPF model object, a PYPOWER options vector and
a dict containing keys (can be empty) for each of the desired
optional output fields.
outputs are a C{results} dict, C{success} flag and C{raw} output dict.
C{results} is a PYPOWER case dict (ppc) with the usual baseMVA, bus
branch, gen, gencost fields, along with the following additional
fields:
- C{order} see 'help ext2int' for details of this field
- C{x} final value of optimization variables (internal order)
- C{f} final objective function value
- C{mu} shadow prices on ...
- C{var}
- C{l} lower bounds on variables
- C{u} upper bounds on variables
- C{nln}
- C{l} lower bounds on nonlinear constraints
- C{u} upper bounds on nonlinear constraints
- C{lin}
- C{l} lower bounds on linear constraints
- C{u} upper bounds on linear constraints
C{success} is C{True} if solver converged successfully, C{False} otherwise
C{raw} is a raw output dict in form returned by MINOS
- xr final value of optimization variables
- pimul constraint multipliers
- info solver specific termination code
- output solver specific output information
@see: L{opf}, L{pips}
"""
##----- initialization -----
## options
verbose = ppopt['VERBOSE']
feastol = ppopt['PDIPM_FEASTOL']
gradtol = ppopt['PDIPM_GRADTOL']
comptol = ppopt['PDIPM_COMPTOL']
costtol = ppopt['PDIPM_COSTTOL']
max_it = ppopt['PDIPM_MAX_IT']
max_red = ppopt['SCPDIPM_RED_IT']
step_control = (ppopt['OPF_ALG'] == 565) ## OPF_ALG == 565, PIPS-sc
if feastol == 0:
feastol = ppopt['OPF_VIOLATION']
opt = { 'feastol': feastol,
'gradtol': gradtol,
'comptol': comptol,
'costtol': costtol,
'max_it': max_it,
'max_red': max_red,
'step_control': step_control,
'cost_mult': 1e-4,
'verbose': verbose }
## unpack data
ppc = om.get_ppc()
baseMVA, bus, gen, branch = \
ppc["baseMVA"], ppc["bus"], ppc["gen"], ppc["branch"]
vv, _, nn, _ = om.get_idx()
## problem dimensions
nb = bus.shape[0] ## number of buses
nl = branch.shape[0] ## number of branches
## linear constraints
A, l, u = om.linear_constraints()
## bounds on optimization vars
_, xmin, xmax = om.getv()
## build admittance matrices
Ybus, Yf, Yt = makeYbus(baseMVA, bus, branch)
## try to select an interior initial point
ll, uu = xmin.copy(), xmax.copy()
ll[xmin == -Inf] = -1e10 ## replace Inf with numerical proxies
uu[xmax == Inf] = 1e10
x0 = (ll + uu) / 2
Varefs = bus[bus[:, BUS_TYPE] == REF, VA] * (pi / 180)
## angles set to first reference angle
# x0[vv["i1"]["Va"]:vv["iN"]["Va"]] = Varefs[0]
## find branches with flow limits
il = find((branch[:, RATE_A] != 0) & (branch[:, RATE_A] < 1e10))
nl2 = len(il) ## number of constrained lines
##----- run opf -----
f_fcn = lambda x, return_hessian=False: opf_costfcn(x, om, rho, nu, z, return_hessian)
gh_fcn = lambda x: opf_consfcn(x, om, Ybus, Yf[il, :], Yt[il,:], ppopt, il)
hess_fcn = lambda x, lmbda, cost_mult: opf_hessfcn(x, lmbda, om, Ybus, Yf[il, :], Yt[il, :], ppopt, rho, il)
solution = pips(f_fcn, x0, A, l, u, xmin, xmax, gh_fcn, hess_fcn, opt)
x, f, info, lmbda, output = solution["x"], solution["f"], \
solution["eflag"], solution["lmbda"], solution["output"]
success = (info > 0)
## update solution data
Va = x[vv["i1"]["Va"]:vv["iN"]["Va"]]
Vm = x[vv["i1"]["Vm"]:vv["iN"]["Vm"]]
Pg = x[vv["i1"]["Pg"]:vv["iN"]["Pg"]]
Qg = x[vv["i1"]["Qg"]:vv["iN"]["Qg"]]
V = Vm * exp(1j * Va)
##----- calculate return values -----
## update voltages & generator outputs
bus[:, VA] = Va * 180 / pi
bus[:, VM] = Vm
gen[:, PG] = Pg * baseMVA
gen[:, QG] = Qg * baseMVA
gen[:, VG] = Vm[ gen[:, GEN_BUS].astype(int) ]
## compute branch flows
Sf = V[ branch[:, F_BUS].astype(int) ] * conj(Yf * V) ## cplx pwr at "from" bus, p["u"].
St = V[ branch[:, T_BUS].astype(int) ] * conj(Yt * V) ## cplx pwr at "to" bus, p["u"].
branch[:, PF] = Sf.real * baseMVA
branch[:, QF] = Sf.imag * baseMVA
branch[:, PT] = St.real * baseMVA
branch[:, QT] = St.imag * baseMVA
## line constraint is actually on square of limit
## so we must fix multipliers
muSf = zeros(nl)
muSt = zeros(nl)
if len(il) > 0:
muSf[il] = \
2 * lmbda["ineqnonlin"][:nl2] * branch[il, RATE_A] / baseMVA
muSt[il] = \
2 * lmbda["ineqnonlin"][nl2:nl2+nl2] * branch[il, RATE_A] / baseMVA
## update Lagrange multipliers
bus[:, MU_VMAX] = lmbda["upper"][vv["i1"]["Vm"]:vv["iN"]["Vm"]]
bus[:, MU_VMIN] = lmbda["lower"][vv["i1"]["Vm"]:vv["iN"]["Vm"]]
gen[:, MU_PMAX] = lmbda["upper"][vv["i1"]["Pg"]:vv["iN"]["Pg"]] / baseMVA
gen[:, MU_PMIN] = lmbda["lower"][vv["i1"]["Pg"]:vv["iN"]["Pg"]] / baseMVA
gen[:, MU_QMAX] = lmbda["upper"][vv["i1"]["Qg"]:vv["iN"]["Qg"]] / baseMVA
gen[:, MU_QMIN] = lmbda["lower"][vv["i1"]["Qg"]:vv["iN"]["Qg"]] / baseMVA
bus[:, LAM_P] = \
lmbda["eqnonlin"][nn["i1"]["Pmis"]:nn["iN"]["Pmis"]] / baseMVA
bus[:, LAM_Q] = \
lmbda["eqnonlin"][nn["i1"]["Qmis"]:nn["iN"]["Qmis"]] / baseMVA
branch[:, MU_SF] = muSf / baseMVA
branch[:, MU_ST] = muSt / baseMVA
## package up results
nlnN = om.getN('nln')
results = ppc
results["bus"], results["branch"], results["gen"], \
results["om"], results["x"], results["f"] = \
bus, branch, gen, om, x, f
return results, success
def gh6(x):
h = dot([[1.0, -1.0, 1.0], [1.0, 1.0, 1.0]], x**2) + [-2.0, -10.0]
dh = 2 * csr_matrix([[x[0], x[0]], [-x[1], x[1]], [x[2], x[2]]],
dtype=float)
g = array([])
dg = None
return h, g, dh, dg
def hess6(x, lam, cost_mult=1):
mu = lam['ineqnonlin']
Lxx = cost_mult * \
csr_matrix([[ 0, -1, 0],
[-1, 0, -1],
[ 0, -1, 0]], dtype=float) + \
csr_matrix([[2 * dot([1, 1], mu), 0, 0],
[0, 2 * dot([-1, 1], mu), 0],
[0, 0, 2 * dot([1, 1], mu)]], dtype=float)
return Lxx
f_fcn = f6
gh_fcn = gh6
hess_fcn = hess6
x0 = array([1.0, 1.0, 0.0])
# solution = pips(f_fcn, x0, gh_fcn=gh_fcn, hess_fcn=hess_fcn, opt={'verbose': 2, 'comptol': 1e-9})
solution = pips(f_fcn, x0, gh_fcn=gh_fcn, hess_fcn=hess_fcn)
x, f, s, lam, out = solution["x"], solution["f"], solution["eflag"], \
solution["lmbda"], solution["output"]
print(x, s)