from PEPit import PEP
from PEPit.functions import SmoothStronglyConvexFunction
from PEPit.primitive_steps import shifted_optimization_step
[docs]
def wc_difference_of_convex_algorithm(mu1, mu2, L1, L2, n, alpha=0, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the minimization problem
.. math:: F_\\star \\triangleq \\min_x f_1(x)-f_2(x),
where :math:`f_1` and :math:`f_2` are convex functions, respectively :math:`L_1`-smooth and
:math:`\\mu_1`-strongly convex and :math:`L_2`-smooth and :math:`\\mu_2`-strongly convex.
This code computes a worst-case guarantee for **DCA** (difference-of-convex algorithm, also known as the
convex-concave procedure). That is, it computes the smallest possible :math:`\\tau(n, \\mu_1, L_1,\\mu_2, L_2)`
such that the guarantee
.. math:: \\min_{t\\leqslant n} \\|\\nabla f_1(x_t)-\\nabla f_2(x_t)\\|^2 \\leqslant \\tau(n, \\mu_1, L_1,\\mu_2, L_2) (f_1(x_0)-f_2(x_0)-F_\\star)
is valid, where :math:`x_n` is the n-th iterates obtained with DCA.
**Algorithm**:
DCA is described as follows, for :math:`t \in \\{ 0, \\dots, n-1\\}`,
.. math:: x_{t+1} \\in \\mathrm{argmin}_x\\,\\{ f_1(x) - \\langle \\nabla f_2(x_t), x\\rangle\\},
**Theoretical guarantee**: The results are compared with [1, Theorem 3];
a more complete picture can be obtained from [2], also by possibly allowing for non-convex functions
:math:`f_1` and :math:`f_2` (i.e., possibly negative values for :math:`\\mu_1`, :math:`\\mu_2`).
**References**:
`[1] H. Abbaszadehpeivasti, E. de Klerk, M. Zamani (2021).
On the rate of convergence of the difference-of-convex algorithm (DCA).
Journal of Optimization Theory and Applications, 202(1), 475-496.
<https://arxiv.org/pdf/2109.13566>`_
`[2] T. Rotaru, P. Patrinos, F. Glineur (2025).
Tight Analysis of Difference-of-Convex Algorithm (DCA) Improves Convergence Rates for Proximal Gradient Descent.
Journal of Optimization Theory and Applications, 202(1), 475-496.
<https://arxiv.org/pdf/2503.04486>`_
Args:
mu1 (float): strong convexity parameter for f1.
mu2 (float): strong convexity parameter for f2.
L1 (float): smoothness parameter for f1.
L2 (float): smoothness parameter for f2.
alpha (float): boosting parameter (defaulted to 0).
n (int): number of iterations.
wrapper (str): the name of the wrapper to be used.
solver (str): the name of the solver the wrapper should use.
verbose (int): level of information details to print.
- -1: No verbose at all.
- 0: This example's output.
- 1: This example's output + PEPit information.
- 2: This example's output + PEPit information + solver details.
Returns:
pepit_tau (float): worst-case value.
theoretical_tau (float): reference theoretical value [1, Theorem 3].
Example:
>>> L1, L2, mu1, mu2 = 2., 5., .15, .1
>>> pepit_tau, theory = wc_difference_of_convex_algorithm(mu1=mu1, mu2=mu2, L1=L1, L2=L2, n=5, alpha=0, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 15x15
(PEPit) Setting up the problem: performance measure is the minimum of 6 element(s)
(PEPit) Setting up the problem: Adding initial conditions and general constraints ...
(PEPit) Setting up the problem: initial conditions and general constraints (7 constraint(s) added)
(PEPit) Setting up the problem: interpolation conditions for 2 function(s)
Function 1 : Adding 42 scalar constraint(s) ...
Function 1 : 42 scalar constraint(s) added
Function 2 : Adding 42 scalar constraint(s) ...
Function 2 : 42 scalar constraint(s) added
(PEPit) Setting up the problem: additional constraints for 0 function(s)
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: prosta.prim_and_dual_feas (wrapper:mosek, solver: MOSEK); optimal value: 0.49113062165416893
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite
All the primal scalar constraints are verified up to an error of 8.798134450149764e-09
(PEPit) Dual feasibility check:
The solver found a residual matrix that is positive semi-definite
All the dual scalar values associated with inequality constraints are nonnegative up to an error of 1.0048489946103415e-08
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 2.102726387109924e-07
(PEPit) Final upper bound (dual): 0.4911306282045956 and lower bound (primal example): 0.49113062165416893
(PEPit) Duality gap: absolute: 6.550426645546281e-09 and relative: 1.3337442946407816e-08
*** Example file: worst-case performance of DCA ***
PEPit guarantee: min_i ||f'(x_i)||^2 <= 0.491131 (f(x_0)-f_*)
Theoretical guarantee: min_i ||f'(x_i)||^2 <= 0.491131 (f(x_0)-f_*)
"""
# Instantiate PEP
problem = PEP()
# Declare a smooth convex function
f1 = problem.declare_function(SmoothStronglyConvexFunction, L=L1, mu=mu1)
f2 = problem.declare_function(SmoothStronglyConvexFunction, L=L2, mu=mu2)
F = f1 - f2
# Start by defining its unique optimal point xs = x_* and corresponding function value fs = f_*
xs = F.stationary_point()
Fs = F(xs)
# Then define the starting point x0 of the algorithm
x0 = problem.set_initial_point()
x = x0
g1x = f1.gradient(x0)
g2x = f2.gradient(x0)
problem.set_initial_condition(f1(x) - f2(x) - Fs <= 1)
for i in range(n):
problem.set_performance_metric((g1x - g2x) ** 2)
problem.add_constraint(Fs <= f1(x) - f2(x) - 1 / 2 / (L1 - mu2) * (g1x - g2x) ** 2)
y, _, _ = shifted_optimization_step(g2x, f1)
x = (1 + alpha) * y - alpha * x
g1x, f1x = f1.oracle(x)
g2x, f1x = f2.oracle(x)
problem.set_performance_metric((g1x - g2x) ** 2)
problem.add_constraint(Fs <= f1(x) - f2(x) - 1 / (2 * (L1 - mu2)) * (g1x - g2x) ** 2)
# Solve the PEP
pepit_verbose = max(verbose, 0)
pepit_tau = problem.solve(wrapper=wrapper, solver=solver, verbose=pepit_verbose)
# Compute theoretical guarantee (for comparison)
if alpha == 0:
Delta = 1
A = 2 * (L1 * L2 - mu1 * L2 * (L1 >= L2) - mu2 * L1 * (L2 > L1))
B = L1 + L2 + mu1 * (L1 / L2 - 3) * (L1 >= L2) + mu2 * (L2 / L1 - 3) * (L2 > L1)
C = (L1 * L2 - mu1 * L2 * (L1 >= L2) - mu2 * L1 * (L2 > L1)) / (L1 - mu2)
theoretical_tau = A * Delta / (B * n + C)
else:
theoretical_tau = None
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of DCA ***')
print('\tPEPit guarantee:\t min_i ||f\'(x_i)||^2 <= {:.6} (f(x_0)-f_*)'.format(pepit_tau))
print('\tTheoretical guarantee:\t min_i ||f\'(x_i)||^2 <= {:.6} (f(x_0)-f_*)'.format(theoretical_tau))
# Return the worst-case guarantee of the evaluated method (and the reference theoretical value)
return pepit_tau, theoretical_tau
if __name__ == "__main__":
L1, L2, mu1, mu2 = 2., 5., .15, .1
pepit_tau, theoretical_tau = wc_difference_of_convex_algorithm(mu1=mu1, mu2=mu2, L1=L1, L2=L2, n=5, alpha=0,
wrapper="mosek", solver=None, verbose=1)