Reference implementation and numerical experiments for prox-convex, a structure-preserving extension of prox-linear methods for composite optimization problems of the form

[ F(x)=g(x)+h(c(x))+s(R(x)), ]

where:

• g and h are convex,
• c and s are smooth,
• each component of R is convex, possibly nonsmooth.

The key idea of prox-convex is to linearize only the smooth maps while preserving the existing convex structure. In particular, instead of fully linearizing s(R(x)), prox-convex keeps the convex inner map R exact and linearizes only the outer smooth coupling s.

Builds the convex subproblem for each method:

• prox_gradient
• prox_linear
• prox_convex

The subproblem has the generic form

[ \min_d ; \text{model}(x_k+d;x_k) + \tfrac12 |d|_{Q_k}^2, ]

implemented with CVXPY variables and parameters.

Provides the main solve loop with:

• optional use of generated code,
• adaptive proximal-weight update via predicted/actual reduction,
• tracking of accepted iterates, subproblem solves, timing, and convergence status.

run_p0.ipynb

This notebook visualizes the local convex models for the nonsmooth objective

[ f(x)=\left|\tfrac12 x^2+x\right|=|x|,\left|1+\tfrac{x}{2}\right|, ]

which admits both a prox-linear representation h(c(x)) and a prox-convex representation s(R(x)).

It generates a figure comparing:

• the true objective,
• the prox-linear local model,
• the prox-convex local model.

Convexified local models for the 1D example. The prox-convex surrogate preserves the kink structure and tracks the true objective more tightly than the prox-linear surrogate.

run_p1_p2.ipynb

This notebook studies

[ F(x)=\log(Q(x))+8, \qquad Q(x)=|A(x-\bar x)|^2+c, ]

which can be represented both as:

• a prox-linear problem via h(c(x)) = -\log(Q(x)^{-1}) + 8, and
• a prox-convex problem via s(R(x)) = \log(Q(x)) + 8.

The notebook compares:

• prox-gradient,
• prox-linear,
• prox-convex,

in both fixed-weight and adaptive-weight settings.

Adaptive convergence history for the log-of-quadratic example. Prox-convex converges in very few accepted iterations and maintains the smallest proximal weight.

Iterate trajectories over the level sets of the log-of-quadratic objective. Prox-convex follows a near-direct path to the minimizer, while prox-gradient and prox-linear require many smaller corrective steps.

run_p3.ipynb

This is the main benchmark used in the paper. It considers the disjunctive-feasibility-type objective

[ F(x)=R_1(x)R_2(x), \qquad R_i(x)=\tfrac12 (x-c_i)^\top P_i (x-c_i), ]

where the condition that at least one nonnegative quadratic should be nonpositive is parameterized by requiring the product to vanish.

The notebook includes:

• single-run trajectory and convergence plots,
• distribution over random initializations,
• condition-number sweep,
• sensitivity to the initial proximal weight.

Single-run convergence comparison on the main benchmark. Prox-convex reaches a minimizer in far fewer accepted iterations than the fully linearized baseline.

Iterate trajectories over the level sets of the product-of-quadratics objective. Prox-convex preserves the quadratic geometry of both channels and follows a more direct route to a global minimizer.

Distribution of accepted iterations and wall-clock time over random initializations. Prox-convex is both faster and more stable across starts.

Effect of increasing anisotropy of the quadratic channels. The advantage of prox-convex grows strongly with the condition number.

Sensitivity to the initial proximal weight. Prox-convex remains substantially better across a wide range of initialization values.

If you use this code, please cite the corresponding paper.

@article{uzun2025proximal,
  title={A Proximal Method for Composite Optimization with Smooth and Convex Components},
  author={Uzun, Samet and Luo, Dayou and A{\c{c}}{\i}kme{\c{s}}e, Beh{\c{c}}et and Aravkin, Aleksandr Y},
  journal={arXiv preprint arXiv:2512.20602},
  year={2025}
}