{"ID":2826143,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.20602","arxiv_id":"2512.20602","title":"A Proximal Method for Composite Optimization with Smooth and Convex Components","abstract":"We introduce prox-convex for minimizing $F(x)=g(x)+h(C(x))+s(R(x))$, where $g$ and $h$ are convex, $C$ and $s$ are smooth, and each component of $R$ is convex (possibly nonsmooth). Here $g$ captures general convex objectives and indicator functions for convex constraints, while the composite template simultaneously models convex penalties on smooth features $(h \\circ C)$ and smooth couplings of convex (possibly nonsmooth) features $(s \\circ R)$. Each prox-convex step forms a convex subproblem by linearizing only the smooth maps while preserving the existing convex structure. The resulting subproblem is made strongly convex with the proximal metric $Q_k=μ_k I+H_k^+ \\succ 0$ where $μ_k$ is adapted using an implicit trust-region strategy, and $H_k^+ \\succeq 0$ is an optional curvature term for local acceleration. Under mild Lipschitz/smoothness and a per-coordinate monotone-or-smooth condition, we prove subdifferential regularity, derive two-sided quadratic model error bounds with explicit constants, and obtain sufficient decrease with $O(\\varepsilon^{-2})$ complexity for driving the norm of the metric prox-gradient below $\\varepsilon$. Furthermore, a local error-bound condition for $F$ guarantees a metric step-size error bound and hence local $Q$-linear convergence of the function values. Using the Taylor-like model framework of Drusvyatskiy, Ioffe, and Lewis, we show that every cluster point of the iterates is limiting-stationary; under our regularity conditions, this further implies Fréchet stationarity. The same framework also establishes robustness to inexact subproblem solves and justifies a model-decrease termination rule.","short_abstract":"We introduce prox-convex for minimizing $F(x)=g(x)+h(C(x))+s(R(x))$, where $g$ and $h$ are convex, $C$ and $s$ are smooth, and each component of $R$ is convex (possibly nonsmooth). Here $g$ captures general convex objectives and indicator functions for convex constraints, while the composite template simultaneously mod...","url_abs":"https://arxiv.org/abs/2512.20602","url_pdf":"https://arxiv.org/pdf/2512.20602v1","authors":"[\"Samet Uzun\",\"Dayou Luo\",\"Behçet Açıkmeşe\",\"Aleksandr Y. Aravkin\"]","published":"2025-12-22T08:13:41Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}