A projected descent method for minimization of weakly semismooth functions over polyhedral sets

Morteza Maleknia; Majid Soleimani-damaneh

Home/Archive/arXiv

arXiv · 2609.36282

A projected descent method for minimization of weakly semismooth functions over polyhedral sets

Morteza Maleknia·Majid Soleimani-damaneh

View original source ↗Read source document ↗Export metadata ↓

Abstract

This study develops a projection-based descent method for minimizing weakly semismooth functions over polyhedral sets, taking an initial step toward extending projection-based descent algorithms to nonsmooth, nonconvex optimization problems. At each iteration, the method considers an inner approximation of the Clarke $\varepsilon$-subdifferential at the current iterate over the feasible region and uses its least-norm element to generate a search direction. An exponential limited backtracking line search is then developed to assess the quality of the generated direction, while a projected variant of Mifflin's line search is proposed to identify nonredundant subgradients that enrich the Clarke $\varepsilon$-subdifferential approximation when necessary. To quantify stationarity, a characterization of stationary points based on the projection operator yields a computable optimality measure for the proposed method. The convergence properties of the proposed method are established under mild assumptions. If the algorithm generates infinitely many serious steps, we identify two subsequences such that every cluster point of each subsequence is stationary. If the number of serious steps is finite, we show that the final serious step generates a stationary point. Numerical experiments demonstrate the efficiency and broad applicability of the proposed method across a wide range of test problems, including applications in image denoising, multiobjective optimization, and data clustering. To extend the applicability of the proposed method to problems with general smooth nonlinear constraints, we propose a heuristic approach based on sequential linearization.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Morteza Maleknia, Majid Soleimani-damaneh. 2026-09-28. A projected descent method for minimization of weakly semismooth functions over polyhedral sets. https://arxiv.org/abs/2609.36282

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

ItsDEAL: Inexact two-level smoothing descent algorithms for weakly convex optimization

This paper addresses nonconvex optimization problems via a two-level smoothing framework, where the high-order Moreau envelope (HOME) provides a smooth approximation of weakly convex cost functions. We study the differentiability and weak smoothness of HOME. Building on the inexact two-level smoothing optimization framework (ItsOPT), we propose an inexact two-level smoothing descent algorithm (ItsDEAL) by: (i) approximately solving the proximal subproblem to obtain an inexact first-order oracle for HOME at the lower level; and (ii) applying an inexact first-order method at the upper level. We derive computable sufficient conditions for lower-level inexactness, enabling approximate high-order proximal-points to be certified without access to the exact proximal-point or HOME value. In particular, we study a parameter-free Hölderian inexact gradient method and an inexact descent method with Armijo line search, both exploiting the weak smoothness of HOME. We establish subsequential convergence under mild inexactness assumptions, and global and linear convergence under the Kurdyka--Łojasiewicz (KL) property. Preliminary experiments on ridge-regularized hinge-loss classification, a high-dimensional MCP-regularized classification model, and robust matrix completion illustrate the numerical behavior of the proposed methods and the role of the HOME order.

math.OC↗

arXiv2026

OFF Newton-type Methods for Unconstrained Optimization with Inexact Derivatives

We study objective-function-free (OFF) Newton-type methods for smooth unconstrained optimization with inexact gradients and Hessians. We propose two methods for first- and second-order stationarity, coupling explicit step-size rules with derivative-accuracy and linear-solver residual conditions to guarantee sufficient decrease without evaluating the objective function. The OFF regularized Newton method finds an $\varepsilon$-first-order stationary point in $\mathcal{O}(\varepsilon^{-2})$ iterations. Under strong convexity, a local Newton variant converges with $Q$-order at least $1+θ$, where $θ\in (0, 1]$. For second-order stationarity, we combine regularized Newton steps with approximate eigenvectors associated with the minimum eigenvalue. With high probability, the resulting method finds an $(\varepsilon, \sqrt{L_h\varepsilon})$-second-order stationary point in $\mathcal{O}(\varepsilon^{-3/2})$ iterations, using a total of $\widetilde{\mathcal{O}}(\varepsilon^{-7/4})$ approximate gradient evaluations and Hessian-vector products. For finite-sum problems, we develop subsampling schemes that yield best-iterate convergence in expectation in the nonconvex first-order setting and local $R$-linear convergence with high probability in the strongly convex setting, followed by $Q$-order at least $1+θ$ once full sampling is reached. The subsampled second-order method achieves a high-probability $\widetilde{\mathcal{O}}(\varepsilon^{-7/2})$ total sample-operation bound, counting component gradient evaluations and component Hessian-vector products.

math.OC↗

arXiv2026

An efficient mixed-integer linear programming formulation for solving influence diagrams

Influence diagrams represent decision-making problems with interdependencies between random events, decisions, and consequences. Traditionally, they have been solved using tailored algorithms that determine the expected utility-maximizing decision strategy. Towards this end, most recent state-of-the-art solution approaches convert them into mixed-integer linear programming (MILP) models that can be solved using off-the-shelf MILP solvers. Computationally, these approaches are most efficient for influence diagrams that represent periodic (or sequential) decision processes, which can be cast as partially observable Markov Decision Processes. However, they are inefficient for problems that lack this periodic structure or whose influence diagrams have large state spaces. The efficient MILP formulation presented in this paper is specifically designed for influence diagrams that are challenging for earlier MILP formulations. Additionally, this formulation can be adapted to optimize conditional value-at-risk and to incorporate chance and logical constraints, thereby retaining the modeling flexibility of MILP-based methods. Finally, computational experiments on a range of problems from the literature are reported to show that the proposed formulation is much more efficient than earlier MILP formulations, particularly in problems which cannot be modeled as partially observable Markov decision processes and which lead to influence diagrams whose decision or chance nodes may have large state spaces.

math.OC↗