Convex and Nonsmooth Optimization

Approximating inequality systems within probability functions: studying implications for problems and consistency of first-order information

  • Wim van Ackooij
  • Pedro Pérez-Aros
  • David Villacís
    • Categories Nonsmooth Optimization, Stochastic Programming Tags , , , ,

    In this work, we are concerned with the study of optimization problems featuring so-called probability or chance constraints. Probability constraints measure the level of satisfaction of an underlying random inequality system and ensure that this level is high enough. Such an underlying inequality system could be expressed by an abstractly known or perhaps costly to … Read more

    Bregman Douglas-Rachford Splitting Method

  • Shiqian Ma
    • Categories Convex Optimization Tags , , ,

    In this paper, we propose the Bregman Douglas-Rachford splitting (BDRS) method and its variant Bregman Peaceman-Rachford splitting method for solving maximal monotone inclusion problem. We show that BDRS is equivalent to a Bregman alternating direction method of multipliers (ADMM) when applied to the dual of the problem. A special case of the Bregman ADMM is … Read more

    Gradient Methods with Online Scaling Part II. Practical Aspects

  • Wenzhi Gao
    • Categories Convex Optimization, Nonlinear Optimization Tags , ,

    Part I of this work [Gao25] establishes online scaled gradient methods (OSGM), a framework that utilizes online convex optimization to adapt stepsizes in gradient methods. This paper focuses on the practical aspects of OSGM. We leverage the OSGM framework to design new adaptive first-order methods and provide insights into their empirical behavior. The resulting method, … Read more

    On the convergence rate of the Douglas-Rachford splitting algorithm

  • Hadi Abbaszadehpeivasti
  • Moslem Zamani
    • Categories Complementarity and Variational Inequalities, Convex Optimization, Nonsmooth Optimization Tags , ,

    This work is concerned with the convergence rate analysis of the Dou- glas–Rachford splitting (DRS) method for finding a zero of the sum of two maximally monotone operators. We obtain an exact rate of convergence for the DRS algorithm and demonstrate its sharpness in the setting of convex feasibility problems. Further- more, we investigate the … Read more

    Alternating Iteratively Reweighted \(\ell_1\) and Subspace Newton Algorithms for Nonconvex Sparse Optimization

  • Xiangyu Yang
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , ,

    This paper presents a novel hybrid algorithm for minimizing the sum of a continuously differentiable loss function and a nonsmooth, possibly nonconvex, sparsity‑promoting regularizer. The proposed method adaptively switches between solving a reweighted \(\ell_1\)-regularized subproblem and performing an inexact subspace Newton step. The reweighted \(\ell_1\)-subproblem admits an efficient closed-form solution via the soft-thresholding operator, thereby … Read more

    Polyconvex double well functions

  • Didier Henrion
  • Martin Kružík
    • Categories Generalized Convexity/Monoticity, Systems governed by Differential Equations Optimization Tags ,

    We investigate polyconvexity of the double well function $f(X) := |X-X_1|^2|X-X_2|^2$ for given matrices $X_1, X_2 \in \R^{n \times n}$. Such functions are fundamental in the modeling of phase transitions in materials, but their non-convex nature presents challenges for the analysis of variational problems. We prove that $f$ is polyconvex if and only if the … Read more

    A First Order Algorithm on an Optimization Problem with Improved Convergence when Problem is Convex

  • Chee-Khian Sim
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , ,

    We propose a first order algorithm, a modified version of FISTA, to solve an optimization problem with an objective function that is the sum of a possibly nonconvex function, with Lipschitz continuous gradient, and a convex function which can be nonsmooth. The algorithm is shown to have an iteration complexity of \(\mathcal{O}(\epsilon^{-2})\) to find an … Read more

    Online Convex Optimization with Heavy Tails: Old Algorithms, New Regrets, and Applications

  • Zijian Liu
    • Categories Convex Optimization, Nonsmooth Optimization Tags , ,

    In Online Convex Optimization (OCO), when the stochastic gradient has a finite variance, many algorithms provably work and guarantee a sublinear regret. However, limited results are known if the gradient estimate has a heavy tail, i.e., the stochastic gradient only admits a finite \(\mathsf{p}\)-th central moment for some \(\mathsf{p}\in\left(1,2\right]\). Motivated by it, this work examines … Read more

    The Optimal Smoothings of Sublinear Functions and Convex Cones

  • Thabo Samakhoana
  • Benjamin Grimmer
    • Categories Convex and Nonsmooth Optimization Tags , , , ,

    This paper considers the problem of smoothing convex functions and sets, seeking the nearest smooth convex function or set to a given one. For convex cones and sublinear functions, a full characterization of the set of all optimal smoothings is given. These provide if and only if characterizations of the set of optimal smoothings for … Read more

    On the Resolution of Ties in Fair Convex Allocation Problems

  • Bismark Singh
    • Categories Convex and Nonsmooth Optimization, Convex Optimization Tags , , , ,

    We study the emergence of indistinguishable, but structurally distinct, allocation outcomes in convex resource allocation models. Such outcomes occur when different users receive proportionally identical allocations despite differences in initial conditions, eligibility sets, or priority weights. We formalize this behavior and analyze the structural conditions under which it arises, with a focus on fairness-oriented objectives. … Read more