Convex and Nonsmooth Optimization

The Convexity Zoo: A Taxonomy of Function Classes in Optimization

  • Abbas Khademi
    • Categories Constrained Nonlinear Optimization, Generalized Convexity/Monoticity, Nonlinear Optimization Tags , , , ,

    The tractability of optimization problems depends critically on structural properties of the objective function. Convexity guarantees global optimality of local solutions and enables polynomial-time algorithms under mild assumptions, but many problems arising in modern applications—particularly in machine learning—are inherently nonconvex. Remarkably, a large class of such problems remains amenable to efficient optimization due to additional … Read more

    New Location Science Models with Applications to UAV-Based Disaster Relief

  • Sina Kazemdehbashi
  • Boris S. Mordukhovich
  • Yanchao Liu
    • Categories Convex and Nonsmooth Optimization, Facility Planning and Design Tags , , , ,

    Natural and human-made disasters can cause severe devastation and claim thousands of lives worldwide. Therefore, developing efficient methods for disaster response and management is a critical task for relief teams. One of the most essential components of effective response is the rapid collection of information about affected areas, damages, and victims. More data translates into … Read more

    A Proximal-Gradient Method for Solving Regularized Optimization Problems with General Constraints

  • Daniel P. Robinson
  • Frank E. Curtis
  • Xiaoyi Qu
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , ,

    We propose, analyze, and test a proximal-gradient method for solving regularized optimization problems with general constraints. The method employs a decomposition strategy to compute trial steps and uses a merit function to determine step acceptance or rejection. Under various assumptions, we establish a worst-case iteration complexity result, prove that limit points are first-order KKT points, … Read more

    Primal-dual resampling for solution validation in convex stochastic programming

  • Yi Chu
  • Susan R. Hunter
  • Raghu Pasupathy
    • Categories Convex Optimization, Optimization in Data Science, Stochastic Programming Tags , ,

    Suppose we wish to determine the quality of a candidate solution to a convex stochastic program in which the objective function is a statistical functional parameterized by the decision variable and known deterministic constraints may be present. Inspired by stopping criteria in primal-dual and interior-point methods, we develop cancellation theorems that characterize the convergence of … Read more

    Facial reduction for nice (and non-nice) convex programs

  • Ying Lin
  • Tianxiang Liu
  • Bruno F. Lourenço
    • Categories Convex and Nonsmooth Optimization, Convex Optimization Tags , , ,

    Consider the primal problem of minimizing the sum of two closed proper convex functions \(f\) and \(g\). If the relative interiors of the domains of \(f\) and \(g\) intersect, then the primal problem and its corresponding Fenchel dual satisfy strong duality. When these relative interiors fail to intersect, pathologies and numerical difficulties may occur. In … Read more

    Stability analysis of parameterized models relative to nonconvex constraints

  • Zijian Shi
  • Miantao Chao
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization Tags , , , , ,

    For solution mappings of parameterized models (such as optimization problems, variational inequalities, and generalized equations), standard stability inevitably fails as the parameter approaches the boundary of the feasible domain. One remedy is relative stability restricted to a constraint set (e.g., the feasible domain), which is our focus in this paper. We establish generalized differentiation criteria … Read more

    An Elementary Proof of the Near Optimality of LogSumExp Smoothing

  • Thabo Samakhoana
  • Benjamin Grimmer
    • Categories Convex Optimization, Data Science Algorithms, Nonsmooth Optimization Tags , , , ,

    We consider the design of smoothings of the (coordinate-wise) max function in $\mathbb{R}^d$ in the infinity norm. The LogSumExp function $f(x)=\ln(\sum^d_i\exp(x_i))$ provides a classical smoothing, differing from the max function in value by at most $\ln(d)$. We provide an elementary construction of a lower bound, establishing that every overestimating smoothing of the max function must … Read more

    Robust optimality for nonsmooth mathematical programs with equilibrium constraints under data uncertainty

  • Vivek Laha
    • Categories Applications - OR and Management Sciences, Constrained Nonlinear Optimization, Convex and Nonsmooth Optimization, Convex Optimization, Nonlinear Optimization, Nonsmooth Optimization, Robust Optimization Tags

    We develop a unified framework for robust nonsmooth optimization problems with equilibrium constraints (UNMPEC). As a foundation, we study a robust nonsmooth nonlinear program with uncertainty in both the objective function and the inequality constraints (UNP). Using Clarke subdifferentials, we establish Karush–Kuhn–Tucker (KKT)–type necessary optimality conditions under an extended no–nonzero–abnormal–multiplier constraint qualification (ENNAMCQ). When the … Read more

    New Results on the Polyak Stepsize: Tight Convergence Analysis and Universal Function Classes

  • Chang He
  • Wenzhi Gao
  • Bo Jiang
  • Madeleine Udell
  • Shuzhong Zhang
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags ,

    In this paper, we revisit a classical adaptive stepsize strategy for gradient descent: the Polyak stepsize (PolyakGD), originally proposed in Polyak (1969). We study the convergence behavior of PolyakGD from two perspectives: tight worst-case analysis and universality across function classes. As our first main result, we establish the tightness of the known convergence rates of … Read more

    Subsampled cubic regularization method with distinct sample sizes for function, gradient, and Hessian

  • Max L. N. Gonçalves
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    We develop and study a subsampled cubic regularization method for finite-sum composite optimization problems, in which the function, gradient, and Hessian are estimated using possibly different sample sizes. By allowing each quantity to have its own sampling strategy, the proposed method offers greater flexibility to control the accuracy of the model components and to better … Read more