Convex and Nonsmooth Optimization

Iteration Bounds for Finding the $\epsilonhBcStationary Points for Structured Nonconvex Optimization

  • Bo Jiang
  • Shuzhong Zhang
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    In this paper we study proximal conditional-gradient (CG) and proximal gradient-projection type algorithms for a block-structured constrained nonconvex optimization model, which arises naturally from tensor data analysis. First, we introduce a new notion of $\epsilon$-stationarity, which is suitable for the structured problem under consideration. %, compared with other similar solution concepts. We then propose two … Read more

    Variational analysis and full stability of optimal solutions to constrained and minimax problems

  • Boris S. Mordukhovich
  • Ebrahim Sarabi
    • Categories Convex and Nonsmooth Optimization

    The main goal of this paper is to develop applications of advanced tools of first-order and second-order variational analysis and generalized differentiation to the fundamental notion of full stability of local minimizers of general classes of constrained optimization and minimax problems. In particular, we derive second-order characterizations of full stability and investigate its relationships with … Read more

    An induction theorem and nonlinear regularity models

  • Phan Q. Khanh
  • Alexander Y. Kruger
  • Nguyen H. Thao
    • Categories Convex and Nonsmooth Optimization Tags , ,

    A general nonlinear regularity model for a set-valued mapping $F:X\times\R_+\rightrightarrows Y$, where $X$ and $Y$ are metric spaces, is considered using special iteration procedures, going back to Banach, Schauder, Lusternik and Graves. Namely, we revise the \emph{induction theorem} from Khanh, \emph{J. Math. Anal. Appl.}, 118 (1986) and employ it to obtain basic estimates for studying … Read more

    The global weak sharp minima with explicit exponents in polynomial vector optimization problems

  • Tiên-So'n Pham
  • Xuan Duc Ha Truong
  • Jen-Chih Yao
    • Categories Convex and Nonsmooth Optimization, Multi-Criteria Optimization Tags , ,

    In this paper we discuss the global weak sharp minima property for vector optimization problems with polynomial data. Exploiting the imposed polynomial structure together with tools of variational analysis and a quantitative version of \L ojasiewicz’s gradient inequality due to D’Acunto and Kurdyka, we establish the H\”older type global weak sharp minima with explicitly calculated … Read more

    Convergence rate analysis of the forward-Douglas-Rachford splitting scheme

  • Damek Davis
    • Categories Convex and Nonsmooth Optimization Tags , , , , , , ,

    Operator splitting schemes are a class of powerful algorithms that solve complicated monotone inclusion and convex optimization problems that are built from many simpler pieces. They give rise to algorithms in which all simple pieces of the decomposition are processed individually. This leads to easily implementable and highly parallelizable or distributed algorithms, which often obtain … Read more

    Interior-point solver for convex separable block-angular problems

  • Jordi Castro
    • Categories Convex Optimization, Optimization Software and Modeling Systems

    Constraints matrices with block-angular structures are pervasive in Optimization. Interior-point methods have shown to be competitive for these structured problems by exploiting the linear algebra. One of these approaches solved the normal equations using sparse Cholesky factorizations for the block constraints, and a preconditioned conjugate gradient (PCG) for the linking constraints. The preconditioner is based … Read more

    A Parallel Line Search Subspace Correction Method for Composite Convex Optimization

  • Qian Dong
  • Xin Liu
  • Zaiwen Wen
  • Ya-xiang Yuan
    • Categories Convex and Nonsmooth Optimization, Parallel Algorithms Tags , , , ,

    In this paper, we investigate a parallel subspace correction framework for composite convex optimization. The variables are first divided into a few blocks based on certain rules. At each iteration, the algorithms solve a suitable subproblem on each block simultaneously, construct a search direction by combining their solutions on all blocks, then identify a new … Read more

    An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions

  • Radu Ioan Bot
  • Ernö Robert Csetnek
  • Szilard Csaba Laszlo
    • Categories Nonsmooth Optimization Tags , , , ,

    We propose a forward-backward proximal-type algorithm with inertial/memory effects for minimizing the sum of a nonsmooth function with a smooth one in the nonconvex setting. The sequence of iterates generated by the algorithm converges to a critical point of the objective function provided an appropriate regularization of the objective satisfies the Kurdyka-Lojasiewicz inequality, which is … Read more

    Weak and Strong Superiorization: Between Feasibility-Seeking and Minimization

  • Yair Censor
    • Categories Constrained Nonlinear Optimization, Convex Optimization

    We review the superiorization methodology, which can be thought of, in some cases, as lying between feasibility-seeking and constrained minimization. It is not quite trying to solve the full fledged constrained minimization problem; rather, the task is to find a feasible point which is superior (with respect to an objective function value) to one returned … Read more

    Douglas-Rachford splitting for nonconvex feasibility problems

  • Guoyin Li
  • Ting Kei Pong
    • Categories Nonsmooth Optimization

    We adapt the Douglas-Rachford (DR) splitting method to solve nonconvex feasibility problems by studying this method for a class of nonconvex optimization problem. While the convergence properties of the method for convex problems have been well studied, far less is known in the nonconvex setting. In this paper, for the direct adaptation of the method … Read more