Nonsmooth Optimization

Sequential Threshold Control in Descent Splitting Methods for Decomposable Optimization Problems

  • Igor V. Konnov
    • Categories Constrained Nonlinear Optimization, Data-Mining, Nonsmooth Optimization

    We suggest a modification of the descent splitting methods for decomposable composite optimization problems, which maintains the basic convergence properties, but enables one to reduce the computational expenses per iteration and to provide computations in a distributed manner. It consists in making coordinate-wise steps together with a special threshold control. CitationKazan Federal University, Kazan 420008, … Read more

    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

    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

    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

    Normally admissible stratifications and calculation of normal cones to a finite union of polyhedral sets

  • Lukáš Adam
  • Michal Cervinka
  • Miroslav Pistek
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , ,

    This paper considers computation of Fr\’echet and limiting normal cones to a finite union of polyhedra. To this aim, we introduce a new concept of normally admissible stratification which is convenient for calculations of such cones and provide its basic properties. We further derive formulas for the above mentioned cones and compare our approach to … Read more

    Differential properties of Euclidean projection onto power cone

  • Le Thi Khanh Hien
    • Categories Nonsmooth Optimization

    In this paper, we study differential properties of Euclidean projection onto the power cone $K^{(p,q)}_n=\{(x,y,z)\in \mathbb{R}_+\times \mathbb{R}_+\times \mathbb{R}^n,\norm{z} \leq x^p y^q\}$, where $0< p,q < 1, p+q=1$. Projections onto certain power cones are examples of semismooth but non-strongly-semismooth projection onto a convex cone. CitationDivision of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological ... Read more

    An improved algorithm for L2-Lp minimization problem

  • Dongdong Ge
  • He Rongchuan
  • He Simai
    • Categories Global Optimization Theory, Nonsmooth Optimization, Unconstrained Optimization

    In this paper we consider a class of non-Lipschitz and non-convex minimization problems which generalize the L2−Lp minimization problem. We propose an iterative algorithm that decides the next iteration based on the local convexity/concavity/sparsity of its current position. We show that our algorithm finds an epsilon-KKT point within O(log(1/epsilon)) iterations. The same result is also … Read more

    The Principle of Hamilton for Mechanical Systems with Impacts and Unilateral Constraints

  • Kerim Yunt
    • Categories Complementarity and Variational Inequalities, Nonsmooth Optimization Tags , , ,

    An action integral is presented for Hamiltonian mechanics in canonical form with unilateral constraints and/or impacts. The transition conditions on generalized impulses and the energy are presented as variational inequalities, which are obtained by the application of discontinuous transversality conditions. The energetical behavior for elastic, plastic and blocking type impacts are analyzed. A general impact … Read more

    Local Linear Convergence of Forward–Backward under Partial Smoothness

  • Jalal Fadili
  • Jingwei Liang
  • Gabriel Peyré
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , ,

    In this paper, we consider the Forward–Backward proximal splitting algorithm to minimize the sum of two proper closed convex functions, one of which having a Lipschitz–continuous gradient and the other being partly smooth relatively to an active manifold $\mathcal{M}$. We propose a unified framework in which we show that the Forward–Backward (i) correctly identifies the … Read more