Convex and Nonsmooth Optimization

Stochastic Compositional Gradient Descent: Algorithms for Minimizing Compositions of Expected-Value Functions

  • Ethan Fang
  • Mengdi Wang
  • Han Liu
    • Categories Convex Optimization, Statistics, Stochastic Programming Tags , , , ,

    Classical stochastic gradient methods are well suited for minimizing expected-value objective functions. However, they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected-value functions, i.e., problems of the form $\min_x \E_v\[f_v\big(\E_w [g_w(x)]\big) \]$. In order to solve this stochastic composition problem, we propose a class … Read more

    On the Information-Adaptive Variants of the ADMM: an Iteration Complexity Perspective

  • Xiang Gao
  • Bo Jiang
  • Shuzhong Zhang
    • Categories Convex Optimization Tags , , , ,

    Designing algorithms for an optimization model often amounts to maintaining a balance between the degree of information to request from the model on the one hand, and the computational speed to expect on the other hand. Naturally, the more information is available, the faster one can expect the algorithm to converge. The popular algorithm of … Read more

    Directional H”older metric subregularity and application to tangent cones

  • Nguyen Huu Tron
  • Huynh Van Ngai
  • Phan Nhat Tinh
    • Categories Convex and Nonsmooth Optimization Tags , , , , ,

    In this work, we study directional versions of the H\”olderian/Lipschitzian metric subregularity of multifunctions. Firstly, we establish variational characterizations of the H\”olderian/Lipschitzian directional metric subregularity by means of the strong slopes and next of mixed tangency-coderivative objects . By product, we give second-order conditions for the directional Lipschitzian metric subregularity and for the directional metric … Read more

    HIPAD – A Hybrid Interior-Point Alternating Direction algorithm for knowledge-based SVM and feature selection

  • Ioannis G. Akrotirianakis
  • Zhiwei (Tony) Qin
  • Amit Chakraborty
  • Xiaocheng Tang
    • Categories Convex and Nonsmooth Optimization, Data-Mining, Quadratic Programming Tags , , , ,

    We consider classification tasks in the regime of scarce labeled training data in high dimensional feature space, where specific expert knowledge is also available. We propose a new hybrid optimization algorithm that solves the elastic-net support vector machine (SVM) through an alternating direction method of multipliers in the first phase, followed by an interior-point method … Read more

    Nonlinear local error bounds via a change of metric

  • Dominique Azé
  • Jean-Noël Corvellec
    • Categories Convex and Nonsmooth Optimization, Infinite Dimensional Optimization, Nonlinear Optimization Tags , ,

    In this work, we improve the approach of Corvellec-Motreanu to nonlinear error bounds for lowersemicontinuous functions on complete metric spaces, an approach consisting in reducing the nonlinear case to the linear one through a change of metric. This improvement is basically a technical one, and allows dealing with local error bounds in an appropriate way. … Read more

    Conditional Gradient Sliding for Convex Optimization

  • Guanghui Lan
  • Yi Zhou
    • Categories Convex and Nonsmooth Optimization Tags , , , ,

    In this paper, we present a new conditional gradient type method for convex optimization by utilizing a linear optimization (LO) oracle to minimize a series of linear functions over the feasible set. Different from the classic conditional gradient method, the conditional gradient sliding (CGS) algorithm developed herein can skip the computation of gradients from time … 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

    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