Constrained Nonlinear Optimization

Primal-Dual Optimization Algorithms over Riemannian Manifolds: an Iteration Complexity Analysis

  • Shiqian Ma
  • Junyu Zhang
  • Shuzhong Zhang
    • Categories Constrained Nonlinear Optimization Tags , , , ,

    In this paper we study nonconvex and nonsmooth multi-block optimization over Riemannian manifolds with coupled linear constraints. Such optimization problems naturally arise from machine learning, statistical learning, compressive sensing, image processing, and tensor PCA, among others. We develop an ADMM-like primal-dual approach based on decoupled solvable subroutines such as linearized proximal mappings. First, we introduce … Read more

    Globally Solving the Trust Region Subproblem Using Simple First-Order Methods

  • Amir Beck
  • Yakov Vaisbourd
    • Categories Constrained Nonlinear Optimization Tags , , , ,

    We consider the trust region subproblem which is given by a minimization of a quadratic, not necessarily convex, function over the Euclidean ball. Based on the well-known second-order necessary and sufficient optimality conditions for this problem, we present two sufficient optimality conditions defined solely in terms of the primal variables. Each of these conditions corresponds … Read more

    Convergence Analysis of Processes with Valiant Projection Operators in Hilbert Space

  • Yair Censor
  • Rafiq Mansour
    • Categories Constrained Nonlinear Optimization, Convex Optimization

    Convex feasibility problems require to find a point in the intersection of a finite family of convex sets. We propose to solve such problems by performing set-enlargements and applying a new kind of projection operators called valiant projectors. A valiant projector onto a convex set implements a special relaxation strategy, proposed by Goffin in 1971, … Read more

    A sequential optimality condition related to the quasinormality constraint qualification and its algorithmic consequences

  • Roberto Andreani
  • N. S. Fazzio
  • María Laura Schuverdt
  • Leonardo D. Secchin
    • Categories Constrained Nonlinear Optimization Tags , , ,

    In the present paper, we prove that the augmented Lagrangian method converges to KKT points under the quasinormality constraint qualification, which is associated with the external penalty theory. For this purpose, a new sequential optimality condition for smooth constrained optimization, called PAKKT, is defined. The new condition takes into account the sign of the dual … Read more

    Portfolio Optimization with Entropic Value-at-Risk

  • Ahmadi-Javid Amir
  • Fallah-Tafti Malihe
    • Categories Constrained Nonlinear Optimization, Finance and Economics, Stochastic Programming

    The entropic value-at-risk (EVaR) is a new coherent risk measure, which is an upper bound for both the value-at-risk (VaR) and conditional value-at-risk (CVaR). As important properties, the EVaR is strongly monotone over its domain and strictly monotone over a broad sub-domain including all continuous distributions, while well-known monotone risk measures, such as VaR and … Read more

    Randomized Similar Triangles Method: A Unifying Framework for Accelerated Randomized Optimization Methods (Coordinate Descent, Directional Search, Derivative-Free Method)

  • Pavel Dvurechensky
  • Alexander Gasnikov
  • Alexander Tiurin
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , , , , , , ,

    In this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as value, partial or directional derivatives of the objective function. We introduce a unifying framework, which allows to construct different types of accelerated randomized methods for such problems and to prove convergence rate theorems for them. … Read more

    New algorithms for discrete vector optimization based on the Graef-Younes method and cone-monotone sorting functions

  • Christian Günther
  • Nicolae Popovici
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization

    The well-known Jahn-Graef-Younes algorithm, proposed by Jahn in 2006, generates all minimal elements of a finite set with respect to an ordering cone. It consists of two Graef-Younes procedures, namely the forward iteration, which eliminates a part of the non-minimal elements, followed by the backward iteration, which is applied to the reduced set generated by … Read more

    Optimality conditions for minimizers at infinity in polynomial programming

  • Tiên-So'n Pham
    • Categories Constrained Nonlinear Optimization, Global Optimization Tags , , , , , , ,

    In this paper we study necessary optimality conditions for the optimization problem $$\textrm{infimum}f_0(x) \quad \textrm{ subject to } \quad x \in S,$$ where $f_0 \colon \mathbb{R}^n \rightarrow \mathbb{R}$ is a polynomial function and $S \subset \mathbb{R}^n$ is a set defined by polynomial inequalities. Assume that the problem is bounded below and has the Mangasarian–Fromovitz property … Read more

    Two New Weak Constraint Qualifications for Mathematical Programs with Equilibrium Constraints and Applications

  • Alberto Ramos
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags ,

    We introduce two new weaker Constraint Qualifications (CQs) for Mathematical Programs with Equilibrium (or Complementarity) Constraints, MPEC for short. One of them is a tailored version of the Constant Rank of Subspace Component (CRSC) and the other is a relaxed version of the MPEC-No Nonzero Abnormal Multiplier Constraint Qualification (MPEC-NNAMCQ). Both incorporate the exact set … Read more

    Vector Transport-Free SVRG with General Retraction for Riemannian Optimization: Complexity Analysis and Practical Implementation

  • Bo Jiang
  • Shiqian Ma
  • Anthony Man-Cho So
  • Shuzhong Zhang
    • Categories Constrained Nonlinear Optimization, Stochastic Programming Tags , , , ,

    In this paper, we propose a vector transport-free stochastic variance reduced gradient (SVRG) method with general retraction for empirical risk minimization over Riemannian manifold. Existing SVRG methods on manifold usually consider a specific retraction operation, and involve additional computational costs such as parallel transport or vector transport. The vector transport-free SVRG with general retraction we … Read more