Constrained Nonlinear Optimization

Iteration-complexity of a Rockafellar’s proximal method of multipliers for convex programming based on second-order approximations

  • M. Marques Alves
  • Renato D.C. Monteiro
  • Benar Fux Svaiter
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Global Optimization Tags , , , ,

    This paper studies the iteration-complexity of a new primal-dual algorithm based on Rockafellar’s proximal method of multipliers (PMM) for solving smooth convex programming problems with inequality constraints. In each step, either a step of Rockafellar’s PMM for a second-order model of the problem is computed or a relaxed extragradient step is performed. The resulting algorithm … Read more

    Numerical Solution of Linear-Quadratic Optimal Control Problems for Switching System

  • Deniz Hasan Gucoglu
  • Shahlar Meherrem
  • Samir Guliyev
    • Categories Applications - OR and Management Sciences, Constrained Nonlinear Optimization, Control Applications Tags , , ,

    In this paper we obtained an approach to the optimal switching control problem with unknown switching points which it is described in reference [1, 2]. In reference [1], the authors studied the Decomposition of Linear-Quadratic Optimal Control Problems for Two-Steps Systems. In [1], the authors assumed the switching point t1 is xed in the interval … Read more

    A Riemannian rank-adaptive method for low-rank optimization

  • Pierre-Antoine Absil
  • Kyle A. Gallivan
  • Wen Huang
  • Paul Van Dooren
  • Guifang Zhou
    • Categories Constrained Nonlinear Optimization, Data-Mining

    This paper presents an algorithm that solves optimization problems on a matrix manifold $\mathcal{M} \subseteq \mathbb{R}^{m \times n}$ with an additional rank inequality constraint. The algorithm resorts to well-known Riemannian optimization schemes on fixed-rank manifolds, combined with new mechanisms to increase or decrease the rank. The convergence of the algorithm is analyzed and a weighted … Read more

    A multiplier method with a class of penalty functions for convex programming

  • Romulo A Castillo
  • Clavel Quintana
  • Luiz C Matioli
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , ,

    We consider a class of augmented Lagrangian methods for solving convex programming problems with inequality constraints. This class involves a family of penalty functions and specific values of parameters $p,q,\tilde y \in R$ and $c>0$. The penalty family includes the classical modified barrier and the exponential function. The associated proximal method for solving the dual … Read more

    Improved pointwise iteration-complexity of a regularized ADMM and of a regularized non-Euclidean HPE framework

  • Max L. N. Gonçalves
  • Jefferson Gonçalves Melo
  • Renato D.C. Monteiro
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , , , , , ,

    This paper describes a regularized variant of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex programs. It is shown that the pointwise iteration-complexity of the new method is better than the corresponding one for the standard ADMM method and that, up to a logarithmic term, is identical to the ergodic iteration-complexity … Read more

    Regularized Interior Proximal Alternating Direction Method for Separable Convex Optimization Problems

  • Felipe Álvarez
  • Julio López
  • Natalia Ruiz
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , ,

    In this article we present a version of the proximal alternating direction method for a convex problem with linear constraints and a separable objective function, in which the standard quadratic regularizing term is replaced with an interior proximal metric for those variables that are required to satisfy some additional convex constraints. Moreover, the proposed method … Read more

    Local Nonglobal Minima for Solving Large Scale Extended Trust Region Subproblems

  • Maziar Salahi
  • Henry Wolkowicz
  • Akram Taati
    • Categories Constrained Nonlinear Optimization, Global Optimization, Optimization Software and Modeling Systems Tags , , ,

    We study large scale extended trust region subproblems (eTRS) i.e., the minimization of a general quadratic function subject to a norm constraint, known as the trust region subproblem (TRS) but with an additional linear inequality constraint. It is well known that strong duality holds for the TRS and that there are efficient algorithms for solving … Read more

    Strengthening the SDP Relaxation of AC Power Flows with Convex Envelopes, Bound Tightening, and Lifted Nonlinear Cuts

  • Carleton Coffrin
  • Hassan L. Hijazi
  • Pascal Van Hentenryck
    • Categories Applications - OR and Management Sciences, Constrained Nonlinear Optimization, Semi-definite Programming Tags , , ,

    This paper considers state-of-the-art convex relaxations for the AC power flow equations and introduces new valid cuts based on convex envelopes and lifted nonlinear constraints. These valid linear inequalities strengthen existing semidefinite and quadratic programming relaxations and dominate existing cuts proposed in the litterature. Together with model intersections and bound tightening, the new linear cuts … Read more

    Solutions of a constrained Hermitian matrix-valued function optimization problem with applications

  • Yongge Tian
    • Categories Constrained Nonlinear Optimization Tags

    Let $f(X) =\left( XC + D\right)M\left(XC + D \right)^{*} – G$ be a given nonlinear Hermitian matrix-valued function with $M = M^*$ and $G = G^*$, and assume that the variable matrix $X$ satisfies the consistent linear matrix equation $XA = B$. This paper shows how to characterize the semi-definiteness of $f(X)$ subject to all … Read more