Nonlinear Optimization

Second-order convergence properties of trust-region methods using incomplete curvature information, with an application to multigrid optimization

  • Serge Gratton
  • Annick Sartenaer
  • Philippe L. Toint
    • Categories Optimization of Systems modeled by PDEs, Systems governed by Differential Equations Optimization, Unconstrained Optimization Tags , ,

    Convergence properties of trust-region methods for unconstrained nonconvex optimization is considered in the case where information on the objective function’s local curvature is incomplete, in the sense that it may be restricted to a fixed set of “test directions” and may not be available at every iteration. It is shown that convergence to local “weak” … Read more

    Continuous Optimization Methods for Structure Alignments

  • Roberto Andreani
  • Leandro Martínez
  • José Mario Martínez
  • Flavio Yano
    • Categories Basic Sciences Applications, Nonlinear Optimization Tags , , , ,

    Structural Alignment is an important tool for fold identification of proteins, structural screening on ligand databases, pharmacophore identification and other applications. In the general case, the optimization problem of superimposing two structures is nonsmooth and nonconvex, so that most popular methods are heuristic and do not employ derivative information. Usually, these methods do not admit … Read more

    The Rate of Convergence of the Augmented Lagrangian Method for Nonlinear Semidefinite Programming

  • Jie Sun
  • Defeng Sun
  • Liwei Zhang
    • Categories Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , ,

    We analyze the rate of local convergence of the augmented Lagrangian method for nonlinear semidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and certain variational analysis on the projection operator in the symmetric-matrix space. Without … Read more

    Kantorovich’s Majorants Principle for Newton’s Method

  • Orizon Pereira Ferreira
  • Benar Fux Svaiter
    • Categories Nonlinear Systems and Least-Squares Tags ,

    We prove Kantorovich’s theorem on Newton’s method using a convergence analysis which makes clear, with respect to Newton’s Method, the relationship of the majorant function and the non-linear operator under consideration. This approach enable us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate … Read more

    A Note on Sparse SOS and SDP Relaxations for Polynomial Optimization Problems over Symmetric Cones

  • Masakazu Kojima
  • Masakazu Muramatsu
    • Categories Constrained Nonlinear Optimization, Global Optimization, Semi-definite Programming Tags , , , , , ,

    This short note extends the sparse SOS (sum of squares) and SDP (semidefinite programming) relaxation proposed by Waki, Kim, Kojima and Muramatsu for normal POPs (polynomial optimization problems) to POPs over symmetric cones, and establishes its theoretical convergence based on the recent convergence result by Lasserre on the sparse SOS and SDP relaxation for normal … Read more

    On the behavior of the conjugate-gradient method on ill-conditioned problems

  • Anders Forsgren
    • Categories Quadratic Programming, Unconstrained Optimization Tags , ,

    We study the behavior of the conjugate-gradient method for solving a set of linear equations, where the matrix is symmetric and positive definite with one set of eigenvalues that are large and the remaining are small. We characterize the behavior of the residuals associated with the large eigenvalues throughout the iterations, and also characterize the … Read more

    An interior Newton-like method for nonnegative least-squares problems with degenerate solution

  • Stefania Bellavia
  • Benedetta Morini
  • Maria Macconi
    • Categories Nonlinear Optimization Tags , , , ,

    An interior point approach for medium and large nonnegative linear least-squares problems is proposed. Global and locally quadratic convergence is shown even if a degenerate solution is approached. Viable approaches for implementation are discussed and numerical results are provided. Citation Technical Report 1/2005, Dipartimento di Energetica ‘S. Stecco’, Universita di Firenze, Italia Article Download View … Read more

    On Self-Regulated Swarms, Societal Memory, Speed and Dynamics

  • Carlos Fernandes
  • Vitorino Ramos
  • Agostinho C. Rosa
    • Categories Control Applications, Nonlinear Optimization, Stochastic Approaches Tags , , , ,

    Wasps, bees, ants and termites all make effective use of their environment and resources by displaying collective “swarm” intelligence. Termite colonies – for instance – build nests with a complexity far beyond the comprehension of the individual termite, while ant colonies dynamically allocate labor to various vital tasks such as foraging or defense without any … Read more

    An efficient conjugate direction method with orthogonalization for large-scale quadratic optimization problems

  • Edouard Boudinov
  • Arkadiy Manevich
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , , , ,

    A new conjugate direction method is proposed, which is based on an orthogonalization procedure and does not make use of line searches for the conjugate vector set construction. This procedure prevents the set of conjugate vectors from degeneracy and eliminates high sensitivity to computation errors pertinent to methods of conjugate directions, resulting in an efficient … Read more

    Societal Implicit Memory and his Speed on Tracking Extrema over Dynamic Environments using Self-Regulatory Swarms

  • Carlos Fernandes
  • Vitorino Ramos
  • Agostinho C. Rosa
    • Categories Multidisciplinary Design Optimization, Nonlinear Optimization, Stochastic Approaches Tags , , , , ,

    In order to overcome difficult dynamic optimization and environment extrema tracking problems, we propose a Self-Regulated Swarm (SRS) algorithm which hybridizes the advantageous characteristics of Swarm Intelligence as the emergence of a societal environmental memory or cognitive map via collective pheromone laying in the landscape (properly balancing the exploration/exploitation nature of the search strategy), with … Read more