Nonlinear Optimization

On an Elliptical Trust-Region Procedure for Ill-Posed Nonlinear Least-Squares Problems

  • Stefania Bellavia
  • Elisa Riccietti
    • Categories Nonlinear Systems and Least-Squares Tags , , , ,

    In this paper we address the stable numerical solution of ill-posed nonlinear least-squares problems with small residual. We propose an elliptical trust-region reformulation of a Levenberg-Marquardt procedure. Thanks to an appropriate choice of the trust-region radius, the proposed procedure guarantees an automatic choice of the free regularization parameters that, together with a suitable stopping criterion, … Read more

    Inexact Successive Quadratic Approximation for Regularized Optimization

  • Stephen Wright
  • Ching-pei Lee
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , ,

    Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration complexity focus on the special case of proximal gradient method, or accelerated variants thereof. There have been only a few studies of … Read more

    A Branch-and-Bound based Algorithm for Nonconvex Multiobjective Optimization

  • Gabriele Eichfelder
  • Julia Niebling
    • Categories Global Optimization, Multi-Criteria Optimization, Nonlinear Optimization Tags , , , ,

    A new branch-and-bound based algorithm for smooth nonconvex multiobjective optimization problems with convex constraints is presented. The algorithm computes an $(\varepsilon,\delta)$-approximation of all globally optimal solutions. We introduce the algorithm which uses selection rules, discarding and termination tests. The discarding tests are the most important aspect, as they examine in different ways whether a box … Read more

    ADMM for Multiaffine Constrained Optimization

  • Frank E. Curtis
  • Wenbo Gao
  • Donald Goldfarb
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    We propose an expansion of the scope of the alternating direction method of multipliers (ADMM). Specifically, we show that ADMM, when employed to solve problems with multiaffine constraints that satisfy certain easily verifiable assumptions, converges to the set of constrained stationary points if the penalty parameter in the augmented Lagrangian is sufficiently large. When the … Read more

    A Riemannian Conjugate Gradient Algorithm with Implicit Vector Transport for Optimization on the Stiefel Manifold

  • Hugo Lara
  • Harry Oviedo
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

    In this paper, a reliable curvilinear search algorithm for solving optimization problems over the Stiefel manifold is presented. This method is inspired by the conjugate gradient method, with the purpose of obtain a new direction search that guarantees descent of the objective function in each iteration. The merit of this algorithm lies in the fact … Read more

    Algorithms and Convergence Results of Projection Methods for Inconsistent Feasibility Problems: A Review

  • Yair Censor
  • Maroun Zaknoon
    • Categories Convex Optimization, Nonlinear Optimization Tags , ,

    The convex feasibility problem (CFP) is to find a feasible point in the intersection of finitely many convex and closed sets. If the intersection is empty then the CFP is inconsistent and a feasible point does not exist. However, algorithmic research of inconsistent CFPs exists and is mainly focused on two directions. One is oriented … Read more

    Cubic Regularization Method based on Mixed Factorizations for Unconstrained Minimization

  • Ernesto G. Birgin
  • José Mario Martínez
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , , ,

    Newton’s method for unconstrained optimization, subject to proper regularization or special trust-region procedures, finds first-order stationary points with precision $\varepsilon$ employing, at most, $O(\varepsilon^{-3/2})$ functional and derivative evaluations. However, the computer work per iteration of the best-known implementations may need several factorizations per iteration or may use rather expensive matrix decompositions. In this paper, we … Read more

    Concise Complexity Analyses for Trust-Region Methods

  • Frank E. Curtis
  • Daniel P. Robinson
  • Zachary Lubberts
    • Categories Unconstrained Optimization Tags , , , , , ,

    Concise complexity analyses are presented for simple trust region algorithms for solving unconstrained optimization problems. In contrast to a traditional trust region algorithm, the algorithms considered in this paper require certain control over the choice of trust region radius after any successful iteration. The analyses highlight the essential algorithm components required to obtain certain complexity … Read more

    Local attractors of newton-type methods for constrained equations and complementarity problems with nonisolated solutions

  • Andreas Fischer
  • Alexey F. Izmailov
  • Mikhail V. Solodov
    • Categories Complementarity and Variational Inequalities, Nonlinear Systems and Least-Squares

    For constrained equations with nonisolated solutions, we show that if the equation mapping is 2-regular at a given solution with respect to a direction in the null space of the Jacobian, and this direction is interior feasible, then there is an associated domain of starting points from which a family of Newton-type methods is well-de ned … Read more

    Combinatorial Integral Approximation Decompositions for Mixed-Integer Optimal Control

  • Sebastian Sager
  • Tobias Weber
  • Clemens Zeile
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming, Systems governed by Differential Equations Optimization Tags , , , , ,

    Solving mixed-integer nonlinear programs (MINLPs) is hard in theory and practice. Decomposing the nonlinear and the integer part seems promising from a computational point of view. In general, however, no bounds on the objective value gap can be guaranteed and iterative procedures with potentially many subproblems are necessary. The situation is different for mixed-integer optimal … Read more