Geovani Nunes Grapiglia

Complexity of quadratic penalty methods with adaptive accuracy under a PL condition for the constraints

  • Florentin Goyens
  • Geovani Nunes Grapiglia
    • Categories Constrained Nonlinear Optimization Tags , , ,

    We study the quadratic penalty method (QPM) for smooth nonconvex optimization problems with equality constraints. Assuming the constraint violation satisfies the PL condition near the feasible set, we derive sharper worst-case complexity bounds for obtaining approximate first-order KKT points. When the objective and constraints are twice continuously differentiable, we show that QPM equipped with a … Read more

    A Finite-Difference Trust-Region Method for Convexly Constrained Smooth Optimization

  • Dânâ Davar
  • Geovani Nunes Grapiglia
    • Categories Bound-constrained Optimization, Nonlinear Optimization Tags , , , ,

    We propose a derivative-free trust-region method based on finite-difference gradient approximations for smooth optimization problems with convex constraints. The proposed method does not require computing an approximate stationarity measure. For nonconvex problems, we establish a worst-case complexity bound of \(\mathcal{O}\!\left(n\left(\frac{L}{\sigma}\epsilon\right)^{-2}\right)\) function evaluations for the method to reach an \(\left(\frac{L}{\sigma}\epsilon\right)\)-approximate stationary point, where \(n\) is the … Read more

    On the Complexity of Lower-Order Implementations of Higher-Order Methods

  • Nikita Doikov
  • Geovani Nunes Grapiglia
    • Categories Nonlinear Optimization Tags , , , ,

    In this work, we propose a method for minimizing non-convex functions with Lipschitz continuous \(p\)th-order derivatives, starting from \(p \geq 1\). The method, however, only requires derivative information up to order \((p-1)\), since the \(p\)th-order derivatives are approximated via finite differences. To ensure oracle efficiency, instead of computing finite-difference approximations at every iteration, we reuse … Read more

    A Riemannian AdaGrad-Norm Method

  • Glaydston Bento
  • Geovani Nunes Grapiglia
  • Mauricio Louzeiro
  • Daoping Zhang
    • Categories Generalized Convexity/Monoticity, Global Optimization Applications, Nonlinear Optimization Tags , , , ,

    We propose a manifold AdaGrad-Norm method (\textsc{MAdaGrad}), which extends the norm version of AdaGrad (AdaGrad-Norm) to Riemannian optimization. In contrast to line-search schemes, which may require several exponential map computations per iteration, \textsc{MAdaGrad} requires only one. Assuming the objective function $f$ has Lipschitz continuous Riemannian gradient, we show that the method requires at most $\mathcal{O}(\varepsilon^{-2})$ … Read more

    Active-set Newton-MR methods for nonconvex optimization problems with bound constraints

  • Ernesto G. Birgin
  • Geovani Nunes Grapiglia
  • Diaulas S. Marcondes
    • Categories Bound-constrained Optimization, Nonlinear Optimization, Unconstrained Optimization Tags , , , ,

    This paper presents active-set methods for minimizing nonconvex twice-continuously differentiable functions subject to bound constraints. Within the faces of the feasible set, we employ descent methods with Armijo line search, utilizing approximated Newton directions obtained through the Minimum Residual (MINRES) method. To escape the faces, we investigate the use of the Spectral Projected Gradient (SPG) … Read more

    Sub-sampled Trust-Region Methods with Deterministic Worst-Case Complexity Guarantees

  • Max L. N. Gonçalves
  • Geovani Nunes Grapiglia
    • Categories Unconstrained Optimization Tags , , ,

    In this paper, we develop and analyze sub-sampled trust-region methods for solving finite-sum optimization problems. These methods employ subsampling strategies to approximate the gradient and Hessian of the objective function, significantly reducing the overall computational cost. We propose a novel adaptive procedure for deterministically adjusting the sample size used for gradient (or gradient and Hessian) … Read more

    Fully Adaptive Zeroth-Order Method for Minimizing Functions with Compressible Gradients

  • Geovani Nunes Grapiglia
  • Daniel McKenzie
    • Categories Unconstrained Optimization Tags , ,

    We propose an adaptive zeroth-order method for minimizing differentiable functions with L-Lipschitz continuous gradients. The method is designed to take advantage of the eventual compressibility of the gradient of the objective function, but it does not require knowledge of the approximate sparsity level s or the Lipschitz constant L of the gradient. We show that … Read more

    Universal nonmonotone line search method for nonconvex multiobjective optimization problems with convex constraints

  • Maria Eduarda Pinheiro
  • Geovani Nunes Grapiglia
    • Categories Multi-Criteria Optimization Tags

    In this work we propose a general nonmonotone line-search method for nonconvex multi-objective optimization problems with convex constraints. At the \(k\)th iteration, the degree of nonmonotonicity is controlled by a vector \(\nu_{k}\) with nonnegative components. Different choices for \(\nu_{k}\) lead to different nonmonotone step-size rules. Assuming that the sequence \(\left\{\nu_{k}\right\}_{k\geq 0}\) is summable, and that … Read more

    TRFD: A Derivative-Free Trust-Region Method Based on Finite Differences for Composite Nonsmooth Optimization

  • Dânâ Davar
  • Geovani Nunes Grapiglia
    • Categories Nonsmooth Optimization Tags , , ,

    In this work we present TRFD, a derivative-free trust-region method based on finite differences for minimizing composite functions of the form \(f(x)=h(F(x))\), where \(F\) is a black-box function assumed to have a Lipschitz continuous Jacobian, and \(h\) is a known convex Lipschitz function, possibly nonsmooth. The method approximates the Jacobian of \(F\) via forward finite … Read more

    Worst-case evaluation complexity of a derivative-free quadratic regularization method

  • Geovani Nunes Grapiglia
    • Categories Nonlinear Optimization Tags , , ,

    This short paper presents a derivative-free quadratic regularization method for unconstrained minimization of a smooth function with Lipschitz continuous gradient. At each iteration, trial points are computed by minimizing a quadratic regularization of a local model of the objective function. The models are based on forward finite-difference gradient approximations. By using a suitable acceptance condition … Read more