Nonlinear Optimization

A primal-dual modified log-barrier method for inequality constrained nonlinear optimization

  • Joshua D. Griffin
  • Riadh Omheni
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , ,

    We present a primal-dual modified log-barrier algorithm to solve inequality constrained nonlinear optimization problems. Basically, the algorithm is a Newton-like method applied to a perturbation of the optimality system that follows from a reformulation of the initial problem by introducing a modified log-barrier function to handle inequality constraints. The algorithm uses an outer/inner iteration scheme … Read more

    Strong Evaluation Complexity Bounds for Arbitrary-Order Optimization of Nonconvex Nonsmooth Composite Functions

  • Coralia Cartis
  • Nicholas I. M. Gould
  • Philippe L. Toint
    • Categories Bound-constrained Optimization, Constrained Nonlinear Optimization, Unconstrained Optimization Tags , ,

    We introduce the concept of strong high-order approximate minimizers for nonconvex optimization problems. These apply in both standard smooth and composite non-smooth settings, and additionally allow convex or inexpensive constraints. An adaptive regularization algorithm is then proposed to find such approximate minimizers. Under suitable Lipschitz continuity assumptions, whenever the feasible set is convex, it is … Read more

    Weakly Homogeneous Optimization Problems

  • Vu Trung Hieu
    • Categories Nonlinear Optimization Tags , , , , , ,

    This paper investigates a new class of optimization problems whose objective functions are weakly homogeneous relative to the constrain sets. Two sufficient conditions for nonemptiness and boundedness of solution sets are established. We also study linear parametric problems and upper semincontinuity of the solution map. ArticleDownload View PDF

    Nonconvex Constrained Optimization by a Filtering Branch and Bound

  • Gabriele Eichfelder
  • Kathrin Klamroth
  • Julia Niebling
    • Categories Constrained Nonlinear Optimization, Global Optimization Theory, Multi-Criteria Optimization

    A major difficulty in optimization with nonconvex constraints is to find feasible solutions. As simple examples show, the alphaBB-algorithm for single-objective optimization may fail to compute feasible solutions even though this algorithm is a popular method in global optimization. In this work, we introduce a filtering approach motivated by a multiobjective reformulation of the constrained … Read more

    Exploiting problem structure in derivative free optimization

  • Margherita Porcelli
  • Philippe L. Toint
    • Categories Nonlinear Optimization, Optimization Software and Modeling Systems Tags , , ,

    A structured version of derivative-free random pattern search optimization algorithms is introduced which is able to exploit coordinate partially separable structure (typically associated with sparsity) often present in unconstrained and bound-constrained optimization problems. This technique improves performance by orders of magnitude and makes it possible to solve large problems that otherwise are totally intractable by … Read more

    On the convexification of constrained quadratic optimization problems with indicator variables

  • Andrés Gómez
  • Simge Küçükyavuz
  • Linchuan Wei
    • Categories (Mixed) Integer Nonlinear Programming, Quadratic Programming Tags , , , ,

    Motivated by modern regression applications, in this paper, we study the convexification of quadratic optimization problems with indicator variables and combinatorial constraints on the indicators. Unlike most of the previous work on convexification of sparse regression problems, we simultaneously consider the nonlinear objective, indicator variables, and combinatorial constraints. We prove that for a separable quadratic … Read more

    Proximal splitting algorithms: Relax them all!

  • Laurent Condat
  • Andres Contreras
  • Akira Hirabayashi
  • Daichi Kitahara
    • Categories Nonlinear Optimization, Nonsmooth Optimization

    Convex optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, proximal splitting algorithms are particularly adequate: they consist of simple operations, by handling the terms in the objective function separately. We present several existing proximal splitting algorithms and we derive new ones, within a unified framework, which consists … Read more

    A Regularized Smoothing Method for Fully Parameterized Convex Problems with Applications to Convex and Nonconvex Two-Stage Stochastic Programming

  • Pedro Borges de Melo
  • Claudia Sagastizábal
  • Mikhail V. Solodov
    • Categories Nonlinear Optimization, Nonsmooth Optimization, Stochastic Programming Tags , , , , ,

    We present an approach to regularize and approximate solution mappings of parametric convex optimization problems that combines interior penalty (log-barrier) solutions with Tikhonov regularization. Because the regularized mappings are single-valued and smooth under reasonable conditions, they can be used to build a computationally practical smoothing for the associated optimal value function. The value function in … Read more

    On the acceleration of the Barzilai-Borwein method

  • Yu-Hong Dai
  • Yakui Huang
  • Xin-Wei Liu
  • Hongchao Zhang
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization

    The Barzilai-Borwein (BB) gradient method is efficient for solving large-scale unconstrained problems to the modest accuracy and has a great advantage of being easily extended to solve a wide class of constrained optimization problems. In this paper, we propose a new stepsize to accelerate the BB method by requiring finite termination for minimizing two-dimensional strongly … Read more