Constrained Nonlinear Optimization

Optimizing hypergraph-based polynomials modeling job-occupancy in queueing with redundancy scheduling

  • Daniel Brosch
  • Monique Laurent
  • Andries Steenkamp
    • Categories Constrained Nonlinear Optimization Tags , , ,

    We investigate two classes of multivariate polynomials with variables indexed by the edges of a uniform hypergraph and coefficients depending on certain patterns of union of edges. These polynomials arise naturally to model job-occupancy in some queuing problems with redundancy scheduling policy. The question, posed by Cardinaels, Borstand van Leeuwaarden (arXiv:2005.14566, 2020), is to decide … Read more

    Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy

  • Didier Henrion
  • Martin Kružík
  • Marek Tyburec
  • Jan Zeman
    • Categories Civil and Environmental Engineering, Constrained Nonlinear Optimization, Semi-definite Programming Tags , , , , ,

    The design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is … Read more

    Regret Minimization in Stochastic Non-Convex Learning via a Proximal-Gradient Approach

  • Volkan Cevher
  • Nadav Hallak
  • Panayotis Mertikopoulos
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization

    Motivated by applications in machine learning and operations research, we study regret minimization with stochastic first-order oracle feedback in online constrained, and possibly non-smooth, non-convex problems. In this setting, the minimization of external regret is beyond reach, so we focus on a local regret measures defined via a proximal-gradient residual mapping. To achieve no (local) … Read more

    Inexact Variable Metric Method for Convex-Constrained Optimization Problems

  • Douglas S. Gonçalves
  • Max L. N. Gonçalves
  • Tiago Menezes
    • Categories Constrained Nonlinear Optimization

    This paper is concerned with the inexact variable metric method for solving convex-constrained optimization problems. At each iteration of this method, the search direction is obtained by inexactly minimizing a strictly convex quadratic function over the closed convex feasible set. Here, we propose a new inexactness criterion for the search direction subproblems. Under mild assumptions, … Read more

    Largest small polygons: A sequential convex optimization approach

  • Christian Bingane
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Quadratic Programming Tags , , , , ,

    A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m\ge 7$. Finding the largest small $n$-gon for a given number $n\ge 3$ can be formulated as a nonconvex quadratically constrained quadratic optimization problem. We propose to solve this problem with a … Read more

    A Two-level ADMM Algorithm for AC OPF with Convergence Guarantees

  • Xu Andy Sun
  • Kaizhao Sun
    • Categories Constrained Nonlinear Optimization, Network Optimization, Parallel Algorithms Tags , , ,

    This paper proposes a two-level distributed algorithmic framework for solving the AC optimal power flow (OPF) problem with convergence guarantees. The presence of highly nonconvex constraints in OPF poses significant challenges to distributed algorithms based on the alternating direction method of multipliers (ADMM). In particular, convergence is not provably guaranteed for nonconvex network optimization problems … Read more

    Iteration-complexity of a proximal augmented Lagrangian method for solving nonconvex composite optimization problems with nonlinear convex constraints

  • Weiwei Kong
  • Jefferson Gonçalves Melo
  • Renato D.C. Monteiro
    • Categories Constrained Nonlinear Optimization, Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , ,

    This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving smooth nonconvex composite optimization problems with nonlinear K-convex constraints, i.e., the constraints are convex with respect to the order given by a closed convex cone K. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite … Read more

    On the best achievable quality of limit points of augmented Lagrangian schemes

  • Roberto Andreani
  • Gabriel Haeser
  • Alberto Ramos
  • Leonardo M. Mito
  • Leonardo D. Secchin
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , ,

    The optimization literature is vast in papers dealing with improvements on the global convergence of augmented Lagrangian schemes. Usually, the results are based on weak constraint qualifications, or, more recently, on sequential optimality conditions obtained via penalization techniques. In this paper we propose a somewhat different approach, in the sense that the algorithm itself is … Read more

    Exact Penalty Function for L21 Norm Minimization over the Stiefel Manifold

  • Nachuan Xiao
  • Xin Liu
  • Ya-xiang Yuan
    • Categories Constrained Nonlinear Optimization, Statistics Tags , , ,

    L21 norm minimization with orthogonality constraints, feasible region of which is called Stiefel manifold, has wide applications in statistics and data science. The state-of-the-art approaches adopt proximal gradient technique on either Stiefel manifold or its tangent spaces. The consequent subproblem does not have closed-form solution and hence requires an iterative procedure to solve which is … Read more

    A Nonmonotone Matrix-Free Algorithm for Nonlinear Equality-Constrained Least-Squares Problems

  • Elhoucine Bergou
  • Youssef Diouane
  • Vyacheslav Kungurtsev
  • Clément W. Royer
    • Categories Constrained Nonlinear Optimization, Nonlinear Systems and Least-Squares

    Least squares form one of the most prominent classes of optimization problems, with numerous applications in scientific computing and data fitting. When such formulations aim at modeling complex systems, the optimization process must account for nonlinear dynamics by incorporating constraints. In addition, these systems often incorporate a large number of variables, which increases the difficulty … Read more