Nonlinear Optimization

Safeguarded augmented Lagrangian algorithms with scaled stopping criterion for the subproblems

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

    At each iteration of the Safeguarded Augmented Lagrangian algorithm Algencan, a bound-constrained subproblem consisting of the minimization of the Powell-Hestenes-Rockafellar augmented Lagrangian function is considered, for which a minimizer with tolerance tending to zero is sought. More precisely, a point that satisfies a subproblem first-order necessary optimality condition with tolerance tending to zero is required. … Read more

    Fixed point continuation algorithm with extrapolation for Schatten p-quasi-norm regularized matrix optimization problems

  • Peng Dingtao
    • Categories Nonsmooth Optimization, Unconstrained Optimization Tags , , , ,

    In this paper, we consider a general low-rank matrix optimization problem which is modeled by a general Schatten p-quasi-norm (${\rm 0<p<1}$) regularized matrix optimization. For this nonconvex nonsmooth and non-Lipschitz matrix optimization problem, based on the matrix p-thresholding operator, we first propose a fixed point continuation algorithm with extrapolation (FPCAe) for solving it. Secondly, we … Read more

    DC programming approach for solving a class of bilevel partial facility interdiction problems

  • Pham Thi Hoai
    • Categories (Mixed) Integer Linear Programming, Applications - OR and Management Sciences, Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags

    We propose a new approach based DC programming for fnding a solution of the partial facility interdiction problem that belongs to the class of bilevel programming. This model was frst considered in the work of Aksen et al. [1] with a heuristic algorithm named multi-start simplex search (MSS). However, because of the big number of … Read more

    M-stationarity of Local Minimizers of MPCCs and Convergence of NCP-based Methods

  • Kexin Wang
  • Lorenz T. Biegler
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , ,

    This paper focuses on solving mathematical programs with complementarity constraints (MPCCs) without assuming MPCC-LICQ or lower level strict complementarity at a solution. We show that a local minimizer of an MPCC is “piecewise M-stationary” un- der MPCC-GCQ; furthermore, every weakly stationary point of an MPCC is B-stationary if MPCC-ACQ holds. For the Bounding Algorithm proposed … Read more

    Bi-level multi-criteria optimization to include linear energy transfer into proton treatment planning

  • Katrin Teichert
  • Mara Schubert
    • Categories Biomedical Applications, Multi-Criteria Optimization, Nonlinear Optimization Tags , , , ,

    In proton therapy treatment planning, the aim is to ensure tumor control while sparing the various surrounding risk structures. The biological effect of the irradiation depends on both physical dose and linear energy transfer (LET). In order to include LET alongside physical dose in plan creation, we propose to formulate the proton treatment planning problem … Read more

    Trajectory Optimization of Unmanned Aerial Vehicles in the Electromagnetic Environment

  • Alain Zemkoho
  • Jörg Fliege
    • Categories Control Applications, Nonlinear Optimization, Telecommunications

    We consider a type of routing problems common in defence and security, in which we control a fleet of unmanned aerial vehicles (UAVs) that have to reach one or more target locations without being detected by an adversary. Detection can be carried out by a variety of sensors (radio receivers, cameras, personnel, etc) placed by … Read more

    Continuous exact relaxation and alternating proximal gradient algorithm for partial sparse and partial group sparse optimization problems

  • Peng Dingtao
    • Categories Nonsmooth Optimization, Unconstrained Optimization Tags

    In this paper, we consider a partial sparse and partial group sparse optimization problem, where the loss function is a continuously differentiable function (possibly nonconvex), and the penalty term consists of two parts associated with sparsity and group sparsity. The first part is the $\ell_0$ norm of ${\bf x}$, the second part is the $\ell_{2,0}$ … Read more

    Zeroth-order Riemannian Averaging Stochastic Approximation Algorithms

  • Jiaxiang Li
  • Krishnakumar Balasubramanian
  • Shiqian Ma
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Stochastic Programming

    We present Zeroth-order Riemannian Averaging Stochastic Approximation (\texttt{Zo-RASA}) algorithms for stochastic optimization on Riemannian manifolds. We show that \texttt{Zo-RASA} achieves optimal sample complexities for generating $\epsilon$-approximation first-order stationary solutions using only one-sample or constant-order batches in each iteration. Our approach employs Riemannian moving-average stochastic gradient estimators, and a novel Riemannian-Lyapunov analysis technique for convergence analysis. … Read more

    On Tractable Convex Relaxations of Standard Quadratic Optimization Problems under Sparsity Constraints

  • Immanuel M. Bomze
  • Bo Peng
  • Yuzhou Qiu
  • E. Alper Yildirim
    • Categories (Mixed) Integer Nonlinear Programming, Quadratic Programming, Semi-definite Programming Tags , , , ,

    Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex relaxations of sparse StQPs arising from a mixed-binary quadratic formulation, namely, the linear optimization relaxation given by the reformulation-linearization technique, … Read more