A Newton-CG Algorithm with Complexity Guarantees for Smooth Unconstrained Optimization

  • Michael J. O'Neill
  • Clément W. Royer
  • Stephen Wright
    • Categories Unconstrained Optimization Tags , , ,

    We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton’s method and linear conjugate gradient, with explicit detection and use of negative curvature directions for the Hessian of the objective function. The algorithm tracks Newton-conjugate gradient procedures developed in the 1980s closely, but includes enhancements that allow worst-case complexity … Read more

    Outer Approximation for Integer Nonlinear Programs via Decision Diagrams

  • Danial Davarnia
  • Willem-Jan van Hoeve
    • Categories (Mixed) Integer Nonlinear Programming, Cutting Plane Approaches, Dynamic Programming Tags , , , ,

    As an alternative to traditional integer programming (IP), decision diagrams (DDs) provide a new solution technology for discrete problems based on their combinatorial structure and dynamic programming representation. While the literature mainly focuses on the competitive aspects of DDs as a stand-alone solver, we investigate their complementary role by studying IP techniques that can be … Read more

    Branch-Cut-and-Price for the Vehicle Routing Problem with Time Windows and Convex Node Costs

  • Qie He
  • Yongjia Song
  • Stefan Irnich
    • Categories Transportation Tags , , , ,

    Two critical yet frequently conflicting objectives for logistics and transportation service companies are improving customer satisfaction and reducing transportation cost. In particular, given a net- work of customer requests with preferred service times, it is very challenging to find vehicle routes and service schedules simultaneously that respect all operating constraints and minimize the total transportation … Read more

    A comparison of methods for traversing non-convex regions in optimization problems

  • Michael Bartholomew-Biggs
  • Salah Beddiaf
  • Bruce Christianson
    • Categories Unconstrained Optimization Tags , ,

    This paper considers again the well-known problem of dealing with non-convex regions during the minimization of a nonlinear function F(x) by Newton-like methods. The proposal made here involves a curvilinear search along an approximation to the continuous steepest descent path defined by the solution of the ODE dx/dt = -grad F(x). The algorithm we develop … Read more

    Exploring the Numerics of Branch-and-Cut for Mixed Integer Linear Optimization

  • Matthias Miltenberger
  • Ted K. Ralphs
  • Daniel E. Steffy
    • Categories (Mixed) Integer Linear Programming, Branch and Cut Algorithms, Optimization Software and Modeling Systems

    We investigate how the numerical properties of the LP relaxations evolve throughout the solution procedure in a solver employing the branch-and-cut algorithm. The long-term goal of this work is to determine whether the effect on the numerical conditioning of the LP relaxations resulting from the branching and cutting operations can be effectively predicted and whether … Read more

    An Improved Method of Total Variation Superiorization Applied to Reconstruction in Proton Computed Tomography

  • Yair Censor
  • Paniz Karbasi
  • Keith E. Schubert
  • Reinhard W. Schulte
  • Blake Schultze
    • Categories Biomedical Applications, Constrained Nonlinear Optimization, Convex Optimization Tags , , , , ,

    Previous work showed that total variation superiorization (TVS) improves reconstructed image quality in proton computed tomography (pCT). The structure of the TVS algorithm has evolved since then and this work investigated if this new algorithmic structure provides additional benefits to pCT image quality. Structural and parametric changes introduced to the original TVS algorithm included: (1) … Read more

    Revisiting the Hamiltonian p-median problem: a new formulation on directed graphs and a branch-and-cut algorithm

  • Tolga Bektas
  • Luís Gouveia
  • Daniel Santos
    • Categories (Mixed) Integer Linear Programming, Branch and Cut Algorithms, Transportation

    This paper studies the Hamiltonian p-median problem defined on a directed graph, which consists of finding p mutually disjoint circuits of minimum total cost, such that each node of the graph is included in one of the circuits. Earlier formulations are based on viewing the problem as one resulting from the intersection of two subproblems. … Read more

    SOS-Convex Lyapunov Functions and Stability of Difference Inclusions

  • Amir Ali Ahmadi
  • Raphaël Jungers
    • Categories Control Applications, Convex Optimization, Semi-definite Programming Tags , , , , ,

    We introduce the concept of sos-convex Lyapunov functions for stability analysis of both linear and nonlinear difference inclusions (also known as discrete-time switched systems). These are polynomial Lyapunov functions that have an algebraic certificate of convexity and that can be efficiently found via semidefinite programming. We prove that sos-convex Lyapunov functions are universal (i.e., necessary … Read more

    On Algebraic Proofs of Stability for Homogeneous Vector Fields

  • Amir Ali Ahmadi
  • Bachir El Khadir
    • Categories Control Applications, Semi-definite Programming, Systems governed by Differential Equations Optimization Tags , , ,

    We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sum of squares … Read more

    On stochastic auctions in risk-averse electricity markets with uncertain supply

  • Ryan Cory-Wright
  • Golbon Zakeri
    • Categories Finance and Economics, Game Theory, Stochastic Programming Tags , , ,

    This paper studies risk in a stochastic auction which facilitates the integration of renewable generation in electricity markets. We model market participants who are risk averse and reflect their risk aversion through coherent risk measures. We uncover a closed-form characterization of a risk-averse generator’s optimal pre-commitment behaviour for a given real-time policy, both with and … Read more