Convex Optimization

BBCPOP: A Sparse Doubly Nonnegative Relaxation of Polynomial Optimization Problems with Binary, Box and Complementarity Constraints

  • Naoki Ito
  • Sunyoung Kim
  • Masakazu Kojima
  • Kim-Chuan Toh
  • Akiko Takeda
    • Categories Convex Optimization, Global Optimization, Linear, Cone and Semidefinite Programming Tags , , , , , , ,

    The software package BBCPOP is a MATLAB implementation of a hierarchy of sparse doubly nonnegative (DNN) relaxations of a class of polynomial optimization (minimization) problems (POPs) with binary, box and complementarity (BBC) constraints. Given a POP in the class and a relaxation order, BBCPOP constructs a simple conic optimization problem (COP), which serves as a … Read more

    User Manual for BBCPOP: A Sparse Doubly Nonnegative Relaxation of Polynomial Optimization Problems with Binary, Box and Complementarity Constraints

  • Naoki Ito
  • Sunyoung Kim
  • Masakazu Kojima
  • Kim-Chuan Toh
  • Akiko Takeda
    • Categories Convex Optimization, Global Optimization, Linear, Cone and Semidefinite Programming Tags , , , ,

    BBCPOP proposed in [4] is a MATLAB implementation of a hierarchy of sparse doubly nonnegative (DNN) relaxations of a class of polynomial optimization (minimization) problems (POPs) with binary, box and complementarity constraints. Given a POP in the class and a relaxation order (or a hierarchy level), BBCPOP constructs a simple conic optimization problem (COP), which … 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

    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

    A Simple Nearly-Optimal Restart Scheme For Speeding-Up First-Order Methods

  • Benjamin Grimmer
  • James Renegar
    • Categories Convex Optimization, Nonsmooth Optimization Tags , ,

    We present a simple scheme for restarting first-order methods for convex optimization problems. Restarts are made based only on achieving specified decreases in objective values, the specified amounts being the same for all optimization problems. Unlike existing restart schemes, the scheme makes no attempt to learn parameter values characterizing the structure of an optimization problem, … Read more

    Algorithms and Convergence Results of Projection Methods for Inconsistent Feasibility Problems: A Review

  • Yair Censor
  • Maroun Zaknoon
    • Categories Convex Optimization, Nonlinear Optimization Tags , ,

    The convex feasibility problem (CFP) is to find a feasible point in the intersection of finitely many convex and closed sets. If the intersection is empty then the CFP is inconsistent and a feasible point does not exist. However, algorithmic research of inconsistent CFPs exists and is mainly focused on two directions. One is oriented … Read more

    Uniqueness of DRS as the 2 Operator Resolvent-Splitting and Impossibility of 3 Operator Resolvent-Splitting

  • Ernest K. Ryu
    • Categories Convex Optimization

    Given the success of Douglas-Rachford splitting (DRS), it is natural to ask whether DRS can be generalized. Are there are other 2 operator splittings? Can DRS be generalized to 3 operators? This work presents the answers: no and no. In a certain sense, DRS is the unique 2 operator resolvent-splitting, and generalizing DRS to 3 … Read more

    A Progressive Batching L-BFGS Method for Machine Learning

  • Raghu Bollapragada
  • Jorge Nocedal
  • Hao-Jun Shi
  • Ping Tak Peter Tang
  • Dheevatsa Mudigere
    • Categories Convex Optimization, Nonlinear Optimization, Stochastic Programming Tags , , ,

    The standard L-BFGS method relies on gradient approximations that are not dominated by noise, so that search directions are descent directions, the line search is reliable, and quasi-Newton updating yields useful quadratic models of the objective function. All of this appears to call for a full batch approach, but since small batch sizes give rise … Read more

    The condition of a function relative to a polytope

  • David H. Gutman
  • Javier F. Peña
    • Categories Convex and Nonsmooth Optimization, Convex Optimization Tags , ,

    The condition number of a smooth convex function, namely the ratio of its smoothness to strong convexity constants, is closely tied to fundamental properties of the function. In particular, the condition number of a quadratic convex function is precisely the square of the diameter-to-width ratio of a canonical ellipsoid associated to the function. Furthermore, the … Read more

    On self-concordant barriers for generalized power cones

  • Scott Roy
  • Lin Xiao
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming Tags , ,

    In the study of interior-point methods for nonsymmetric conic optimization and their applications, Nesterov introduced the power cone, together with a 4-self-concordant barrier for it. In his PhD thesis, Chares found an improved 3-self-concordant barrier for the power cone. In addition, he introduced the generalized power cone, and conjectured a nearly optimal self-concordant barrier for … Read more