A primal-dual interior-point method based on various selections of displacement step for second-order cone programming

  • Baha Alzalg
    • Categories Second-Order Cone Programming Tags , , , ,

    In this paper, a primal-dual interior-point method equipped with various selections of the displacement step are derived for solving second-order cone programming problems. We first establish the existence and uniqueness of the optimal solution of the corresponding perturbed problem and then demonstrate its convergence to the optimal solution of the original problem. Next, we present … Read more

    Analyzing Random Permutations for Cyclic Coordinate Descent

  • Stephen Wright
  • Ching-pei Lee
    • Categories Unconstrained Optimization Tags , , ,

    We consider coordinate descent methods on convex quadratic problems, in which exact line searches are performed at each iteration. (This algorithm is identical to Gauss-Seidel on the equivalent symmetric positive definite linear system.) We describe a class of convex quadratic problems for which the random-permutations version of cyclic coordinate descent (RPCD) outperforms the standard cyclic … Read more

    A New Voltage Stability-Constrained Optimal Power Flow Model: Sufficient Condition, SOCP Representation, and Relaxation

  • Bai Cui
  • Xu Andy Sun
    • Categories Applications - Science and Engineering Tags , , ,

    A simple characterization of the solvability of power flow equations is of great importance in the monitoring, control, and protection of power systems. In this paper, we introduce a sufficient condition for power flow Jacobian nonsingularity. We show that this condition is second-order conic representable when load powers are fixed. Through the incorporation of the … Read more

    Regularized Nonlinear Acceleration

  • Francis Bach
  • Alexandre d'Aspremont
  • Damien Scieur
    • Categories Convex Optimization

    We describe a convergence acceleration technique for generic optimization problems. Our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method. The weights in this average are computed via a simple linear system, whose solution can be updated online. This acceleration scheme runs in parallel to the … Read more

    Sharpness, Restart and Acceleration.

  • Alexandre d'Aspremont
  • Vincent Roulet
    • Categories Convex Optimization

    The Lojasievicz inequality shows that sharpness bounds on the minimum of convex optimization problems hold almost generically. Here, we show that sharpness directly controls the performance of restart schemes. The constants quantifying sharpness are of course unobservable, but we show that optimal restart strategies are fairly robust, and searching for the best scheme only increases … Read more

    Integration Methods and Accelerated Optimization Algorithms

  • Francis Bach
  • Alexandre d'Aspremont
  • Vincent Roulet
  • Damien Scieur
    • Categories Convex Optimization

    We show that accelerated optimization methods can be seen as particular instances of multi-step integration schemes from numerical analysis, applied to the gradient flow equation. In comparison with recent advances in this vein, the differential equation considered here is the basic gradient flow and we show that multi-step schemes allow integration of this differential equation … Read more

    The Adaptive Sampling Gradient Method: Optimizing Smooth Functions with an Inexact Oracle

  • Fatemeh Hashemi
  • Raghu Pasupathy
  • Michael Taaffe
    • Categories Nonlinear Optimization, Optimization of Simulated Systems, Stochastic Programming Tags , ,

    Consider settings such as stochastic optimization where a smooth objective function $f$ is unknown but can be estimated with an \emph{inexact oracle} such as quasi-Monte Carlo (QMC) or numerical quadrature. The inexact oracle is assumed to yield function estimates having error that decays with increasing oracle effort. For solving such problems, we present the Adaptive … Read more

    A Stochastic MPC Framework for Stationary Battery Systems

  • Kirk Drees
  • Victor M. Zavala
  • Mohammad ElBsat
  • Matthew Ellis
  • Ranjeet Kumar
  • Michael Wenzel
    • Categories Control Applications, Linear Programming, Stochastic Programming

    We present a stochastic model predictive control (MPC) framework to determine real-time commitments in energy and frequency regulation markets for a stationary battery system while simultaneously mitigating long-term demand charges for an attached load. The framework solves a two-stage stochastic program over a receding horizon that maximizes the expected profit and that factors in uncertainty … Read more

    Globally Solving a Class of Optimal Power Flow Problems in Radial Networks by Tree Reduction

  • Yuval Beck
  • Amir Beck
  • Alex Shtof
  • Yoash Levron
  • Luba Tetruashvili
    • Categories Nonlinear Optimization

    We devise an algorithm for finding the global optimal solution of the so-called optimal power flow problem (OPF) for a class of power networks with a tree topology, also called radial networks, for which an efficient and reliable algorithm was not previously known. The algorithm we present is called the tree reduction/expansion method, and is … Read more

    Two New Weak Constraint Qualifications for Mathematical Programs with Equilibrium Constraints and Applications

  • Alberto Ramos
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags ,

    We introduce two new weaker Constraint Qualifications (CQs) for Mathematical Programs with Equilibrium (or Complementarity) Constraints, MPEC for short. One of them is a tailored version of the Constant Rank of Subspace Component (CRSC) and the other is a relaxed version of the MPEC-No Nonzero Abnormal Multiplier Constraint Qualification (MPEC-NNAMCQ). Both incorporate the exact set … Read more