Second-Order Cone Programming

An easily computable upper bound on the Hoffman constant for homogeneous inequality systems

  • Javier F. Peña
    • Categories Linear Programming, Polyhedra, Second-Order Cone Programming Tags , ,

    \(\)Let $A\in \mathbb{R}^{m\times n}\setminus \{0\}$ and $P:=\{x:Ax\le 0\}$. This paper provides a procedure to compute an upper bound on the following {\em homogeneous Hoffman constant} \[ H_0(A) := \sup_{u\in \mathbb{R}^n \setminus P} \frac{\text{dist}(u,P)}{\text{dist}(Au, \mathbb{R}^m_-)}. \] In sharp contrast to the intractability of computing more general Hoffman constants, the procedure described in this paper is entirely … Read more

    Weighted Geometric Mean, Minimum Mediated Set, and Optimal Second-Order Cone Representation

  • Jie Wang
    • Categories Cone Programming, Convex Optimization, Second-Order Cone Programming Tags , , , ,

    We study optimal second-order cone representations for weighted geometric means, which turns out to be closely related to minimum mediated sets. Several lower bounds and upper bounds on the size of optimal second-order cone representations are proved. In the case of bivariate weighted geometric means (equivalently, one dimensional mediated sets), we are able to prove … Read more

    A Quadratically Convergent Sequential Programming Method for Second-Order Cone Programs Capable of Warm Starts

  • Xinyi Luo
  • Andreas Wächter
    • Categories Constrained Nonlinear Optimization, Second-Order Cone Programming Tags , ,

    We propose a new method for linear second-order cone programs. It is based on the sequential quadratic programming framework for nonlinear programming. In contrast to interior point methods, it can capitalize on the warm-start capabilities of active-set quadratic programming subproblem solvers and achieve a local quadratic rate of convergence. In order to overcome the non-differentiability … Read more

    Hub Network Design Problem with Capacity, Congestion and Stochastic Demand Considerations

  • Vedat Bayram
  • Barış Yıldız
  • M. Saleh Farham
    • Categories Second-Order Cone Programming, Stochastic Programming, Transportation Tags , , , , , , ,

    We introduce the hub network design problem with congestion, capacity, and stochastic demand considerations (HNDC), which generalizes the classical hub location problem in several directions. In particular, we extend state-of-the-art by integrating capacity acquisition decision and congestion cost effect into the problem and allowing dynamic routing for origin-destination pairs. Connecting strategic and operational level decisions, … Read more

    Global Convergence of Algorithms Under Constant Rank Conditions for Nonlinear Second-Order Cone Programming

  • Roberto Andreani
  • Gabriel Haeser
  • Leonardo M. Mito
  • Héctor Ramírez
  • Thiago P. Silveira
    • Categories Constrained Nonlinear Optimization, Second-Order Cone Programming Tags , , , ,

    In [R. Andreani, G. Haeser, L. M. Mito, H. Ramírez C., Weak notions of nondegeneracy in nonlinear semidefinite programming, arXiv:2012.14810, 2020] the classical notion of nondegeneracy (or transversality) and Robinson’s constraint qualification have been revisited in the context of nonlinear semidefinite programming exploiting the structure of the problem, namely, its eigendecomposition. This allows formulating the … Read more

    Applications of stochastic mixed-integer second-order cone optimization

  • Baha Alzalg
  • Hadjer Alioui
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Second-Order Cone Programming Tags , , ,

    Second-order cone programming problems are a tractable subclass of convex optimization problems and there are known polynomial algorithms for solving them. Stochastic second-order cone programming problems have also been studied in the past decade and efficient algorithms for solving them exist. A new class of interest to optimization community and practitioners is the mixed-integer version … Read more

    Bishop-Phelps cones given by an equation in Banach spaces

  • Xuan Duc Ha Truong
  • Johannes Jahn
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization, Second-Order Cone Programming Tags , , , ,

    In this work, we study Bishop-Phelps cones (briefly, BP cones) given by an equation in Banach spaces. Due to the special form, these cones enjoy interesting properties. We show that nontrivial BP cones given by an equation form a “large family” in some sense in any Banach space and they can be used to characterize … Read more

    First- and second-order optimality conditions for second-order cone and semidefinite programming under a constant rank condition

  • Roberto Andreani
  • Gabriel Haeser
  • Héctor Ramírez
  • Leonardo M. Mito
  • Thiago P. Siqueira
    • Categories Constrained Nonlinear Optimization, Second-Order Cone Programming, Semi-definite Programming

    The well known constant rank constraint qualification [Math. Program. Study 21:110–126, 1984] introduced by Janin for nonlinear programming has been recently extended to a conic context by exploiting the eigenvector structure of the problem. In this paper we propose a more general and geometric approach for defining a new extension of this condition to the … Read more

    The Promise of EV-Aware Multi-Period OPF Problem: Cost and Emission Benefits

  • Sezen Ece Kayacık
  • Burak Kocuk
  • Tuğçe Yüksel
    • Categories Applications - OR and Management Sciences, Global Optimization, Second-Order Cone Programming Tags , ,

    In this paper, we study the Multi-Period Optimal Power Flow problem (MOPF) with electric vehicles (EV) under emission considerations. We integrate three different real-world datasets: household electricity consumption, marginal emission factors, and EV driving profiles. We present a systematic solution approach based on second-order cone programming to find globally optimal solutions for the resulting nonconvex … Read more

    Sequential constant rank constraint qualifications for nonlinear semidefinite programming with applications

  • Roberto Andreani
  • Gabriel Haeser
  • Héctor Ramírez
  • Leonardo M. Mito
    • Categories Constrained Nonlinear Optimization, Second-Order Cone Programming, Semi-definite Programming Tags , , ,

    We present new constraint qualification conditions for nonlinear semidefinite programming that extend some of the constant rank-type conditions from nonlinear programming. As an application of these conditions, we provide a unified global convergence proof of a class of algorithms to stationary points without assuming neither uniqueness of the Lagrange multiplier nor boundedness of the Lagrange … Read more