Quadratic Programming

Relaxations of KKT Conditions do not Strengthen Finite RLT and SDP-RLT Bounds for Nonconvex Quadratic Programs

  • E. Alper Yildirim
    • Categories Global Optimization, Linear, Cone and Semidefinite Programming, Quadratic Programming Tags , , , ,

    We consider linear and semidefinite programming relaxations of nonconvex quadratic programs given by the reformulation-linearization technique (RLT relaxation), and the Shor relaxation combined with the RLT relaxation (SDP-RLT relaxation). By incorporating the first-order optimality conditions, a quadratic program can be formulated as an optimization problem with complementarity constraints. We investigate the effect of incorporating optimality … Read more

    Tight Semidefinite Relaxations for Verifying Robustness of Neural Networks

  • Godai Azuma
  • Sunyoung Kim
  • Makoto Yamashita
    • Categories Optimization in Data Science, Quadratic Programming, Semi-definite Programming Tags , , , ,

    For verifying the safety of neural networks (NNs), Fazlyab et al. (2019) introduced a semidefinite programming (SDP) approach called DeepSDP. This formulation can be viewed as the dual of the SDP relaxation for a problem formulated as a quadratically constrained quadratic program (QCQP). While SDP relaxations of QCQPs generally provide approximate solutions with some gaps, … Read more

    On the Semidefinite Representability of Continuous Quadratic Submodular Minimization With Applications to Moment Problems

  • Samuel Burer
  • Karthik Natarajan
    • Categories Quadratic Programming, Robust Optimization, Semi-definite Programming Tags , ,

    We show that continuous quadratic submodular minimization with bounds is solvable in polynomial time using semidefinite programming, and we apply this result to two moment problems arising in distributionally robust optimization and the computation of covariance bounds. Accordingly, this research advances the ongoing study of continuous submodular minimization and opens new application areas therein. ArticleDownload … Read more

    qpBAMM: a parallelizable ADMM approach for block-structured quadratic programs

  • Michel Lahmann
  • Christian Kirches
  • Martin T. Köhler
    • Categories Convex Optimization, Quadratic Programming Tags , , ,

    Block-structured quadratic programs (QPs) frequently arise in the context of the direct approach to solving optimal control problems. For successful application of direct optimal control algorithms to many real-world problems it is paramount that these QPs can be solved efficiently and reliably. Besides interior-point methods and active-set methods, ADMM-based quadratic programming approaches have gained popularity. … Read more

    Spherical Support Vector Machine for Interval-Valued Data

  • Rui Malha
  • Paula Alexandra Amaral
    • Categories Data Science Algorithms, Nonlinear Optimization, Quadratic Programming Tags , ,

    In this work we propose a generalization of the Spherical Support Vector Machine method, in which the separator is a sphere, applied to Interval-valued data. This type of data belongs to a more general class, known as Symbolic Data, for which features are described by sets, intervals or histograms instead of classic arrays. This paradigm … Read more

    Constructing QCQP Instances Equivalent to Their SDP Relaxations

  • Masakazu Kojima
  • Naohiko Arima
  • Sunyoung Kim
    • Categories Global Optimization Applications, Quadratic Programming, Semi-definite Programming Tags , ,

    General quadratically constrained quadratic programs (QCQPs) are challenging to solve as they are known to be NP-hard. A popular approach to approximating QCQP solutions is to use semidefinite programming (SDP) relaxations. It is well-known that the optimal value η of the SDP relaxation problem bounds the optimal value ζ  of the QCQP from below, i.e., … Read more

    A class of diagonal quasi-Newton penalty decomposition algorithms for sparse bound-constrained nonconvex optimization

  • Ahmad Mousavi
  • Morteza Kimiaei
  • Saman Babaie-Kafaki
  • Vyacheslav Kungurtsev
    • Categories Bound-constrained Optimization, Constrained Nonlinear Optimization, Quadratic Programming Tags , , ,

    This paper discusses an improved quasi-Newton penalty decomposition algorithm for the cardinality bound-constrained optimization problems whose simple bounds on the variables are assumed to be finite. Until an approximate stationary point is found, this algorithm approximates the solutions of a sequence of penalty subproblems by a two-block decomposition scheme. This scheme finds an approximate solution … Read more

    Extended Triangle Inequalities for Nonconvex Box-Constrained Quadratic Programming

  • Kurt M. Anstreicher
  • Diane Puges
    • Categories Bound-constrained Optimization, Global Optimization Theory, Quadratic Programming Tags , ,

    Let Box_n = {x in R^n : 0<= x <= e }, and let QPB_n denote the convex hull of {(1, x’)'(1, x’) : x  in Box_n}. The quadratic programming problem min{x’Q x + q’x : x in Box_n} where Q is not positive semidefinite (PSD), is equivalent to a linear optimization problem over QPB_n … Read more

    Addressing Estimation Errors through Robust Portfolio Optimization

  • Gérard Cornuéjols
  • Ozgun Elci
  • Vrishabh Patil
    • Categories Convex Optimization, Quadratic Programming, Robust Optimization Tags , ,

    It is well known that the performance of the classical Markowitz model for portfolio optimization is extremely sensitive to estimation errors on the expected asset returns. Robust optimization mitigates this issue. We focus on ellipsoidal uncertainty sets around a point estimate of the expected asset returns. An important issue is the choice of the parameters … Read more

    An analytical lower bound for a class of minimizing quadratic integer optimization problems

  • Christian Schmitt
  • Bismark Singh
    • Categories Combinatorial Optimization, Integer Programming, Quadratic Programming Tags , , ,

    Lower bounds on minimization problems are essential for convergence of both branching-based and iterative solution methods for optimization problems. They are also required for evaluating the quality of feasible solutions by providing conservative optimality gaps. We provide an analytical lower bound for a class of quadratic optimization problems with binary decision variables. In contrast to … Read more