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  • 1. Chen, Ming-Wen
    et al.
    Wang, Xin-Feng
    Wang, Fei
    Lin, Guo-Biao
    Wang, Zi-Zong
    The effect of interfacial kinetics on the morphological stability of a spherical particle2013In: Journal of Crystal Growth, ISSN 0022-0248, E-ISSN 1873-5002Article in journal (Refereed)
  • 2. Corless, Rob
    et al.
    Jeffrey, David
    Wang, Fei
    The asymptotic analysis of some interpolated nonlinear recurrence relations2014In: Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, 2014, p. 41-42Conference paper (Refereed)
  • 3. Fampa, Marcia
    et al.
    Luke, Daniela
    Wang, Fei
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Wolkowicz, Henry
    Parametric Convex Quadratic Relaxation of the Quadratic Knapsack Problem2019In: European Journal of Operational Research, ISSN 0377-2217, E-ISSN 1872-6860, Vol. 281, no 1, p. 36-49Article in journal (Refereed)
    Abstract [en]

    We consider a parametric convex quadratic programming (CQP) relaxation for the quadratic knapsack problem (QKP). This relaxation maintains partial quadratic information from the original QKP by perturbing the objective function to obtain a concave quadratic term. The nonconcave part generated by the perturbation is then linearized by a standard approach that lifts the problem to matrix space. We present a primal-dual interior point method to optimize the perturbation of the quadratic function, in a search for the tightest upper bound for the QKP. We prove that the same perturbation approach, when applied in the context of semidefinite programming (SDP) relaxations of the QKP, cannot improve the upper bound given by the corresponding linear SDP relaxation. The result also applies to more general integer quadratic problems. Finally, we propose new valid inequalities on the lifted matrix variable, derived from cover and knapsack inequalities for the QKP, and present separation problems to generate cuts for the current solution of the CQP relaxation. Our best bounds are obtained alternating between optimizing the parametric quadratic relaxation over the perturbation and applying cutting planes generated by the valid inequalities proposed.

  • 4.
    Forsgren, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Wang, Fei
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    On the existence of a short pivoting sequence for a linear program2019Manuscript (preprint) (Other academic)
  • 5. Ma, Shiqian
    et al.
    Wang, Fei
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Wei, Linchuan
    Wolkowicz, Henry
    Robust Principal Component Analysis using Facial Reduction2019In: Optimization and Engineering, ISSN 1389-4420, E-ISSN 1573-2924Article in journal (Refereed)
  • 6. Reid, Greg
    et al.
    Wang, Fei
    Wu, Wenyuan
    A note on geometric involutive bases for positive dimensional polynomial ideals and SDP methods2014In: Proceedings of the 2014 Symposium on Symbolic-Numeric Computation, 2014, p. 41-42Conference paper (Refereed)
  • 7.
    Sremac, Stefan
    et al.
    Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada.
    Wang, Fei
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Wolkowicz, Henrik
    Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada.
    Pettersson, Lucas
    KTH, School of Engineering Sciences (SCI), Physics.
    Noisy Euclidean Distance Matrix Completion with a Single Missing Node2019In: Journal of Global Optimization, ISSN 0925-5001, E-ISSN 1573-2916Article in journal (Refereed)
    Abstract [en]

    We present several solution techniques for the noisy single source localization problem, i.e. the Euclidean distance matrix completion problem with a single missing node to locate under noisy data. For the case that the sensor locations are fixed, we show that this problem is implicitly convex, and we provide a purification algorithm along with the SDP relaxation to solve it efficiently and accurately. For the case that the sensor locations are relaxed, we study a model based on facial reduction. We present several approaches to solve this problem efficiently, and we compare their performance with existing techniques in the literature. Our tools are semidefinite programming, Euclidean distance matrices, facial reduction, and the generalized trust region subproblem. We include extensive numerical tests.

  • 8.
    Wang, Fei
    Department of Applied Mathematics, University of Western Ontario, London, ON, Canada.
    Proof of a series solution for Euler's trinomial equation2016In: ACM Communications in Computer Algebra, ISSN 1932-2232, E-ISSN 1932-2240, Vol. 50, no 4, p. 136-144Article in journal (Refereed)
    Abstract [en]

    In 1779, Leonhard Euler published a paper about Lambert's transcendental equation in the symmetric form x(alpha) - x(beta) = (alpha -beta) vx(alpha+beta). In the paper, he studied the series solution of this equation and other results based on an assumption which was not proved in the paper. Euler's paper gave the first series expanion for the so-called Lambert W function. In this work, we briefly review Euler's results and give a proof to modern standards of rigor of the series solution of Lambert's transcendental equation.

  • 9.
    Wang, Fei
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Reid, Greg
    Wolkowicz, Henry
    An SDP-based method for the real radical ideal membership test2017In: Proceedings of the 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, Institute of Electrical and Electronics Engineers (IEEE), 2017, p. 86-93Conference paper (Refereed)
    Abstract [en]

    Let V be the set of real solutions of a system of multivariate polynomial equations with real coefficients. The real radical ideal (RRI) of V is the infinite set of multivariate polynomials that vanish on V. We give theoretical results that yield a finite step numerical algorithm for testing if a given polynomial is a member of this RRI. The paper exploits recent work that connects solution sets of such real polynomial systems with solution sets of semidefinite programming, SDP, problems involving moment matrices. We take advantage of an SDP technique called facial reduction. This technique regularizes our problem by projecting the feasible set onto the so-called minimal face. In addition, we use the Douglas-Rachford iterative approach which has advantages over traditional interior point methods for our application. If V has finitely many real solutions, then our method yields a finite set of polynomials in the form of a geometric involutive basis that are generators of the RRI and form an RRI membership test. In the case where the set V has real solution components of positive dimension, and given an input polynomial of degree d, our method can also decide RRI membership via a truncated geometric involutive basis of degree d. Examples are given to illustrate our approach and its advantages that remove multiplicities and sums of squares that cause illconditioning for real solutions of polynomial systems.

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