Complementarity and Variational Inequalities

Solving Binary-Constrained Mixed Complementarity Problems Using Continuous Reformulations

  • Steven A. Gabriel
  • Martin Schmidt
  • Marina Leal
    • Categories (Mixed) Integer Nonlinear Programming, Complementarity and Variational Inequalities, Transportation Tags , , ,

    Mixed complementarity problems are of great importance in practice since they appear in various fields of applications like energy markets, optimal stopping, or traffic equilibrium problems. However, they are also very challenging due to their inherent, nonconvex structure. In addition, recent applications require the incorporation of integrality constraints. Since complementarity problems often model some kind … Read more

    A hybrid projection-proximal point algorithm for solving nonmonotone variational inequality problems

  • Lei Ming
  • He Yiran
    • Categories Complementarity and Variational Inequalities

    A hybrid projection-proximal point algorithm is proposed for variational inequality problems. Though the usual proximal point method and its variants require that the mapping involved be monotone, at least pseudomonotone, we assume only that the so-called Minty variational inequality has a solution, in order to ensure the global convergence. This assumption is less stringent than … Read more

    Dimension in Polynomial Variational Inequalities

  • Vu Trung Hieu
    • Categories Complementarity and Variational Inequalities Tags , , , ,

    The aim of the paper is twofold. Firstly, by using the constant rank level set theorem from differential geometry, we establish sharp upper bounds for the dimensions of the solution sets of polynomial variational inequalities under mild conditions. Secondly, a classification of polynomial variational inequalities based on dimensions of their solution sets is introduced and … Read more

    Equilibrium selection for multi-portfolio optimization

  • Lorenzo Lampariello
  • Christoph Neumann
  • Simone Sagratella
  • Oliver Stein
  • Jacopo Maria Ricci
    • Categories Complementarity and Variational Inequalities, Finance and Economics Tags , , , ,

    This paper studies a Nash game arising in portfolio optimization. We introduce a new general multi-portfolio model and state sufficient conditions for the monotonicity of the underlying Nash game. This property allows us to treat the problem numerically and, for the case of nonunique equilibria, to solve hierarchical problems of equilibrium selection. We also give … Read more

    Gaddum’s test for symmetric cones

  • Michael Orlitzky
    • Categories Complementarity and Variational Inequalities, Game Theory, Linear, Cone and Semidefinite Programming Tags , , , ,

    A real symmetric matrix “A” is copositive if the inner product if Ax and x is nonnegative for all x in the nonnegative orthant. Copositive programming has attracted a lot of attention since Burer showed that hard nonconvex problems can be formulated as completely-positive programs. Alas, the power of copositive programming is offset by its … Read more

    Understanding Limitation of Two Symmetrized Orders by Worst-case Complexity

  • Ruoyu Sun
  • Zhisheng Xiao
  • Peijun Xiao
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , ,

    It was recently found that the standard version of multi-block cyclic ADMM diverges. Interestingly, Gaussian Back Substitution ADMM (GBS-ADMM) and symmetric Gauss-Seidel ADMM (sGS-ADMM) do not have the divergence issue. Therefore, it seems that symmetrization can improve the performance of the classical cyclic order. In another recent work, cyclic CD (Coordinate Descent) was shown to … Read more

    Gamma-Robust Linear Complementarity Problems with Ellipsoidal Uncertainty Sets

  • Vanessa Krebs
  • Martin Schmidt
  • Michael Müller
    • Categories Complementarity and Variational Inequalities, Robust Optimization, Transportation Tags , , ,

    We study uncertain linear complementarity problems (LCPs), i.e., problems in which the LCP vector q or the LCP matrix M may contain uncertain parameters. To this end, we use the concept of Gamma-robust optimization applied to the gap function formulation of the LCP. Thus, this work builds upon [16]. There, we studied Gamma-robustified LCPs for … Read more

    Dynamic Optimization with Complementarity Constraints: Smoothing for Direct Shooting

  • Lorenz T. Biegler
  • Adrian Caspari
  • Alexander Mitsos
  • Pascal Schäfer
  • Lukas Lüken
  • Adel Mhamdi
  • Yannic Vaupel
    • Categories Complementarity and Variational Inequalities, Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , ,

    We consider optimization of differential-algebraic equations (DAEs) with complementarity constraints (CCs) of algebraic state pairs. Formulating the CCs as smoothed nonlinear complementarity problem (NCP) functions leads to a smooth DAE, allowing for the solution in direct shooting. We provide sufficient conditions for well-posedness. Thus, we can prove that with the smoothing parameter going to zero, … Read more

    Continuous selections of solutions for locally Lipschitzian equations

  • A.V. Arutyunov
  • Alexey F. Izmailov
  • S.E. Zhukovskiy
    • Categories Complementarity and Variational Inequalities, Nonlinear Systems and Least-Squares, Nonsmooth Optimization

    This paper answers in affirmative the long-standing question of nonlinear analysis, concerning the existence of a continuous single-valued local selection of the right inverse to a locally Lipschitzian mapping. Moreover, we develop a much more general result, providing conditions for the existence of a continuous single-valued selection not only locally, but rather on any given … Read more

    Complementary problems with polynomial data

  • Canh Hung Nguyen
  • Tiên-So'n Pham
    • Categories Complementarity and Variational Inequalities Tags , , , , ,

    Given polynomial maps $f, g \colon \mathbb{R}^n \to \mathbb{R}^n,$ we consider the {\em polynomial complementary problem} of finding a vector $x \in \mathbb{R}^n$ such that \begin{equation*} f(x) \ \ge \ 0, \quad g(x) \ \ge \ 0, \quad \textrm{ and } \quad \langle f(x), g(x) \rangle \ = \ 0. \end{equation*} In this paper, we … Read more