Complementarity and Variational Inequalities

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

    Near-optimal Robust Bilevel Optimization

  • Miguel F. Anjos
  • Mathieu Besançon
  • Luce Brotcorne
    • Categories (Mixed) Integer Nonlinear Programming, Complementarity and Variational Inequalities, Robust Optimization Tags , , , , , ,

    Bilevel optimization studies problems where the optimal response to a second mathematical optimization problem is integrated in the constraints. Such structure arises in a variety of decision-making problems in areas such as market equilibria, policy design or product pricing. We introduce near-optimal robustness for bilevel problems, protecting the upper-level decision-maker from bounded rationality at the … Read more

    Single-Forward-Step Projective Splitting: Exploiting Cocoercivity

  • Jonathan Eckstein
  • Patrick R. Johnstone
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization Tags , ,

    This work describes a new variant of projective splitting for monotone inclusions, in which cocoercive operators can be processed with a single forward step per iteration. This result establishes a symmetry between projective splitting algorithms, the classical forward backward splitting method (FB), and Tseng’s forward-backward-forward method (FBF). Another symmetry is that the new procedure allows … Read more

    There’s No Free Lunch: On the Hardness of Choosing a Correct Big-M in Bilevel Optimization

  • Thomas Kleinert
  • Martine Labbé
  • Martin Schmidt
  • Fränk Plein
    • Categories (Mixed) Integer Linear Programming, Complementarity and Variational Inequalities Tags , , , ,

    One of the most frequently used approaches to solve linear bilevel optimization problems consists in replacing the lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions and by reformulating the KKT complementarity conditions using techniques from mixed-integer linear optimization. The latter step requires to determine some big-M constant in order to bound the lower level’s dual feasible … Read more

    An Enhanced Logical Benders Approach for Linear Programs with Complementarity

  • Francisco Jara-Moroni
  • John E. Mitchell
  • Jong-Shi Pang
    • Categories Complementarity and Variational Inequalities, Global Optimization, Nonlinear Optimization Tags , ,

    This work extends the ones of Hu et al. (2008) and Bai et al. (2013) of a logical Benders approach for globally solving Linear Programs with Complementarity Constraints. By interpreting the logical Benders method as a reversed branch-and-bound method, where the whole exploration procedure starts from the leaf nodes in an enumeration tree, we provide … Read more

    Γ-Robust Linear Complementarity Problems

  • Vanessa Krebs
  • Martin Schmidt
    • Categories Complementarity and Variational Inequalities, Game Theory, Robust Optimization Tags , , , ,

    Complementarity problems are often used to compute equilibria made up of specifically coordinated solutions of different optimization problems. Specific examples are game-theoretic settings like the bimatrix game or energy market models like for electricity or natural gas. While optimization under uncertainties is rather well-developed, the field of equilibrium models represented by complementarity problems under uncertainty … Read more

    Active-set Newton methods and partial smoothness

  • Adrian Lewis
  • Calvin Wylie
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , ,

    Diverse optimization algorithms correctly identify, in finite time, intrinsic constraints that must be active at optimality. Analogous behavior extends beyond optimization to systems involving partly smooth operators, and in particular to variational inequalities over partly smooth sets. As in classical nonlinear programming, such active-set structure underlies the design of accelerated local algorithms of Newton type. … Read more

    An extragradient method for solving variational inequalities without monotonicity

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

    A new extragradient projection method is devised in this paper, which does not obviously require generalized monotonicity and assumes only that the so-called dual variational inequality has a solution in order to ensure its global convergence. In particular, it applies to quasimonotone variational inequality having a nontrivial solution. ArticleDownload View PDF