bilevel optimization

Bilevel optimization with a multi-objective lower-level problem: Risk-neutral and risk-averse formulations

  • Tommaso Giovannelli
  • Griffin D. Kent
  • Luis Nunes Vicente
    • Categories Data-Mining, Multi-Criteria Optimization, Stochastic Programming Tags , , ,

    In this work, we propose different formulations and gradient-based algorithms for deterministic and stochastic bilevel problems with conflicting objectives in the lower level. Such problems have received little attention in the deterministic case and have never been studied from a stochastic approximation viewpoint despite the recent advances in stochastic methods for single-level, bilevel, and multi-objective … Read more

    Analyzing Inexact Hypergradients for Bilevel Learning

  • Matthias J. Ehrhardt
  • Lindon Roberts
    • Categories Nonlinear Optimization, Optimization in Data Science Tags , ,

    Estimating hyperparameters has been a long-standing problem in machine learning. We consider the case where the task at hand is modeled as the solution to an optimization problem. Here the exact gradient with respect to the hyperparameters cannot be feasibly computed and approximate strategies are required. We introduce a unified framework for computing hypergradients that … Read more

    First-order penalty methods for bilevel optimization

  • Zhaosong Lu
  • Sanyou Mei
    • Categories Constrained Nonlinear Optimization Tags , , , ,

    \(\) In this paper we study a class of unconstrained and constrained bilevel optimization problems in which the lower-level part is a convex optimization problem, while the upper-level part is possibly a nonconvex optimization problem. In particular, we propose penalty methods for solving them, whose subproblems turn out to be a structured minimax problem and … Read more

    A Brief Introduction to Robust Bilevel Optimization

  • Yasmine Beck
  • Ivana Ljubić
  • Martin Schmidt
    • Categories Robust Optimization Tags , ,

    Bilevel optimization is a powerful tool for modeling hierarchical decision making processes. However, the resulting problems are challenging to solve – both in theory and practice. Fortunately, there have been significant algorithmic advances in the field so that we can solve much larger and also more complicated problems today compared to what was possible to … Read more

    An Exact Method for Nonlinear Network Flow Interdiction Problems

  • Martin Schmidt
  • Johannes Thürauf
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization, Network Optimization Tags , , ,

    We study network flow interdiction problems with nonlinear and nonconvex flow models. The resulting model is a max-min bilevel optimization problem in which the follower’s problem is nonlinear and nonconvex. In this game, the leader attacks a limited number of arcs with the goal to maximize the load shed and the follower aims at minimizing … Read more

    A Fast Combinatorial Algorithm for the Bilevel Knapsack Problem with Interdiction Constraints

  • Noah Weninger
  • Ricardo Fukasawa
    • Categories Combinatorial Optimization, Dynamic Programming Tags , , , ,

    \(\) We consider the bilevel knapsack problem with interdiction constraints, a fundamental bilevel integer programming problem which generalizes the 0-1 knapsack problem. In this problem, there are two knapsacks and \(n\) items. The objective is to select some items to pack into the first knapsack such that the maximum profit attainable from packing some of … Read more

    Fixed-Point Automatic Differentiation of Forward–Backward Splitting Algorithms for Partly Smooth Functions

  • Sheheryar Mehmood
  • Peter Ochs
    • Categories Convex and Nonsmooth Optimization, Optimization in Data Science Tags , , , ,

    A large class of non-smooth practical optimization problems can be written as minimization of a sum of smooth and partly smooth functions. We consider such structured problems which also depend on a parameter vector and study the problem of differentiating its solution mapping with respect to the parameter which has far reaching applications in sensitivity … Read more

    A Machine Learning Approach to Solving Large Bilevel and Stochastic Programs: Application to Cycling Network Design

  • Bo Lin
    • Categories Integer Programming, Stochastic Programming, Transportation Tags , , , ,

    We present a novel machine learning-based approach to solving bilevel programs that involve a large number of independent followers, which as a special case include two-stage stochastic programming. We propose an optimization model that explicitly considers a sampled subset of followers and exploits a machine learning model to estimate the objective values of unsampled followers. … Read more

    A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities

  • Julia Grübel
  • Richard Krug
  • Martin Schmidt
  • Winnifried Wollner
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Global Optimization Tags , , , ,

    We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate … Read more

    An Improved Unconstrained Approach for Bilevel Optimization

  • Xiaoyin Hu
  • Nachuan Xiao
  • Xin Liu
  • Kim-Chuan Toh
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    In this paper, we focus on the nonconvex-strongly-convex bilevel optimization problem (BLO). In this BLO, the objective function of the upper-level problem is nonconvex and possibly nonsmooth, and the lower-level problem is smooth and strongly convex with respect to the underlying variable $y$. We show that the feasible region of BLO is a Riemannian manifold. … Read more