Mixed-integer nonlinear optimization

An adaptive relaxation-refinement scheme for multi-objective mixed-integer nonconvex optimization

  • Gabriele Eichfelder
  • Moritz Link
  • Stefan Volkwein
  • Leo Warnow
    • Categories (Mixed) Integer Nonlinear Programming, Multi-Criteria Optimization Tags , , , ,

    In this work, we present an algorithm for computing an enclosure for multi-objective mixed-integer nonconvex optimization problems. In contrast to existing solvers for this type of problem, this algorithm is not based on a branch-and-bound scheme but rather relies on a relax-and-refine approach. While this is an established technique in single-objective optimization, several adaptions to … Read more

    Exact Augmented Lagrangian Duality for Nonconvex Mixed-Integer Nonlinear Optimization

  • Henri Lefebvre
  • Martin Schmidt
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming Tags , ,

    In the context of mixed-integer nonlinear problems (MINLPs), it is well-known that strong duality does not hold in general if the standard Lagrangian dual is used. Hence, we consider the augmented Lagrangian dual (ALD), which adds a nonlinear penalty function to the classic Lagrangian function. For this setup, we study conditions under which the ALD … Read more

    A Toll-Setting Problem with Robust Wardrop Equilibrium Conditions Under Budgeted Uncertainty

  • Yasmine Beck
  • Martine Labbé
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Robust Optimization, Transportation Tags , , , ,

    We consider the problem of determining optimal tolls in a traffic network in which a toll-setting authority aims to maximize revenues and the users of the network act in the sense of Wardrop’s user equilibrium. The setting is modeled as a mathematical problem with equilibrium constraints and a mixed-integer, nonlinear, and nonconvex reformulation is presented … Read more

    Adjustable Robust Nonlinear Network Design Without Controllable Elements under Load Scenario Uncertainties

  • Johannes Thürauf
  • Julia Grübel
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Energy, Robust Optimization Tags , , , ,

    We study network design problems for nonlinear and nonconvex flow models without controllable elements under load scenario uncertainties, i.e., under uncertain injections and withdrawals. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, load scenarios within a given … Read more

    Using Disjunctive Cuts in a Branch-and-Cut Method to Solve Convex Integer Nonlinear Bilevel Problems

  • Horländer Andreas
  • Ivana Ljubić
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Bilevel Optimization, Cutting Plane Approaches Tags , , ,

    We present a branch-and-cut method for solving convex integer nonlinear bilevel problems, i.e., bilevel models with nonlinear but jointly convex objective functions and constraints in both the upper and the lower level. To this end, we generalize the idea of using disjunctive cuts to cut off integer-feasible but bilevel-infeasible points. These cuts can be obtained … Read more

    Robust support vector machines via conic optimization

  • Shaoning Han
  • Andrés Gómez
    • Categories (Mixed) Integer Nonlinear Programming, Cone Programming, Optimization in Data Science Tags , , , ,

    We consider the problem of learning support vector machines robust to uncertainty. It has been established in the literature that typical loss functions, including the hinge loss, are sensible to data perturbations and outliers, thus performing poorly in the setting considered. In contrast, using the 0-1 loss or a suitable non-convex approximation results in robust … Read more

    Test Instances for Multiobjective Mixed-Integer Nonlinear Optimization

  • Leo Warnow
  • Gabriele Eichfelder
  • Tobias Gerlach
    • Categories (Mixed) Integer Nonlinear Programming, Multi-Criteria Optimization, Nonlinear Optimization Tags , , ,

    A suitable set of test instances, also known as benchmark problems, is a key ingredient to systematically evaluate numerical solution algorithms for a given class of optimization problems. While in recent years several solution algorithms for the class of multiobjective mixed-integer nonlinear optimization problems have been proposed, there is a lack of a well-established set … Read more

    Gas Transport Network Optimization: Mixed-Integer Nonlinear Models

  • Falk M. Hante
  • Martin Schmidt
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering Tags , , ,

    Although modern societies strive towards energy systems that are entirely based on renewable energy carriers, natural gas is still one of the most important energy sources. This became even more obvious in Europe with Russia’s 2022 war against the Ukraine and the resulting stop of gas supplies from Russia. Besides that it is very important … Read more

    A Consensus-Based Alternating Direction Method for Mixed-Integer and PDE-Constrained Gas Transport Problems

  • Richard Krug
  • Günter Leugering
  • Alexander Martin
  • Martin Schmidt
  • Dieter Weninger
    • Categories (Mixed) Integer Nonlinear Programming, Network Optimization, Optimization of Systems modeled by PDEs Tags , , , ,

    We consider dynamic gas transport optimization problems, which lead to large-scale and nonconvex mixed-integer nonlinear optimization problems (MINLPs) on graphs. Usually, the resulting instances are too challenging to be solved by state-of-the-art MINLP solvers. In this paper, we use graph decompositions to obtain multiple optimization problems on smaller blocks, which can be solved in parallel … 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