Peter Kirst

A branch-and-bound algorithm for non-convex Nash equilibrium problems

  • Peter Kirst
  • Stefan Schwarze
  • Oliver Stein
    • Categories Game Theory Tags , , , ,

    This paper introduces a spatial branch-and-bound method for the approximate computation of the set of all epsilon-Nash equilibria of continuous box-constrained non-convex Nash equilibrium problems. We explain appropriate discarding and fathoming techniques, provide a termination proof for a prescribed approximation tolerance, and report our computational experience. Article Download View A branch-and-bound algorithm for non-convex Nash … Read more

    A general branch-and-bound framework for continuous global multiobjective optimization

  • Gabriele Eichfelder
  • Peter Kirst
  • Oliver Stein
  • Laura Meng
    • Categories Global Optimization, Multi-Criteria Optimization, Nonlinear Optimization

    Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and … Read more

    Deterministic upper bounds in global minimization with nonlinear equality constraints

  • Christian Füllner
  • Peter Kirst
  • Oliver Stein
    • Categories Constrained Nonlinear Optimization, Global Optimization Theory

    We address the problem of deterministically determining upper bounds in continuous non-convex global minimization of box-constrained problems with equality constraints. These upper bounds are important for the termination of spatial branch-and-bound algorithms. Our method is based on the theorem of Miranda which helps to ensure the existence of feasible points in certain boxes. Then, the … Read more

    Solving disjunctive optimization problems by generalized semi-infinite optimization techniques

  • Peter Kirst
  • Oliver Stein
    • Categories Nonsmooth Optimization, Semi-infinite Programming Tags , , , ,

    We describe a new possibility to model disjunctive optimization problems as generalized semi-infinite programs. In contrast to existing methods, for our approach neither a conjunctive nor a disjunctive normal form is expected. Applying existing lower level reformulations for the corresponding semi-infinite program we derive conjunctive nonlinear problems without any logical expressions, which can be locally … Read more

    An Enhanced Spatial Branch-and-Bound Method in Global Optimization with Nonconvex Constraints

  • Peter Kirst
  • Oliver Stein
  • Paul Steuermann
    • Categories Global Optimization Tags , , ,

    We discuss some difficulties in determining valid upper bounds in spatial branch-and-bound methods for global minimization in the presence of nonconvex constraints. In fact, two examples illustrate that standard techniques for the construction of upper bounds may fail in this setting. Instead, we propose to perturb infeasible iterates along Mangasarian-Fromovitz directions to feasible points whose … Read more