17,437 research outputs found
Approximating Multiobjective Optimization Problems: How exact can you be?
It is well known that, under very weak assumptions, multiobjective
optimization problems admit -approximation
sets (also called -Pareto sets) of polynomial cardinality (in the
size of the instance and in ). While an approximation
guarantee of for any is the best one can expect
for singleobjective problems (apart from solving the problem to optimality),
even better approximation guarantees than
can be considered in the multiobjective case since the approximation might be
exact in some of the objectives.
Hence, in this paper, we consider partially exact approximation sets that
require to approximate each feasible solution exactly, i.e., with an
approximation guarantee of , in some of the objectives while still obtaining
a guarantee of in all others. We characterize the types of
polynomial-cardinality, partially exact approximation sets that are guaranteed
to exist for general multiobjective optimization problems. Moreover, we study
minimum-cardinality partially exact approximation sets concerning (weak)
efficiency of the contained solutions and relate their cardinalities to the
minimum cardinality of a -approximation
set
ND-Tree-based update: a Fast Algorithm for the Dynamic Non-Dominance Problem
In this paper we propose a new method called ND-Tree-based update (or shortly
ND-Tree) for the dynamic non-dominance problem, i.e. the problem of online
update of a Pareto archive composed of mutually non-dominated points. It uses a
new ND-Tree data structure in which each node represents a subset of points
contained in a hyperrectangle defined by its local approximate ideal and nadir
points. By building subsets containing points located close in the objective
space and using basic properties of the local ideal and nadir points we can
efficiently avoid searching many branches in the tree. ND-Tree may be used in
multiobjective evolutionary algorithms and other multiobjective metaheuristics
to update an archive of potentially non-dominated points. We prove that the
proposed algorithm has sub-linear time complexity under mild assumptions. We
experimentally compare ND-Tree to the simple list, Quad-tree, and M-Front
methods using artificial and realistic benchmarks with up to 10 objectives and
show that with this new method substantial reduction of the number of point
comparisons and computational time can be obtained. Furthermore, we apply the
method to the non-dominated sorting problem showing that it is highly
competitive to some recently proposed algorithms dedicated to this problem.Comment: 15 pages, 21 figures, 3 table
Algorithms for generalized potential games with mixed-integer variables
We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixed-integer variables, i.e., games in which some variables are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss–Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches
Mechanism Design for Team Formation
Team formation is a core problem in AI. Remarkably, little prior work has
addressed the problem of mechanism design for team formation, accounting for
the need to elicit agents' preferences over potential teammates. Coalition
formation in the related hedonic games has received much attention, but only
from the perspective of coalition stability, with little emphasis on the
mechanism design objectives of true preference elicitation, social welfare, and
equity. We present the first formal mechanism design framework for team
formation, building on recent combinatorial matching market design literature.
We exhibit four mechanisms for this problem, two novel, two simple extensions
of known mechanisms from other domains. Two of these (one new, one known) have
desirable theoretical properties. However, we use extensive experiments to show
our second novel mechanism, despite having no theoretical guarantees,
empirically achieves good incentive compatibility, welfare, and fairness.Comment: 12 page
- …