592 research outputs found
A New Heuristic for Feature Selection by Consistent Biclustering
Given a set of data, biclustering aims at finding simultaneous partitions in
biclusters of its samples and of the features which are used for representing
the samples. Consistent biclusterings allow to obtain correct classifications
of the samples from the known classification of the features, and vice versa,
and they are very useful for performing supervised classifications. The problem
of finding consistent biclusterings can be seen as a feature selection problem,
where the features that are not relevant for classification purposes are
removed from the set of data, while the total number of features is maximized
in order to preserve information. This feature selection problem can be
formulated as a linear fractional 0-1 optimization problem. We propose a
reformulation of this problem as a bilevel optimization problem, and we present
a heuristic algorithm for an efficient solution of the reformulated problem.
Computational experiments show that the presented algorithm is able to find
better solutions with respect to the ones obtained by employing previously
presented heuristic algorithms
Tropical analogues of a Dempe-Franke bilevel optimization problem
We consider the tropical analogues of a particular bilevel optimization
problem studied by Dempe and Franke and suggest some methods of solving these
new tropical bilevel optimization problems. In particular, it is found that the
algorithm developed by Dempe and Franke can be formulated and its validity can
be proved in a more general setting, which includes the tropical bilevel
optimization problems in question. We also show how the feasible set can be
decomposed into a finite number of tropical polyhedra, to which the tropical
linear programming solvers can be applied.Comment: 11 pages, 1 figur
A regularization method for ill-posed bilevel optimization problems
We present a regularization method to approach a solution of the pessimistic
formulation of ill -posed bilevel problems . This allows to overcome the
difficulty arising from the non uniqueness of the lower level problems
solutions and responses. We prove existence of approximated solutions, give
convergence result using Hoffman-like assumptions. We end with objective value
error estimates.Comment: 19 page
The robust bilevel continuous knapsack problem with uncertain follower's objective
We consider a bilevel continuous knapsack problem where the leader controls
the capacity of the knapsack and the follower chooses an optimal packing
according to his own profits, which may differ from those of the leader. To
this bilevel problem, we add uncertainty in a natural way, assuming that the
leader does not have full knowledge about the follower's problem. More
precisely, adopting the robust optimization approach and assuming that the
follower's profits belong to a given uncertainty set, our aim is to compute a
solution that optimizes the worst-case follower's reaction from the leader's
perspective. By investigating the complexity of this problem with respect to
different types of uncertainty sets, we make first steps towards better
understanding the combination of bilevel optimization and robust combinatorial
optimization. We show that the problem can be solved in polynomial time for
both discrete and interval uncertainty, but that the same problem becomes
NP-hard when each coefficient can independently assume only a finite number of
values. In particular, this demonstrates that replacing uncertainty sets by
their convex hulls may change the problem significantly, in contrast to the
situation in classical single-level robust optimization. For general polytopal
uncertainty, the problem again turns out to be NP-hard, and the same is true
for ellipsoidal uncertainty even in the uncorrelated case. All presented
hardness results already apply to the evaluation of the leader's objective
function
Fuzzy Random Noncooperative Two-level Linear Programming through Absolute Deviation Minimization Using Possibility and Necessity
This paper considers fuzzy random two-level linear programming problems under noncooperative behaviorof the decision makers. Having introduced fuzzy goals of decision makers together with the possibiliy and necessity measure, following absolute deviation minimization, fuzzy random two-level programin problems are transformed into deterministic ones. Extended Stackelberg solutions are introduced andcomputational methods are also presented
The robust bilevel continuous knapsack problem with uncertain coefficients in the follower’s objective
We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack and the follower chooses an optimal packing according to his own profits, which may differ from those of the leader. To this bilevel problem, we add uncertainty in a natural way, assuming that the leader does not have full knowledge about the follower’s problem. More precisely, adopting the robust optimization approach and assuming that the follower’s profits belong to a given uncertainty set, our aim is to compute a solution that optimizes the worst-case follower’s reaction from the leader’s perspective. By investigating the complexity of this problem with respect to different types of uncertainty sets, we make first steps towards better understanding the combination of bilevel optimization and robust combinatorial optimization. We show that the problem can be solved in polynomial time for both discrete and interval uncertainty, but that the same problem becomes NP-hard when each coefficient can independently assume only a finite number of values. In particular, this demonstrates that replacing uncertainty sets by their convex hulls may change the problem significantly, in contrast to the situation in classical single-level robust optimization. For general polytopal uncertainty, the problem again turns out to be NP-hard, and the same is true for ellipsoidal uncertainty even in the uncorrelated case. All presented hardness results already apply to the evaluation of the leader’s objective function
On SOCP-based disjunctive cuts for solving a class of integer bilevel nonlinear programs
We study a class of integer bilevel programs with second-order cone
constraints at the upper-level and a convex-quadratic objective function and
linear constraints at the lower-level. We develop disjunctive cuts (DCs) to
separate bilevel-infeasible solutions using a second-order-cone-based
cut-generating procedure. We propose DC separation strategies and consider
several approaches for removing redundant disjunctions and normalization. Using
these DCs, we propose a branch-and-cut algorithm for the problem class we
study, and a cutting-plane method for the problem variant with only binary
variables.
We present an extensive computational study on a diverse set of instances,
including instances with binary and with integer variables, and instances with
a single and with multiple linking constraints. Our computational study
demonstrates that the proposed enhancements of our solution approaches are
effective for improving the performance. Moreover, both of our approaches
outperform a state-of-the-art generic solver for mixed-integer bilevel linear
programs that is able to solve a linearized version of our binary instances.Comment: arXiv admin note: substantial text overlap with arXiv:2111.0682
Multi-parametric programming : novel theory and algorithmic developments
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