272 research outputs found
Solving Bilevel Knapsack Problem using Graph Neural Networks
The Bilevel Optimization Problem is a hierarchical optimization problem with
two agents, a leader and a follower. The leader make their own decisions first,
and the followers make the best choices accordingly. The leader knows the
information of the followers, and the goal of the problem is to find the
optimal solution by considering the reactions of the followers from the
leader's point of view. For the Bilevel Optimization Problem, there are no
general and efficient algorithms or commercial solvers to get an optimal
solution, and it is very difficult to get a good solution even for a simple
problem. In this paper, we propose a deep learning approach using Graph Neural
Networks to solve the bilevel knapsack problem. We train the model to predict
the leader's solution and use it to transform the hierarchical optimization
problem into a single-level optimization problem to get the solution. Our model
found the feasible solution that was about 500 times faster than the exact
algorithm with optimal gap. Also, our model performed well on problems
of different size from the size it was trained on.Comment: 27 pages, 2 figure
An exact approach for the bilevel knapsack problem with interdiction constraints and extensions
We consider the bilevel knapsack problem with interdiction constraints, an extension of the classic 0–1 knapsack problem formulated as a Stackelberg game with two agents, a leader and a follower, that choose items from a common set and hold their own private knapsacks. First, the leader selects some items to be interdicted for the follower while satisfying a capacity constraint. Then the follower packs a set of the remaining items according to his knapsack constraint in order to maximize the profits. The goal of the leader is to minimize the follower’s total profit. We derive effective lower bounds for the bilevel knapsack problem and present an exact method that exploits the structure of the induced follower’s problem. The approach strongly outperforms the current state-of-the-art algorithms designed for the problem. We extend the same algorithmic framework to the interval min–max regret knapsack problem after providing a novel bilevel programming reformulation. Also for this problem, the proposed approach outperforms the exact algorithms available in the literature
On a Stackelberg Subset Sum Game
This contribution deals with a two-level discrete decision problem, a
so-called Stackelberg strategic game: A Subset Sum setting is addressed with a
set of items with given integer weights. One distinguished player, the
leader, may alter the weights of the items in a given subset , and
a second player, the follower, selects a solution in order to
utilize a bounded resource in the best possible way. Finally, the leader
receives a payoff from those items of its subset that were included in the
overall solution , chosen by the follower. We assume that the follower
applies a publicly known, simple, heuristic algorithm to determine its solution
set, which avoids having to solve NP-hard problems.
Two variants of the problem are considered, depending on whether the leader
is able to control (i.e., change) the weights of its items (i) in the objective
function or (ii) in the bounded resource constraint. The leader's objective is
the maximization of the overall weight reduction, for the first variant, or the
maximization of the weight increase for the latter one. In both variants there
is a trade-off for each item between the contribution value to the leader's
objective and the chance of being included in the follower's solution set.
We analyze the leader's pricing problem for a natural greedy strategy of the
follower and discuss the complexity of the corresponding problems. We show that
setting the optimal weight values for the leader is, in general, NP-hard. It is
even NP-hard to provide a solution within a constant factor of the best
possible solution. Exact algorithms, based on dynamic programming and running
in pseudopolynomial time, are provided. The additional cases, in which the
follower faces a continuous (linear relaxation) version of the above problems,
are shown to be straightforward to solve.Comment: 13 pages, 1 figur
Models and algorithms for decomposition problems
This thesis deals with the decomposition both as a solution method and as a problem itself. A decomposition approach can be very effective for mathematical problems presenting a specific structure in which the associated matrix of coefficients is sparse and it is diagonalizable in blocks. But, this kind of structure may not be evident from the most natural formulation of the problem. Thus, its coefficient matrix may be preprocessed by solving a structure detection problem in order to understand if a decomposition method can successfully be applied. So, this thesis deals with the k-Vertex Cut problem, that is the problem of finding the minimum subset of nodes whose removal disconnects a graph into at least k components, and it models relevant applications in matrix decomposition for solving systems of equations by parallel computing. The capacitated k-Vertex Separator problem, instead, asks to find a subset of vertices of minimum cardinality the deletion of which disconnects a given graph in at most k shores and the size of each shore must not be larger than a given capacity value. Also this problem is of great importance for matrix decomposition algorithms.
This thesis also addresses the Chance-Constrained Mathematical Program that represents a significant example in which decomposition techniques can be successfully applied. This is a class of stochastic optimization problems in which the feasible region depends on the realization of a random variable and the solution must optimize a given objective function while belonging to the feasible region with a probability that must be above a given value. In this thesis, a decomposition approach for this problem is introduced.
The thesis also addresses the Fractional Knapsack Problem with Penalties, a variant of the knapsack problem in which items can be split at the expense of a penalty depending on the fractional quantity
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
Canonical duality theory and algorithm for solving bilevel knapsack problems with applications
A novel canonical duality theory (CDT) is presented for solving general bilevel mixed integer nonlinear optimization governed by linear and quadratic knapsack problems. It shows that the challenging knapsack problems can be solved analytically in term of their canonical dual solutions. The existence and uniqueness of these analytical solutions are proved. NP-hardness of the knapsack problems is discussed. A powerful CDT algorithm combined with an alternative iteration and a volume reduction method is proposed for solving the NP-hard bilevel knapsack problems. Application is illustrated by benchmark problems in optimal topology design. The performance and novelty of the proposed method are compared with the popular commercial codes. © 2013 IEEE
The Stochastic 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, while the follower chooses a feasible packing
maximizing his own profit. The leader's aim is to optimize a linear objective
function in the capacity and in the follower's solution, but with respect to
different item values. We address a stochastic version of this problem where
the follower's profits are uncertain from the leader's perspective, and only a
probability distribution is known. Assuming that the leader aims at optimizing
the expected value of her objective function, we first observe that the
stochastic problem is tractable as long as the possible scenarios are given
explicitly as part of the input, which also allows to deal with general
distributions using a sample average approximation. For the case of
independently and uniformly distributed item values, we show that the problem
is #P-hard in general, and the same is true even for evaluating the leader's
objective function. Nevertheless, we present pseudo-polynomial time algorithms
for this case, running in time linear in the total size of the items. Based on
this, we derive an additive approximation scheme for the general case of
independently distributed item values, which runs in pseudo-polynomial time.Comment: A preliminary version of parts of this article can be found in
Section 8 of arXiv:1903.02810v
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