On the solution of convex bilevel optimization problems

Dempe, Stephan; Franke, Susanne

doi:10.1007/s10589-015-9795-8

Cited by 49 publications

(29 citation statements)

References 35 publications

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“…(ii) Relations (9), (11) and (12) follow from (4) and the fact that w * ∈ U , g(x * , w * ) ≤ 0 and f (x * , w * ) = f (x * , y * ). Moreover, (20) implies (10)…”

Section: Global Solutionsmentioning

confidence: 99%

“…The latter problems are structurally nonconvex and nonsmooth (see [8]); furthermore, it is hard to define suitable constraint qualification conditions for them, see, e.g., [12,37]. In fact, the study of provably convergent and practically implementable algorithms for the solution of even just SBPs is still in its infancy (see, for example, [3,6,9,10,25,27,30,33,35,36,39]), as also witnessed by the scarcity of results in the literature. We remark that suitable reformulations of the SBP have been proposed in order to investigate optimality conditions and constraint qualifications, as well as to devise suitable algorithmic approaches: to date, the most studied and promising are optimal value and KKT one level reformulations (see [13], the references therein and [29,38]).…”

Section: Introductionmentioning

confidence: 99%

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A Bridge Between Bilevel Programs and Nash Games

Lampariello

Sagratella

2017

J Optim Theory Appl

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We study connections between optimistic bilevel programming problems and generalized Nash equilibrium problems. We remark that, with respect to bilevel problems, we consider the general case in which the lower level program is not assumed to have a unique solution. Inspired by the optimal value approach, we propose a Nash game that, transforming the so-called implicit value function constraint into an explicitly defined constraint function, incorporates some taste of hierarchy and turns out to be related to the bilevel programming problem. We provide a complete theoretical analysis of the relationship between the vertical bilevel problem and our "uneven" horizontal model: in particular, we define classes of problems for which solutions of the bilevel program can be computed by finding equilibria of our game. Furthermore, by referring to some applications in economics, we show that our "uneven" horizontal model, in some sense, lies between the vertical bilevel model and a "pure" horizontal game.

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“…(ii) Relations (9), (11) and (12) follow from (4) and the fact that w * ∈ U , g(x * , w * ) ≤ 0 and f (x * , w * ) = f (x * , y * ). Moreover, (20) implies (10)…”

Section: Global Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Bridge Between Bilevel Programs and Nash Games

Lampariello

Sagratella

2017

J Optim Theory Appl

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“…In Dempe and Franke (2016) lower-level problems are replaced with their Fritz-John conditions (John 1948), and an algorithm is presented for solving problems with fully convex lower-levels. This method is applied in Dempe and Franke (2014) to solve a bilevel road pricing problem. Nogales Martín and Miguel (2004) show a relationship between one bilevel decomposition algorithm and a direct interior-point method based on Newton's method.…”

Section: Decomposition Approaches In the Literaturementioning

confidence: 99%

An analytics-based heuristic decomposition of a bilevel multiple-follower cutting stock problem

2021

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This paper presents a new class of multiple-follower bilevel problems and a heuristic approach to solving them. In this new class of problems, the followers may be nonlinear, do not share constraints or variables, and are at most weakly constrained. This allows the leader variables to be partitioned among the followers. We show that current approaches for solving multiple-follower problems are unsuitable for our new class of problems and instead we propose a novel analytics-based heuristic decomposition approach. This approach uses Monte Carlo simulation and k-medoids clustering to reduce the bilevel problem to a single level, which can then be solved using integer programming techniques. The examples presented show that our approach produces better solutions and scales up better than the other approaches in the literature. Furthermore, for large problems, we combine our approach with the use of self-organising maps in place of k-medoids clustering, which significantly reduces the clustering times. Finally, we apply our approach to a real-life cutting stock problem. Here a forest harvesting problem is reformulated as a multiple-follower bilevel problem and solved using our approach.

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“…Learning the parameter of end-to-end trainable rank pooled CNN Now we present how to learn the parameters of the CNN (θ ) and the classifier parameters. Consider again the learning problem defined in Equation 10. The derivative with respect to β , which only appears in the upper-level problem, is straightforward and well known.…”

Section: 3mentioning

confidence: 99%

Discriminatively Learned Hierarchical Rank Pooling Networks

Fernando

Gould

2017

Int J Comput Vis

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Rank pooling is a temporal encoding method that summarizes the dynamics of a video sequence to a single vector which has shown good results in human action recognition in prior work. In this work, we present novel temporal encoding methods for action and activity classification by extending the unsupervised rank pooling temporal encoding method in two ways.First, we present discriminative rank pooling in which the shared weights of our video representation and the parameters of the action classifiers are estimated jointly for a given training dataset of labelled vector sequences using a bilevel optimization formulation of the learning problem. When the frame level features vectors are obtained from a convolutional neural network (CNN), we rank pool the network activations and jointly estimate all parameters of the model, including CNN filters and fully-connected weights, in an end-to-end manner which we coined as end-to-end trainable rank pooled CNN. Importantly, this model can make use of any existing convolutional neural network architecture (e.g., AlexNet or VGG) without modification or introduction of additional parameters.Then, we extend rank pooling to a high capacity video representation, called hierarchical rank pooling. Hierarchical rank pooling consists of a network of rank pooling functions, which encode temporal semantics over arbitrary long video clips based on rich frame level features. By stacking non-linear feature functions and temporal sub-sequence encoders one on top of the other, we build a high capacity encoding network of the dynamic behaviour of the video. The resulting video representation is a fixed-length feature vec-ACRV,

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On the solution of convex bilevel optimization problems

Cited by 49 publications

References 35 publications

A Bridge Between Bilevel Programs and Nash Games

A Bridge Between Bilevel Programs and Nash Games

An analytics-based heuristic decomposition of a bilevel multiple-follower cutting stock problem

Discriminatively Learned Hierarchical Rank Pooling Networks

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