scholarly journals Reformulating Bilevel Problems by SQP Embedding

PAMM ◽  
2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Kai Schäfer ◽  
Jörg Fliege ◽  
Kathrin Flaßkamp ◽  
Christof Büskens
Keyword(s):  
Bilevel Problems

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

OR Spectrum ◽  
2021 ◽  
Author(s):  
Adejuyigbe O. Fajemisin ◽  
Laura Climent ◽  
Steven D. Prestwich
Keyword(s):  
Real Life ◽  
Forest Harvesting ◽  
Cutting Stock Problem ◽  
Cutting Stock ◽  
Decomposition Approach ◽  
New Class ◽  
Bilevel Problem ◽  
Programming Techniques ◽  
Bilevel Problems ◽  
Better Than

AbstractThis 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.

A new equivalent single-level problem for bilevel problems

Optimization ◽  
2012 ◽  
Vol 63 (5) ◽  
pp. 789-798 ◽  
Author(s):  
S. Dempe ◽  
N. Gadhi
Keyword(s):  
Single Level ◽  
Bilevel Problems ◽  
Level Problem

Bilevel problems over polyhedra with extreme point optimal solutions

2011 ◽  
Vol 53 (3) ◽  
pp. 573-586 ◽  
Author(s):  
Herminia I. Calvete ◽  
Carmen Galé ◽  
Stephan Dempe ◽  
Sebastian Lohse
Keyword(s):  
Extreme Point ◽  
Optimal Solutions ◽  
Bilevel Problems

Numerically tractable optimistic bilevel problems

2020 ◽  
Vol 76 (2) ◽  
pp. 277-303 ◽  
Author(s):  
Lorenzo Lampariello ◽  
Simone Sagratella
Keyword(s):  
Bilevel Problems

scholarly journals Optimistic Variants of Single-Objective Bilevel Optimization for Evolutionary Algorithms

2020 ◽  
Vol 19 (03) ◽  
pp. 2050020
Author(s):  
Anuraganand Sharma
Keyword(s):  
Optimization Problems ◽  
Bilevel Optimization ◽  
Benchmark Problems ◽  
New Variant ◽  
Real World Applications ◽  
Bilevel Problems ◽  
Specialized Form ◽  
Comparative Statistical Analysis ◽  
Constraint Optimization Problems ◽  
Single Objective

Single-objective bilevel optimization is a specialized form of constraint optimization problems where one of the constraints is an optimization problem itself. These problems are typically non-convex and strongly NP-Hard. Recently, there has been an increased interest from the evolutionary computation community to model bilevel problems due to its applicability in real-world applications for decision-making problems. In this work, a partial nested evolutionary approach with a local heuristic search has been proposed to solve the benchmark problems and have outstanding results. This approach relies on the concept of intermarriage-crossover in search of feasible regions by exploiting information from the constraints. A new variant has also been proposed to the commonly used convergence approaches, i.e., optimistic and pessimistic. It is called an extreme optimistic approach. The experimental results demonstrate the algorithm converges differently to known optimum solutions with the optimistic variants. Optimistic approach also outperforms pessimistic approach. Comparative statistical analysis of our approach with other recently published partial to complete evolutionary approaches demonstrates very competitive results.

Computing Feasible Points of Bilevel Problems with a Penalty Alternating Direction Method

2020 ◽  
Author(s):  
Thomas Kleinert ◽  
Martin Schmidt
Keyword(s):  
Optimization Problems ◽  
Mathematical Optimization ◽  
Alternating Direction Method ◽  
Convergence Theory ◽  
Mixed Integer ◽  
Solution Quality ◽  
Alternating Direction ◽  
Bilevel Problems ◽  
Primal Heuristics ◽  
Feasible Points

Bilevel problems are highly challenging optimization problems that appear in many applications of energy market design, critical infrastructure defense, transportation, pricing, and so on. Often these bilevel models are equipped with integer decisions, which makes the problems even harder to solve. Typically, in such a setting in mathematical optimization, one develops primal heuristics in order to obtain feasible points of good quality quickly or to enhance the search process of exact global methods. However, there are comparably few heuristics for bilevel problems. In this paper, we develop such a primal heuristic for bilevel problems with a mixed-integer linear or quadratic upper level and a linear or quadratic lower level. The heuristic is based on a penalty alternating direction method, which allows for a theoretical analysis. We derive a convergence theory stating that the method converges to a stationary point of an equivalent single-level reformulation of the bilevel problem and extensively test the method on a test set of more than 2,800 instances—which is one of the largest computational test sets ever used in bilevel programming. The study illustrates the very good performance of the proposed method in terms of both running times and solution quality. This renders the method a suitable subroutine in global bilevel solvers as well as a reasonable standalone approach. Summary of Contribution: Bilevel optimization problems form a very important class of optimization problems in the field of operations research, which is mainly due to their capability of modeling hierarchical decision processes. However, real-world bilevel problems are usually very hard to solve—especially in the case in which additional mixed-integer aspects are included in the modeling. Hence, the development of fast and reliable primal heuristics for this class of problems is very important. This paper presents such a method.

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