scholarly journals Presolving Linear Bilevel Optimization Problems

2021 ◽  
pp. 100020
Author(s):  
Thomas Kleinert ◽  
Julian Manns ◽  
Martin Schmidt ◽  
Dieter Weninger
Keyword(s):  
Optimization Problems ◽  
Bilevel Optimization

scholarly journals R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

2021 ◽  
Author(s):  
Patrick Mehlitz ◽  
Leonid I. Minchenko
Keyword(s):  
Linear Dependence ◽  
Constraint Qualification ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Bilevel Optimization ◽  
Regularity Property ◽  
Constant Rank Constraint Qualification ◽  
Existence Criterion ◽  
Set Valued Mappings

AbstractThe presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

scholarly journals The trouble with the second quantifier

4OR ◽  
2021 ◽  
Author(s):  
Gerhard J. Woeginger
Keyword(s):  
Computational Complexity ◽  
Robust Optimization ◽  
Complexity Theory ◽  
Operational Research ◽  
Optimization Problems ◽  
Bilevel Optimization ◽  
Theoretical Background ◽  
Research Community ◽  
Universal Quantifier ◽  
Computational Complexity Theory

AbstractWe survey optimization problems that allow natural simple formulations with one existential and one universal quantifier. We summarize the theoretical background from computational complexity theory, and we present a multitude of illustrating examples. We discuss the connections to robust optimization and to bilevel optimization, and we explain the reasons why the operational research community should be interested in the theoretical aspects of this area.

scholarly journals A First Order Method for Solving Convex Bilevel Optimization Problems

2017 ◽  
Vol 27 (2) ◽  
pp. 640-660 ◽  
Author(s):  
Shoham Sabach ◽  
Shimrit Shtern
Keyword(s):  
Optimization Problems ◽  
Bilevel Optimization ◽  
Order Method ◽  
First Order ◽  
First Order Method

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.

Solution of bilevel optimization problems using the KKT approach

Optimization ◽  
2019 ◽  
Vol 68 (8) ◽  
pp. 1471-1489 ◽  
Author(s):  
S. Dempe ◽  
S. Franke
Keyword(s):  
Optimization Problems ◽  
Bilevel Optimization
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