Journal ArticleDOI
New necessary optimality conditions in optimistic bilevel programming
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TLDR
This work reduces a basic optimistic model in bilevel programming to a one-level framework of nondifferentiable programs formulated via (nonsmooth) optimal value function of the parametric lower-level problem in the original model and derives new necessary optimality conditions for bileVEL programs reflecting significant phenomena that have never been observed earlier.Abstract:
The article is devoted to the study of the so-called optimistic version of bilevel programming in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for smooth initial data) and can be treated by using appropriate tools of modern variational analysis and generalized differentiation. Considering a basic optimistic model in bilevel programming, we reduce it to a one-level framework of nondifferentiable programs formulated via (nonsmooth) optimal value function of the parametric lower-level problem in the original model. Using advanced formulas for computing basic subgradients of value/marginal functions in variational analysis, we derive new necessary optimality conditions for bilevel programs reflecting significant phenomena that have never been observed earlier. In particular, our optimality conditions for bilevel programs do not depend on the partial derivatives with respect to parameters of the smooth objective function in the parametric lower-level problem. We present ef...read more
Citations
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Exact Algorithms for Mixed-Integer Multilevel Programming Problems
TL;DR: This work contributes exact finite algorithms for bilevel mixed-integer programs, three-stage defender-attacker-defender problems, and two-stage stochastic programs and demonstrates how the proposed algorithms significantly outperform existing state-of-the-art algorithms.
Journal ArticleDOI
Suboptimality conditions for mathematical programs with equilibrium constraints
TL;DR: In this article, the authors study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended form (vspace{-0.5cm} where both mappings $G$ and $Q$ are set-valued, and establish new conditions for MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do not assume the existence of optimal solutions.
Journal ArticleDOI
Bilevel programming problems with simple convex lower level
TL;DR: In this paper, the authors considered a general bilevel programming problem in Banach spaces possessing operator constraints and derived necessary optimality conditions for the latter problem using the lower level optimal value function, ideas from DC-programming and partial penalization.
Book
Random-Like Bi-level Decision Making
TL;DR: This book is an attempt to elucidate random-like bi-level decision making to all aspects of random sets theory, including some fundamental concepts, the definitions of random variables, fuzzy variables, random-overlapped random (Ra-Ra) variables and fuzzy- overlapped Random (RaFu) variables.
Book ChapterDOI
Algorithms for Simple Bilevel Programming
Joydeep Dutta,Tanushree Pandit +1 more
TL;DR: In this article, the authors focus on algorithms for solving simple bilevel programming problems, which consist of minimizing a convex function over the solution set of another convex optimization problem.
References
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Book
Mathematical Programs with Equilibrium Constraints
TL;DR: Results in the book are expected to have significant impacts in such disciplines as engineering design, economics and game equilibria, and transportation planning, within all of which MPEC has a central role to play in the modelling of many practical problems.
Book
Foundations of Bilevel Programming
TL;DR: This paper presents a meta-modelling framework that automates the very labor-intensive and therefore time-heavy and expensive process of solving linear and Discrete Bilevel Problems.
Book
Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results
TL;DR: This chapter discusses the theory and applications of Elliptic Boundary Value problems, and some of the algorithms used to solve these problems have been described.