Journal ArticleDOI
Global optimization of concave functions subject to quadratic constraints: an application in nonlinear bilevel programming
TLDR
This paper proposes different methods for finding the global minimum of a concave function subject to quadratic separable constraints and shows how this constraint can be replaced by an equivalent system of convex and separablequadratic constraints.Abstract:
When the follower's optimality conditions are both necessary and sufficient, the nonlinear bilevel program can be solved as a global optimization problem. The complementary slackness condition is usually the complicating constraint in such problems. We show how this constraint can be replaced by an equivalent system of convex and separable quadratic constraints. In this paper, we propose different methods for finding the global minimum of a concave function subject to quadratic separable constraints. The first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex, the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints, we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.read more
Citations
More filters
Journal ArticleDOI
An overview of bilevel optimization
TL;DR: This paper presents fields of application, focus on solution approaches, and makes the connection with MPECs (Mathematical Programs with Equilibrium Constraints), a branch of mathematical programming of both practical and theoretical interest.
BookDOI
Handbook of Semidefinite Programming
TL;DR: Conditions and an accurate semidefinite programming solver are described in The Journal of the SDPA family for solving large-scale SDPs and in Handbook on Semidefinitely Programming.
Journal ArticleDOI
Bilevel and multilevel programming: A bibliography review
TL;DR: This paper contains a bibliography of all references central to bilevel and multilevel programming that the authors know of and it is hoped that this bibliography facilitates and encourages their research.
Journal ArticleDOI
A Review on Bilevel Optimization: From Classical to Evolutionary Approaches and Applications
TL;DR: A comprehensive review on bilevel optimization from the basic principles to solution strategies is provided in this paper, where a number of potential application problems are also discussed and an automated text-analysis of an extended list of papers has been performed.
Journal ArticleDOI
Bilevel programming: A survey
TL;DR: An introductory survey of a class of optimization problems known as bilevel programming, which considers various cases (linear, linear-quadratic, nonlinear), describe their main properties and give an overview of solution approaches.
References
More filters
Book
Global Optimization: Deterministic Approaches
Reiner Horst,Tuy Hoang +1 more
TL;DR: This study develops a unifying approach to constrained global optimization that provides insight into the underlying concepts and properties of diverse techniques recently proposed to solve a wide variety of problems encountered in the decision sciences, engineering, operations research and other disciplines.
Journal ArticleDOI
A Representation and Economic Interpretation of a Two-Level Programming Problem
TL;DR: A solution procedure is developed that replaces the subproblem by its Kuhn-Tucker conditions and then further transforms it into a mixed integer quadratic programming problem by exploiting the disjunctive nature of the complementary slackness conditions.
Journal ArticleDOI
On the Stackelberg strategy in nonzero-sum games
Marwan A. Simaan,Jose B. Cruz +1 more
TL;DR: The properties of the Stackelberg solution in static and dynamic nonzero-sum two-player games are investigated in this paper, and necessary and sufficient conditions for its existence are derived.
Related Papers (5)
A Branch and Bound Algorithm for the Bilevel Programming Problem
Jonathan F. Bard,James T. Moore +1 more