A note on partial calmness for bilevel optimization problems with linearly structured lower level
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A note on partial calmness for bilevel optimization problems with linearly structured lower level Patrick Mehlitz1
· Leonid I. Minchenko2
· Alain B. Zemkoho3
Received: 13 March 2020 / Accepted: 19 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Partial calmness is a celebrated but restrictive property of bilevel optimization problems whose presence opens a way to the derivation of Karush–Kuhn–Tucker-type necessary optimality conditions in order to characterize local minimizers. In the past, sufficient conditions for the validity of partial calmness have been investigated. In this regard, the presence of a linearly structured lower level problem has turned out to be beneficial. However, the associated literature suffers from inaccurate results. In this note, we clarify some regarding erroneous statements and visualize the underlying issues with the aid of illustrative counterexamples. Keywords Bilevel optimization · Linear programming · Partial calmness Mathematics Subject Classification 49J53 · 90C30
1 Introduction We consider the standard bilevel optimization problem min{F(x, y) | x ∈ X , y ∈ S(x)},
(BPP)
x,y
B
Patrick Mehlitz [email protected] Leonid I. Minchenko [email protected] Alain B. Zemkoho [email protected]
1
Brandenburgische Technische Universität Cottbus–Senftenberg, 03046 Cottbus, Germany
2
Belarus State University of Informatics and Radioelectronics, 6 P. Brovki Street, Minsk 220013, Belarus
3
School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK
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where F : Rn × Rm → R is a locally Lipschitz continuous function, the set X ⊂ Rn is nonempty and closed, and S : Rn ⇒ Rm is the solution mapping of a standard parametric optimization problem, i.e., ∀x ∈ Rn :
S(x) := argmin y { f (x, y) | g(x, y) ≤ 0}.
Above, the data functions f : Rn × Rm → R and g : Rn × Rm → Rq are assumed to be continuous. The components of g will be addressed by g1 , . . . , gq : Rn × Rm → R. Nowadays, bilevel programming is one of the most intensively investigated topics in optimization theory since, on the one hand, there exist numerous underlying applications from economics, finance, chemistry, or engineering while, on the other hand, problems of this type are quite challenging, both theoretically and numerically, see Bard [1], Dempe [6], Dempe et al. [12]. In this note, we are concerned with the concept of partial calmness, fundamental in deriving necessary optimality conditions when the so-called optimal value function reformulation (a precise definition is stated below) is under consideration, see Sect. 3 for details. To introduce this reformulation, we exploit the function ϕ : Rn → R given by ∀x ∈ Rn :
ϕ(x) := inf { f (x, y) | g(x, y) ≤ 0}. y
Clearly, ϕ is the so-called optimal value function of the parametric optimization problem min y { f (x, y) | g(x, y) ≤ 0} which is referred to as the lower level problem of (BPP). It is well known that (BPP) is equivalent to its optimal value function reformulation defined by min{F(x,
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