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Fakultät Für Mathematik Und Informatik FFakultätakultät fürfür MMathematikathematik uundnd IInformatiknformatik Preprint 2016-06 Susanne Franke, Patrick Mehlitz, Maria Pilecka Optimality conditions for the simple convex bilevel programming problem in Banach spaces ISSN 1433-9307 Susanne Franke, Patrick Mehlitz, Maria Pilecka Optimality conditions for the simple convex bilevel programming problem in Banach spaces TU Bergakademie Freiberg Fakultät für Mathematik und Informatik Prüferstraße 9 09599 FREIBERG http://tu-freiberg.de/fakult1 ISSN 1433 – 9307 Herausgeber: Dekan der Fakultät für Mathematik und Informatik Herstellung: Medienzentrum der TU Bergakademie Freiberg TU Bergakademie Freiberg Preprint Optimality conditions for the simple convex bilevel programming problem in Banach spaces Susanne Franke, Patrick Mehlitz, Maria Pilecka Abstract The simple convex bilevel programming problem is a convex minimization problem whose feasible set is the solution set of another convex optimization problem. Such problems appear frequently when searching for the projection of a certain point onto the solution set of another program. Due to the nature of the problem, Slater's constraint qualification generally fails to hold at any feasible point. Hence, one has to formulate weaker constraint qualifications or stationarity notions in order to state optimality conditions. In this paper, we use two different single-level reformulations of the problem, the optimal value and the Karush-Kuhn-Tucker approach, to derive optimality conditions for the original program. Since all these considerations are carried out in Banach spaces, the results are not limited to standard optimization problems in Rn. On the road, we introduce and discuss a certain concept of M-stationarity for mathematical programs with complementarity constraints in Banach spaces. Keywords Bilevel Programming · Convex Programming · Constraint Qualifications · Mathematical Program with Complementarity Constraints · Programming in Banach Spaces Mathematics Subject Classification (2000) 46N10 · 49K27 · 90C25 · 90C33 Dedicated to Professor Stephan Dempe on the occasion of his 60th birthday. 1 Introduction Bilevel programming problems were first formulated by Stackelberg in his investigation of market econ- omy in 1934 [49], whereas the first mathematical model can be found in [10]. Since then, bilevel problems have been studied thoroughly from both a theoretical point of view and with regard to applications as well as solution strategies, see the monographs [6,12,46]. In order to handle a bilevel programming prob- lem mathematically, it is usually transformed into an optimization problem with only one level. Two such possibilities which we will use in our paper are the optimal value reformulation, introduced in [39], and the Karush-Kuhn-Tucker (KKT) approach, see for example [34]. Instead of the standard bilevel programming problem which has one variable for the upper and one for the lower level, we investigate the special (simple) case where both levels share the same variable. Hence, consider the simple convex bilevel programming problem (SCBPP for short) which is given as stated below: F (x) ! min (SCBPP) x 2 Ψ := Argminff(y) j y 2 Θg: Here, F; f : X! R are convex, continuous, and sufficiently smooth functionals of a Banach space X , while Θ ⊆ X is a nonempty, closed, and convex set. Note that these assumptions guarantee that the set Ψ is convex, i.e. (SCBPP) is a convex optimization problem. However, the derivation of optimality S. Franke, P. Mehlitz (corresponding author), and M. Pilecka Technische Universit¨atBergakademie Freiberg, Faculty of Mathematics and Computer Science, 09596 Freiberg, Germany E-mail: fsusanne.franke,mehlitz,[email protected], the work of the first author has been supported by the Deutsche Forschungsgemeinschaft, grant DE-650/7-1 2 Susanne Franke, Patrick Mehlitz, Maria Pilecka conditions is a challenging problem since most of the standard constraint qualifications generally fail to hold at any feasible point of (SCBPP). Problems of this type arise frequently when a best point (in a certain sense) among the optimal solutions of a convex optimization problem has to be found. Note that searching for an efficient solution of a convex biobjective optimization problems can also be reformulated as (SCBPP). However, solving (SCBPP) provides only special efficient solutions of such a vector optimization problem. Regarding the finite-dimensional situation, (SCBPP) was first introduced and investigated in [48] where an application of this problem is given as well. Further discussions and its relationship to standard bilevel programming problems can be found in [13]. In order to stay as close as possible to the terminology of bilevel programming, we call f(x) ! min x 2 Θ the lower level problem of (SCBPP). If the lower level feasible set Θ is described by conic constraints, it is not difficult to show, see Theorem 5.5, that (SCBPP) is equivalent to a certain mathematical program with complementarity constraints (MPCC for short) in Banach spaces, recently introduced and studied in [36, 51, 53]. These problems in finite-dimensional spaces have been the subject of theoretical research for many years. Since the usual KKT conditions for such problems are very restrictive, several different stationarity conditions have been defined, see [18, 41, 45, 60]. Furthermore, since some of the well-known constraint qualifications are violated at every feasible point of such problems, other regularity conditions for MPCCs have been introduced, cf. [17, 19]. The numerical approach to MPCCs is developed as well, see [29] and references therein. For the systematic and comprehensive study on this topic, we refer the interested reader to [34]. Moreover, a study of MPCCs in very general settings can also be found in [38]. Due to the framework of the considered problem in Banach spaces, it is also connected to works on optimal control. Hierarchical programming in function spaces is the subject of discussion in [8,36,55,56]. The special case where the solution of a variational inequality in Sobolev spaces has to be controlled was investigated in [22, 27, 28, 31, 37, 50]. It turns out that the strong stationarity conditions of the surrogate problem are, in general, too strong to hold at the local minimizers of (SCBPP), see Section5. Hence, weaker stationarity notions have to be considered. Therefore, in this paper, we want to consider a generalized concept of Mordukhovich- stationarity (M-stationarity for short) which is known to be weaker than strong stationarity for common finite-dimensional MPCCs. We organized this article as follows: In Section2, we introduce the notation and preliminary results we use throughout the paper. Section3 is dedicated to the study of constraint qualifications for optimization problems in Banach spaces. Afterwards, MPCCs in Banach spaces are discussed in Section4. Here, we recall the notions of weak and strong stationarity already known from [36,51]. Furthermore, we introduce a concept of M-stationarity which is shown to be reasonable in the case where the cone which generates the complementarity constraint is polyhedral. Finally, we consider (SCBPP) in Section5. The lower level optimal objective value and the lower KKT conditions of (SCBPP) are used to state two single-level surrogate problems which are exploited to derive optimality conditions for (SCBPP). The aforementioned KKT approach leads to necessary optimality conditions of M-stationarity-type. 2 Notation and preliminary results 2.1 Notation n n;+ Let us start this section by introducing the notation we exploit in this paper. We use N, R, R, R , R0 , n×m R , and Sn to denote the natural numbers (without zero), the real numbers, the extended real line, the set of all real vectors with n components, the cone of all vectors from Rn possessing nonnegative components, the set of all real matrices with n rows and m columns, and the set of all symmetric matrices n×n + 1;+ n×m from R , respectively. Furthermore, we stipulate R0 := R0 . For an arbitrary matrix Q 2 R and jI|×m an index set I ⊆ f1; : : : ; ng, QI 2 R denotes the matrix composed of the rows of Q whose indices n×m n×n come from I. Furthermore, O 2 R and I 2 R represent the zero matrix and the identity matrix Optimality conditions for the simple convex bilevel programming problem in Banach spaces 3 + − of appropriate dimension, respectively. By Sn and Sn we denote the cone of all positive and negative semidefinite matrices from Sn, respectively. The forthcoming definitions and results in this section are based on [5, 22, 38, 40]. Let X be a real Banach space with norm k·kX and zero vector 0, and let A ⊆ X be nonempty. We denote by lin(A), cone(A), conv(A), int(A), and cl(A) the smallest subspace of X containing A, the smallest convex cone containing A, the convex hull of A, the interior of A, and the closure of A, respectively. The indicator function δA : X! R of the set A is defined by ( 0 if x 2 A; 8x 2 X : δA(x) := +1 if x2 = A: The (topological) dual space of X is denoted by X ?, and h·; ·i: X × X ? ! R is the corresponding dual pairing. Now fix some x 2 A. Then we define the radial cone, the tangent (or Bouligand) cone, and the weak tangent cone to A at x as stated below: RA(x) := fd 2 X j 9α > 0 8t 2 (0; α): x + td 2 Ag; TA(x) := fd 2 X j 9fdkg ⊆ X 9ftkg ⊆ R: dk ! d; tk & 0; x + tkdk 2 A 8k 2 Ng; w TA (x) := fd 2 X j 9fdkg ⊆ X 9ftkg ⊆ R: dk * d; tk & 0; x + tkdk 2 A 8k 2 Ng: Here, ! and * denote the norm and the weak convergence in X , respectively. Furthermore, we will use ? ? ! to denote the weak-?-convergence in X . Observe that the cone TA(x) is always a closed set which satisfies w RA(x) ⊆ TA(x) ⊆ TA (x): " " Let BX be the closed unit ball of X , while for any " > 0 and x 2 X , we use UX (x) and BX (x) to denote the open and closed "-ball around x, respectively.
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