Optimality conditions for pessimistic semivectorial bilevel programming problems
© Liu et al.; licensee Springer. 2014
Received: 13 September 2013
Accepted: 26 December 2013
Published: 24 January 2014
In this paper, a class of pessimistic semivectorial bilevel programming problems is investigated. By using the scalarization method, we transform the pessimistic semivectorial bilevel programming problem into a scalar objective optimization problem with inequality constraints. Furthermore, we derive a generalized minimax optimization problem using the maximization bilevel optimal value function, of which the sensitivity analysis is constructed via the lower-level value function approach. Using the generalized differentiation calculus of Mordukhovich, the first-order necessary optimality conditions are established in the smooth setting. As an application, we take the optimality conditions of the bilevel programming problems with multiobjective lower level problem when the lower level multiobjective optimization problem is linear with respect to the lower-level variables.
MSC:90C26, 90C30, 90C31, 90C46.
Keywordssemivectorial bilevel programming multiobjective optimization weakly efficient solution bilevel optimal value function sensitivity analysis optimality condition
where is the lower-level constrained function, is the lower-level multiobjective function. The term ‘’ in (1.3) is used to symbolize that vector-values in the lower-level problem are in the sense of weak Pareto minima (see Section 2) with respect to an order induced by the positive orthant of .
In order to ensure that the results in this paper are correct, we make some hypotheses throughout the paper as follows.
Hypothesis 1 The set is nonempty and compact.
Hypothesis 2 For any x verifying , the set is nonempty and compact.
Generally speaking, the weakly efficient solution set of the lower-level problem (1.3) and (1.4) is not singleton, i.e., the set in (1.2) has more than one point. In this case, the notion of an optimal solution of the bilevel programming problem may be ambiguous. That is why the word ‘min’ is written in quotes in (1.1). Two ways to deal with this situation are given by the optimistic formulation and the pessimistic formulation in .
For the research papers on the optimistic formulation of the semivectorial bilevel programming problem one is referred to [1, 3–9]. In , a penalty method was given to solve the problem in case of weakly efficient solutions in the lower-level problem (1.2). Zheng and Wan  developed another penalty method consisting of two penalty parameters in the case where the multiobjective lower-level problem is linear. Ankhili and Mansouri  developed an exact penalty method for the problem in the case where the upper-level problem was concave and the lower-level problem was a linear multiobjective optimization problem. Eichfelder  considered the problem in the case where F is also vector-valued. In the latter paper, the induced set of the investigated problem is shown to be the set of minima point (with respect to a cone) of another unperturbed multiobjective optimization problem. Hence, the resulting problem is simply a multiobjective optimization problem over an efficient set. Then it is solved by using a scalarization method by Pascoletti and Serafini combined with an adaptive parameter control method based on sensitivity for the problem. Recently, Calvete and Galé  also considered the problem in the case where the upper-level objective function is quasiconcave and the lower-level problem is a linear multiobjective optimization problem. The problem was reformulated as an optimization problem over a nonconvex region given by a union of faces of the polyhedron defined by all constraints. An extreme point method was showed to deal with the problem. Then, based on the ‘k th’ best method and genetic algorithm, they developed an exact and a metaheuristic algorithm, respectively. The performance of the above two algorithms were also evaluated. In , Nie defined the risk optimal decision, conservative optimal decision and mean optimal decision of the semivectorial bilevel programming problem. Weighting methods were employed to analyze the lower-level multiobjective optimization problem, and some properties about the problem were obtained. In , Bonnel derived necessary optimality conditions for the problem (1.5) in general Banach spaces, while considering efficient and weakly efficient solutions for the lower-level multiobjective optimization the problem (1.3). In the latter paper, the author inserted the weak or properly weak solution set-valued mapping of the lower-level problem in the upper-level objective function to derive a set-valued optimization problem. Using the notion of contingent derivative, necessary optimality conditions, which are abstract in nature, were derived. In , Dempe et al. considered also the optimistic formulation of the semivectorial bilevel programming problem. Considering the scalarization approach for the lower-level multiobjective optimization problem, they transformed the problem into a scalar objective optimization problem with inequality constraints by means of the optimal value reformulation, completely detailed first-order KKT-type necessary optimality conditions were derived in the smooth and nonsmooth settings while using the generalized differentiation calculus of Mordukhovich. It is worth to mention that the method of  was different from that of .
Bonnel and Morgan  developed optimality conditions for the bilevel optimal control problem, which is a special case of semivectorial bilevel programming problem. For two extreme cases, the optimistic case and the pessimistic case, the optimality conditions were presented, respectively. In , Nie defined the conservative optimal decision for the problem (1.1) (i.e. (1.6)). Weighting methods were employed to analyze the lower-level multiobjective optimization problem when the lower-level objective functions were all continuously differentiable and strictly convex and the lower-level constraints were all continuously differentiable, then a minimax optimization problem with constraints was derived. But for the problem (1.6), no detailed optimality conditions and concrete solving methods were found in .
Hence, in this paper, our main work is as follows: Using the scalarization method and the maximization bilevel optimal value function, the pessimistic problem (1.6) is transformed into a generalized minimax optimization problem, i.e. the problem (3.4). Then, we develop a link between the problems (1.6) and (3.4), that is, Proposition 3.1, which shows that these two problems have the same local or global optimal solutions under some mild conditions. The results of Proposition 3.1 is formally similar to that of Proposition 3.1 in , but is different in nature. Based on Proposition 3.1, we transform the problem (3.4) into the problem (3.9) using the bilevel optimal value function formulation. Furthermore, we develop the necessary optimality conditions (see Theorem 4.4) for the problem (3.4) using generalized differentiation calculus of Mordukhovich. By Proposition 3.1, we obtain the necessary optimality conditions (see Corollary 4.1) for the pessimistic problem (1.6). Our results in this paper and the results in  make up together the first-order necessary optimality conditions for the semivectorial bilevel programming problem. It is very important for the development of the optimality theory of the semivectorial bilevel programming problem in the future.
The rest of the paper is organized as follows. In Section 2, we present the definitions of efficient solutions and Pareto minima, and then the relevant notions and properties from variational analysis will be presented as well. The transformation process of the pessimistic semivectorial bilevel programming problem (PSBP) into a single-level generalized minimax optimization problem with constraints by means of the optimal value function reformulation is given in Section 3. In Section 4, we first present the estimation of the lower-level negative value function and the sensitivity analysis of the lower-level optimal solution maps. Based on these, the sensitivity analysis for the maximization bilevel value function is presented. Finally, the necessary optimality conditions are derived for the problem (1.6) while considering the case where all functions involved are strictly differentiable. The special case where the lower-level multiobjective optimization problem is linear in the lower-level variable is studied in Section 5.
In this section, we mainly recall some basic definitions and results.
2.1 Efficient solution and Pareto minima
Definition 2.1 Let be a closed convex cone with nonempty interior, C is said to be pointed convex cone if . We denote a partial order by in induced by C.
where ‘int’ denotes the topological interior of the set in question.
where f represents a vector-valued function and X the nonempty feasible set. For a nonempty set , the image of A by f is defined by .
Definition 2.3 The point is said to be an efficient (resp. weakly efficient) optimal solution of problem (2.2) if is a Pareto (resp. weak Pareto) minima of .
Definition 2.4 The point is said to be a local efficient (resp. weakly local efficient) optimal solution of problem (2.2) if there exists a neighborhood U of such that is a Pareto (resp. weak Pareto) minima of .
2.2 Tools from variational analysis
where ψ is Lipschitz continuous near .
where Ξ is strictly differentiable at point , denotes its Jacobian matrix at and ‘⊤’ stands for transposition.
Definition 2.11 A set-valued mapping Ξ is said to be inner semicompact at with , if for every sequence with , there exists a sequence of which contains a convergent subsequence as .
It follows that inner semicompactness holds whenever Ξ is uniformly bounded near , i.e., there exist a neighborhood U and a bounded set such that for all .
Definition 2.12 A set-valued mapping Ξ is said to be inner semicontinuous at , if for every sequence there exists a sequence of that converges to as .
In addition, the infimum of all for which (2.16) holds is equal to the coderivative norm as a positively homogeneous mapping . Set in (2.16), the resulting weaker property is known as calmness of Ξ at , which is used to derive the sensitivity analysis of the lower-level optimal solution mapping of the problem (3.4) in the sequel. For , the Lipschitz-like property in (2.16) corresponds to the upper Lipschitz property of Robinson .
3 Optimal value function reformulation for the pessimistic semivectorial bilevel programming problem
For a given upper-level variable x, the weakly efficient solution set of the lower-level problem (1.3) is not in general a singleton, hence it is difficult to choose the best point on the set . Furthermore, we consider the set Y (3.2) as a new constraint set for the upper-level problem . For all (where ), we denote by the solution set of the problem (3.1). When the weakly efficient solutions are considered for the lower-level problem (1.3), the relationship (see e.g. ) relates the solution set of this problem and that of (3.1) as follows.
Remark 3.1 The variable y in (3.4) is regarded as an upper-level decision making variable rather than lower-level decision variable. That is why we use the representation ‘’ and not use the representation ‘’. The hierarchically decision making process of the pessimistic bilevel programming problem (3.4) is as follows: The leader announces his variables first and then the follower, bearing in mind, optimizes the objective function of himself and reacts the lower-level decision making variable z which is an optimal solution of the lower-level problem. In essence, we regard y in (3.4) as a weight vector to which the leader attaches the follower rather than the follower gives himself. For the problem (3.4), the existence and approximation of solution, the regularization properties and so on were studied in [11, 12, 17–27]. In , Loridan and Morgan considered the pessimistic formulation (i.e., the weak Stackelberg problem). Based on a method of Molodtsov, they presented an approach to approximate such problem by sequences of the optimistic formulation (i.e., the strong Stackelberg problem). The results related to the convergence of marginal functions and approximated solutions were given and the case of data perturbations was also considered. In , Aboussoror and Mansouri considered a class of weak linear bilevel programming with nonunique lower-level solutions, they gave an existence theorem of solution and a solving algorithm via exact penalty method. In , Lv et al. developed a penalty function method for the weak price control problem. In , Tsoukalas et al. provided an introduction to bilevel programming problems that illustrates some of the applications and computational challenges, and that outlines how bilevel programming problems can be solved. In , they and Kleniati provided a formal justification for the conjectures given in , the computational complexity of pessimistic bilevel programming problems were examined, and a solution scheme was developed and analyzed for the pessimistic programming problems. In , Malyshev and Strekalovsky considered the pessimistic formulation of a quadratic-linear bilevel programming problem, they reduced the problem to a series of bilevel programming problems in its optimistic formulation and then to nonconvex optimization problems by the KKT-optimality condition of the lower-level problem. Global and local search algorithms for the latter problems are developed. In , Dassanayaka studied the pessimistic formulation of the bilevel programming problems in finite dimensional spaces. Using the analysis tools from modern variational and generalized differentiation developed by Mordukhovich, first-order necessary and sufficient optimality conditions were established. A genetic algorithm for the weak linear bilevel programming problem was developed by Xiao and Li in . Very recently, the pessimistic formulation for the bilevel programming problem was considered by Zemkoho in  and by Dempe et al. in  in the case where the functions involved were nonsmooth and smooth, respectively, the necessary optimality conditions were derived via the bilevel optimal value function reformulation.
It is called a global pessimistic solution if can be selected.
Now, we present the theorem of the existence of solution to the problem (3.4) (see [, Theorem 5.3]).
Theorem 3.2 If the set is nonempty and compact, and for each , the Mangasarian-Fromowitz Constraint Qualification (MFCQ) holds. Suppose that the lower-level solution set mapping is lower semicontinuous at all points . Then the problem (3.4) has an optimal solution.
Proof Due to lower semicontinuity of the lower-level solution set mapping , thus, the optimal value function in (3.5) is lower semicontinuous. Hence, this optimal value function attains its minimum on the compact set provided that this set is nonempty. □
The link between the problems (1.6) and (3.4) will be given in the next result. For this purpose, note that a set-valued mapping is closed-valued at a point if for any sequence with , one has . Ξ is said to be closed-valued if it is closed-valued at any point of .
Let be a local (resp. global) optimal solution of the problem (1.6). Then, for all with , the point is a local (resp. global) optimal solution of the problem (3.4).
Let be a local (resp. global) optimal solution of the problem (3.4). Assume the set-valued mapping Ψ is closed-valued. Then is a local (resp. global) optimal solution of the problem (1.6).
- (i)Let be a local optimal solution of the problem (1.6). Then
- (ii)Assume that is a local optimal solution of the problem (3.4). Then we have
Therefore is a local optimal solution of the problem (1.6). This completes the proof. □
where and . Denote the argminimum mapping in (3.14) by . We summarize in the next theorem some known results on general value functions needed in the paper (see [, Corollary 1.109] and [, Theorem 5.2]).
- (i)Let be inner semicompact at . Then μ is lower semicontinuous at and the upper bound for its basic subdifferential is given as follows:
If in addition Ξ is Lipschitz-like around for all vectors , then we also have the Lipschitz continuity of μ around .
- (ii)Let be inner semicontinuous at . Then μ is lower semicontinuous at and the upper bound for its basic subdifferential is given as follows:
If in addition Ξ is Lipschitz-like around , then μ is Lipschitz continuous around .
By Theorem 3.3, we can estimate the upper bound of the subdifferential of the bilevel optimal value function (resp. ) via estimating the subdifferential of (resp. ). In the next section, based on specific structures of the set-valued mapping Ξ, our aim is to give detailed upper bounds for in terms of problem data. Verifiable rules for Ξ to be Lipschitz-like will also be provided. Further, we present the sensitivity analysis for the maximization bilevel optimal value function and . Based on these results, we develop the necessary optimality conditions for the problems (3.4) and (1.6).
4 Main results
Since the basic subdifferential ∂φ does not satisfy the plus symmetry, an appropriate estimate of is needed to proceed with this approach. By the well-known convex hull property (2.8), the estimate of can be done.
It is clear that these are the dual forms of the MFCQ for the lower-level constraints , (for the fixed parameter ) and the upper-level constraints , , respectively. A particularity of the new constraint set Y (3.2), that the related Lagrange multipliers can be completely eliminated from the optimality conditions, is given in the next lemma (see [, Lemma 4.2]).
4.1 Sensitivity analysis of the lower-level negative value function
In this subsection, we shall study the sensitivity analysis of the negative value function in the lower-level problem (3.1).
- (i)If the solution map in (4.1) is inner semicompact at for all satisfying (4.4), then φ is Lipschitz continuous around , and the following inclusion holds:(4.8)
- (ii)Assume that with satisfying (4.4) and that either Ψ is inner semicontinuous at this point or f and g are convex. Then φ is Lipschitz continuous around , and the following inclusion holds:(4.9)
by [, Corollary 4] under the assumptions of (i). The claimed estimate of follows from this by combining (2.8) and the classical Carathéodory’s theorem.
This implies the subdifferential inclusion of (ii) by (2.7) and (2.8). If both f and g are convex, inclusion (4.11) holds without the inner semicontinuity assumption (see [, Theorem 4.2] and [, Corollary 4]). This completes the proof. □
Note that in the fully convex (even nonsmooth) case, the assumption (4.4) in Theorem 4.1 can be replaced by a much weaker qualification condition  requiring that the set is closed on , where denotes the conjugate function for an extended real-valued function f.
4.2 Sensitivity analysis of the lower-level optimal solution maps
with K being semismooth, in particular, convex. The condition (4.15) seems to be especially effective for so-called simple convex bilevel programming problems. For the more details, the readers can be refer to [45, 46]. It is deserved that for the latter case, the condition (4.15) can be further weakened by passing to the boundary of the subdifferential of f . It is also worth mentioning that, except the condition (4.15), another sufficient condition for the validity of the calmness property (4.14) is provided by the notion of uniform weak sharp minima. More details can be found in [34, 43, 47].
Now, we present the coderivative estimate and Lipschitz-like property of lower-level solution maps.
with and for . If in addition the condition (4.16) holds at , then Ψ is Lipschitz-like around this point.
If in addition the condition (4.16) holds at , then Ψ is Lipschitz-like around this point.
by [, Theorem 4.1] taking into account that the constraint is active at point . By (4.12), the equality (4.21) holds. Combining the definition of the coderivative (2.15), we derive the coderivative estimate (4.20). Further, by (4.20) and the coderivative criterion (2.17) for the Lipschitz-like property, the coderivative criterion holds provided the (4.22). This completes the proof. □
Noting that if the functions f and g are convex, the inner semicontinuity of Ψ can be dropped in Theorem 4.2(ii).
4.3 Sensitivity analysis of the maximization bilevel optimal value functions and using the lower-level value function approach
In the rest of this paper, we always assume that the set is nonempty. The following results illustrate the local sensitivity analysis of the bilevel value function defined in (3.5).
- (i)Assume that is inner semicompact at , the condition (4.4) holds at , while the condition (4.14) holds at for all . Then the following inclusion holds:(4.24)
- (ii)Assume that is inner semicontinuous at , the conditions (4.4) and (4.14) hold at this point. Furthermore, assume that the set is closed. Then the following inclusion holds:(4.25)
If in addition (4.16) is satisfied at point , then is Lipschitz continuous around .
under the inner semicompactness assumption on . Since for all , the lower-level optimal solution map Ψ in (4.1) is also inner semicompact at . Hence, by the subdifferential of the lower-level negation value function −φ in Theorem 4.1(i) and the coderivative of Ψ in Theorem 4.2(i), combining with (3.15) and Carathéodory’s theorem, we can derive the upper estimate of .
To prove the local Lipschitz continuity of in (i) under the condition (4.16), the latter condition implies the Lipschitz-like property of Ψ around . Thus the desired result is obtained from Theorem 3.3(i).
The latter inclusion implies that provided the set is closed. Combining the above two results, by (2.7) and (3.15), we can justify (4.25). To justify the local Lipschitz continuity of in (ii) under the condition (4.16), the latter condition implies the Lipschitz-like property of Ψ around . This completes the proof. □
By (3.16), we derive that and is Lipschitz continuous around under the corresponding conditions of Theorem 4.3.
Remark 4.1 Observe that for the subdifferential estimate of in Theorem 4.3(ii), the upper bound of the basic subdifferential does not contain the partial derivative of the lower-level objective function with respect to the upper-level variable x. Such a phenomenon is no longer true if the inner semicontinuity for is replaced by the inner semicompactness in Theorem 4.3(i), this phenomenon can also be found in [6, 40]. We mention that the inner semicompactness of in Theorem 4.3(i) can be replaced by the restrictive uniform boundedness assumption on or even on Ψ. Finally, by Theorem 3.3(ii) and Theorem 4.2(ii), we can derive the subdifferential estimate for , which is different from (4.25). In this case, the gradient of the upper-level objective function F is involved in the convex combinations summation, while that of (4.25) not be. This will be shown in the following Theorem 4.4(ii).
4.4 Necessary optimality conditions using the bilevel optimal value function formulation
In this subsection, we shall establish the necessary optimality conditions for the optimal value reformulation (3.9) of the problem (3.4) using the above sensitivity analysis results.
- (i)Let be inner semicompact at while for all and the point is lower-level regular, let f and g be strictly differentiable at , , and let the conditions (4.14) and (4.16) be satisfied at all point with . Then there exist , α, , , and with such that (4.7) and the following conditions hold:(4.26)
The relationships (4.7) and (4.26) considered together are called the KM-stationarity conditions.
- (ii)Let be inner semicontinuous at , be lower-level regular, f and g be strictly differentiable at , and let the conditions (4.14) and (4.16) be satisfied at and the set be closed. Then there exist , α, , and such that the following conditions hold:(4.27)
The relationships (4.27) are called the KN-stationarity conditions.
Combining with Theorem 4.3(ii) and (4.29), Theorem 4.4(ii) is easily derived. If is inner semicompact around , the condition (4.7) holds at all point with , and that (4.14) and (4.16) are satisfied at all point with . Thus, by Theorem 4.3(i), we obtain the conclusion (i). This completes the proof. □
Remark 4.2 (i) The prefixes ‘KN’ and ‘KM’ in Theorem 4.4 reflect the difference between the KKT-type optimality conditions via the inner semicompactness and inner semicontinuity of the upper-level optimal solution set mapping , respectively. For the notions ‘KM-stationary’ and ‘KM-stationarity’, the readers can be referred to . In (4.27), the gradient of F does not involve the convex combinations summation, in this case, analogously to , we call (4.27) KKT-type necessary optimality (stationarity) conditions for the problem (3.4).
This means that the above framework, the constraints described by Y (3.2) can be dropped and the condition that set is closed is reduced to that the set X is closed, the latter is immediately reached by Hypothesis 1 in Section 1, while deriving the necessary optimality conditions of the problem (1.6), which is a strange phenomenon just as that said in .
By Proposition 3.1(i) and Theorem 4.4, the necessary optimality conditions for the pessimistic semivectorial bilevel programming problem (1.6) are derived when the involved functions are strictly differentiable.
(KM-stationarity conditions) Let be inner semicompact at while for all and the point is lower-level regular. Let f and g be strictly differentiable at , , and let the conditions (4.14) and (4.16) be satisfied at all point with and the set be closed. Then there exist , α, , , and with such that (4.7) and (4.26) hold.
(KN-stationarity conditions) Let be inner semicontinuous at and the point be lower-level regular. Let f and g be strictly differentiable at , be closed, and let the conditions (4.14) and (4.16) be satisfied at and the set be closed. Then there exist , α, and such that (4.27) holds.
5 BP with linear multiobjective optimization lower-level problems
where the real value functions (, ) and (, ) are strictly differentiable.
The following result which is a consequence of Theorem 4.4 is the necessary optimality conditions for the problem (5.1).
Theorem 5.1 (The necessary optimality conditions for the problem (5.1))
- (i)(KM-stationarity conditions) Let (where in (4.1) is replaced by in (5.5)) be inner semicompact at while for all , is lower-level regular in the sense of (5.7), f and g be strictly differentiable at , , and let the conditions (4.14) and (4.16) be satisfied at all point with and the set be closed. Then there exist , α, , , and with such that (4.7) (with ) and the following hold:(5.8)
- (ii)(KN-stationarity conditions) Let (where in (4.1) is replaced by in (5.5)) be inner semicontinuous at , be lower-level regular in the sense of (5.7), f and g be strictly differentiable at and the set be closed and let the conditions (4.14) and (4.16) be satisfied at and the set be closed. Then there exist , α, and such that(5.9)
Proof The proof is similar to that of Theorem 4.4 and so is omitted here. □
note that .
(ii) If F and G are linear functions with respect to their variables, then the calmness condition (4.14) is automatically satisfied in this case by . Then we can obtain the more concise results, which is the particular case of Corollary 4.1. The interested reader can try to give detailed results of the optimality conditions for this case.
In this paper, we develop the necessary optimality conditions for the pessimistic formulation of semivectorial bilevel optimization problem. Firstly, we transform our problem into a scalar objective optimization problem with inequality constraints via the scalarization method for the multiobjective optimization problem. Furthermore, we derive a generalized minimax optimization problem by means of the bilevel optimal value function, of which the sensitivity analysis is constructed via the lower-level value function approach. Considering the special case where the lower-level multiobjective optimization problem is linear, we also give it the necessary optimality conditions. In the further work, we intend to develop the necessary optimality conditions in the nonsmooth setting and develop the solving algorithms for the pessimistic formulation of semivectorial bilevel optimization problem, especially the latter, which is challenging. For the problem (1.1), if the leader is not certain of that the follower cooperates or dose not cooperate fully with him, it would be inappropriate for the leader who considers only the optimistic or pessimistic formulation. Hence, when both the leader and the follower are neither fully cooperative nor fully non-cooperative, it is meaningful to consider a partial cooperation model which combines the optimistic formulation and the pessimistic formulation for the problem (1.1).
This work was supported by the Natural Science Foundation of China (71171150, 11201039, 11301570), the Youth Foundation of Educational Commission of Anhui Province of China (2009SQRZ121, 2011SQRL097), the Doctor Fund of Southwest University (SWU113037) and the Fundamental Research Fund for the Central Universities (XDJK2014C073). The authors would like to thank two anonymous referees for numerous insightful comments and suggestions, which have greatly improved the paper.
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