On Proper Efficiency in Multiobjective Semi-infinite Optimization
This chapter deals with multiobjective semi-infinite optimization problems which are defined by finitely many objective functions and infinitely many inequality constraints in a finite-dimensional space. We discuss constraint qualifications as well as nec
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Abstract This chapter deals with multiobjective semi-infinite optimization problems which are defined by finitely many objective functions and infinitely many inequality constraints in a finite-dimensional space. We discuss constraint qualifications as well as necessary and sufficient conditions for locally weakly efficient solutions. Furthermore, we generalize two concepts of properly efficient solutions to the semi-infinite setting and present corresponding optimality conditions.
1 Introduction This chapter deals with multiobjective semi-infinite optimization problems which are nonlinear problems with finitely many objective functions whose variables belong to a finite-dimensional space and whose feasible sets are defined by infinitely many inequality constraints. There is a wide range of applications of semi-infinite optimization and of multiobjective optimization; both topics, their theory and numerical analysis, became very active research areas in the recent two decades. We refer to several recent books [Gob01, Polak97, RR98]; in particular to the standard book [Ehr05] on vector optimization.
This work was partially supported by SNI (Sistema Nacional de Investigadores, México), grant 14480. F. Guerra-Vázquez (B) Escuela de Ciencias, Universidad de las Américas Puebla, San Andrés Cholula, 72820 Puebla, México e-mail: [email protected] J.-J. Rückmann Department of Informatics, University of Bergen, PO Box 7803, 5020 Bergen, Norway e-mail: [email protected] H. Xu et al. (eds.), Optimization and Control Techniques and Applications, Springer Proceedings in Mathematics & Statistics 86, DOI: 10.1007/978-3-662-43404-8_6, © Springer-Verlag Berlin Heidelberg 2014
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As a starting point of this chapter we consider a multiobjective semi-infinite optimization problem (MOSIP) of the form MOSIP “min” f (x) s.t. x ∈ M with • the vector of objective functions f = f 1 , . . . , f p , where f i : Rn → R, i = 1, . . . , p are continuously differentiable and • the feasible set M = x ∈ Rn | g(x, y) ≤ 0, y ∈ Y , where Y ⊆ Rm is a compact infinite index set and g : Rn × Rm → R is continuous as well as continuously differentiable with respect to x. Obviously, each index y ∈ Y represents a corresponding constraint g (x, y) ≤ 0. For x ∈ M we define the set of active inequality constraints at x ∈ M as Y0 (x) = {y ∈ Y | g (x, y) = 0} . It is obvious that for x ∈ M each index y ∈ Y0 (x) is a global maximizer of the corresponding parameter-dependent (x is the parameter) problem max g (x, y) s.t. y ∈ Y whose non-differentiable optimal value function ϕ (x) = max g (x, y) y∈Y
can be used for describing the feasible set as M = x ∈ Rn | ϕ(x) ≤ 0 (note that the set Y0 (x) can be empty). We mention that some of the results presented in this chapter can be described by using this optimal function (in case that the point under consideration is a boundary point of M). The objective of this chapter is as follows. Although there are many applications which can be modeled
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