An Algorithm for Vectorial Control Approximation Problems

Benker, Hans; Hamel, Andreas H.; Tammer, Christiane

doi:10.1007/978-3-642-59132-7_1

Cited by 7 publications

(6 citation statements)

References 14 publications

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“…oo and L (en, xn) exists and is finite (4) n=l n=l (see Eqs. (18) and (19) in [5]). Condition (4) is an assumption on the whole generated sequence { Xn} and the error term sequence {en}, and thus seems to be slightly stronger, but it can be checked and enforced in practice more easily than those that existed earlier.…”

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confidence: 95%

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Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

Ceng

Mordukhovich

Yao

2010

J Optim Theory Appl

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Abstract. This paper studies the general vector optimization problem of finding weakly efficient points for mappings in a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C C Y with nonempty interior. In order to find a solution of this problem, we introduce an auxiliary variational inequality problem for monotone, Lipschitz-continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem by the combination of extragradient method for finding a solution to the variational inequality problem and approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone, Lipschitz-continuous mapping, and finding weakly efficient points for suitable regularizations of the original mapping. We present both an absolute and a relative version in which the subproblems are solved only approximately. Weak convergence of the generated sequence to a weak efficient point is established under quite mild conditions. In addition, we also discuss an extension to Bregman-function-based hybrid approximate proximal algorithms for finding weakly efficient points for mappings.

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“…It is a generalization of algorithms for specifical instances of a vector optimization problem (VOP): a particular control approximation problem in [19] and certain location problems in [20]. In the presentation given in [18], it deals with a problem more general than VOP, namely, the vector equilibrium problem (VEP).…”

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Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

Ceng

Mordukhovich

Yao

2010

J Optim Theory Appl

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“…The exact version of this method is the one presented in (1), where, instead of ∂ εn f n with f n : x → f (x) + αn 2 x − x n 2 , an adequate enlargement of the operator T n , related to ε n , is used. This type of enlargement was introduced in [4].…”

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“…Given a Hilbert space X and a point-to-set operator T : X → P(X), the proximal point method, in its so-called exact version, is an iterative procedure for finding a zero of T , i.e., a point x * ∈ X such that 0 ∈ T (x * ). The method generates a sequence {x n } ⊂ X, starting from an arbitrary x 0 ∈ X, through the following iteration: given a bounded exogenous sequence of positive real numbers {α n } (called regularization parameters) and the current iterate x n , x n+1 is the unique vector in X such that 0 ∈ T n (x n+1 ), (1) where T n : X → P(X) is defined as T n (x) = T (x) + α n (x − x n ).…”

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Proximal Methods in Vector Optimization

Bonnel¹,

Iusem²,

Svaiter³

2005

SIAM J. Optim.

163

144

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We consider the vector optimization problem of finding weakly efficient points for maps from a Hilbert space X to a Banach space Y , with respect to the partial order induced by a closed, convex, and pointed cone C ⊂ Y with a nonempty interior. We develop for this problem an extension of the proximal point method for scalar-valued convex optimization. In this extension, the subproblems consist of finding weakly efficient points for suitable regularizations of the original map. We present both an exact and an inexact version, in which the subproblems are solved only approximately, within a constant relative tolerance. In both cases, we prove weak convergence of the generated sequence to a weakly efficient point, assuming convexity of the map with respect to C and C-completeness of the initial section. In cases where this last assumption fails, we still establish that the generating sequence is, in a suitable sense, a minimizing one. We also exhibit a particular instance of the algorithm for which, under a mild hypothesis on C, the weak limit of the generated sequence is an efficient, rather than a weakly efficient, point. Introduction.We discuss methods for vector-valued optimization in the following setting. We consider maps F : X → Y , where X is a real Hilbert space and Y is a real Banach space containing a closed, convex, and pointed cone C with nonempty interior, which defines a partial order C in Y , given by y C y if and only if y − y belongs to C, with its associate relation ≺ C , given by y ≺ C y if and only if y − y belongs to the interior int(C) of C. Actually, we admit the possibility that F takes the value +∞; this is made precise in section 2. Our goal is to analyze methods for finding a weakly efficient minimizer of F with respect to C , meaning a point a ∈ X such that there exists no x ∈ X satisfying F (x) ≺ C F (a). This paper is part of a wider research program consisting of the extension to vector-valued optimization of several iterative methods for scalar-valued methods. In these extensions, we attempt to define the iterates in the vector-valued case by considering the order C in Y , mimicking, whenever possible, the role of the usual order in R in the corresponding algorithm for scalar-valued optimization.Several methods already have been extended in this fashion in a finite dimensional setting. The case of the steepest descent method for multiobjective optimization (i.e., when C is the nonnegative orthant of R n ) was dealt with in [7]; the same method for general finite dimensional vector optimization (i.e., for partial orders given by rather general cones in R n ) is analyzed in [10]. An extension of the projected gradient method to the case of convexly constrained vector optimization, for the order given by a rather general cone in R n , can be found in [9].

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