Cone Characterizations of Approximate Solutions in Real Vector Optimization

In multi-objective optimization, one considers optimization problems with more than one objective function, and in general these objectives conflict each other. As the solution set of a multiobjective problem is often rather large and contains points of no interest to the decision-maker, strategies are sought that reduce the size of the solution set. One such strategy is to combine several objectives with each other, i.e. by summing them up, before employing tools to solve the resulting multiobjective optimization problem. This approach can be used to reduce the dimensionality of the solution set as well as to discarde certain unwanted solutions, especially the 'extreme' ones found by minimizing just one of the objectives given in the classical sense while disregarding all others. In this paper, we discuss in detail how the strategy of combining objectives linearly influences the set of optimal, i.e. efficient solutions.

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“…The result generalizes a result given in [5] for C = R p + . In case the matrix A is a componentwise non-negative (i.e.…”

Section: We Have M(mop K) ⊆ M(a-mop C)supporting

confidence: 89%

On the effects of combining objectives in multi-objective optimization

2015

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“…In another paper, Engau and Wiecek (2005a) study the problem of maximizing over the set of ε-efficient solutions in a more abstract setting and propose an alternative characterization of the efficient solution set for the ε-MOP. To clarify terminology, we first define ε-solutions as fully relaxed ε-Pareto outcomes in the following sense.…”

Section: Exact Generating Methodsmentioning

confidence: 99%

Exact generation of epsilon-efficient solutions in multiple objective programming

Engau

Wiecek

2006

OR Spectrum

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Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. AbstractIt is a common characteristic of many multiple objective programming problems that the efficient solution set can only be identified in approximation: since this set often contains an infinite number of points, only a discrete representation can be computed, and due to numerical difficulties, each of these points itself might in general be only approximate to some efficient point. From among the various approximation concepts, this paper considers the notion of epsilon-efficiency which has also been shown to be of relevance other than merely for the purpose to approximate solutions.

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“…We denote the set of e-efficient decisions by EðX; f ; D; eÞ ¼ EðX; f ; D e Þ, where D e ¼ D þ e is the characterizing e-translated domination cone [9], and we use the analogous convention for the other weakly e-efficient and e-nondominated sets. While Definition 2.2 was initially introduced as theoretical generalization of Pareto optimality by [22] and later adjusted for general cones, it is only recently that approximate efficiency has received increasing attention also for practical applications in multiobjective programming [including the author's work in [10,11], and other references therein].…”

Section: Preliminariesmentioning

confidence: 99%

“…It is minimal, however, in the following sense: There does not exist anotherẽ 2 D such thatx isẽ-efficient and e 2ẽ þ D ¼ Dẽ [9]. 2.…”

Section: Similar For a Vector D ¼ ðDmentioning

confidence: 99%