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Managing Groundwater Resources in a Complex Environment: Challenges and Opportunities
Published in M. Thangarajan, Vijay P. Singh, Groundwater Assessment, Modeling, and Management, 2016
In order to take into consideration stakeholders' objectives, we transform the above single objective into a multiobjective problem. It is worth noting that in multiobjective optimization one is interested in optimizing (minimizing or maximizing) several objectives simultaneously (an approach also known as vector optimization). This approach recognizes the fact that not all the objectives can achieve their optimal values simultaneously unless the objectives are not competing (conflicting). This means that there is no unique solution to such problems. However, one may establish a specific numeric goal (also known as aspiration level) for each of the objectives and then seek a solution that minimizes the sum of deviations of the objective functions from their respective goals. This solution process is known as goal programming.
Mathematical methods for structural synthesis
Published in József Farkas, Károly Jármai, Analysis and Optimum Design of Metal Structures, 2020
The basis of these methods is the transformation of the vector optimization problem into a sequence of single objective optimization problems by retaining one selected objective as the primary criterion to be optimized and treating the remaining criteria as some predetermined constraints. These constants are then altered within their defined ranges, and the subset of Pareto optima is systematically generated. This approach has gained wide acceptance because it is more practical and rational than the weighting objectives method, if there is a dominant objective.
Optimality characterizations for approximate nondominated and Fermat rules for nondominated solutions
Published in Optimization, 2022
Vector optimization is an important branch in optimization, which arises widely in functional analysis, multiobjective programming, multi-criteria decision making, statistics, approximation theory and cooperative game theory. Naturally, optimization with set-valued objective maps is a substantial extension of vector optimization theory. Because optimality criteria may change during the decision-making process, the idea of optimal elements related to variable ordering structures called nondominated elements was proposed in the earliest mathematical publication [1] in the 1970s, and refer to [2–4] for some interesting applications of this topic in image registration, portfolio optimization and location theory. Since then, vector optimization with variable ordering structures has been studied extensively on the theories of scalarizations, optimality conditions, duality and numerical approaches, see [5–7] for more details.
Robust efficiency and well-posedness in uncertain vector optimization problems
Published in Optimization, 2023
Most real-world optimization problems are usually modelled as a vector optimization problem because they involve multiple (often conflicting) optimization criteria during the decision-making process. Moreover, we rarely find a feasible solution that optimizes all criteria at once, and hence various concepts for an efficient solution along with aspects concerning vector optimization problems have been extensively studied in the literature. We would like to refer the reader to [1,2] for further information on the development of this field.
Density and connectedness of optimal points with respect to improvement sets
Published in Optimization, 2023
Vector optimization has found many important applications in decision-making problems such as those in economics theory, management science and engineering design. Since proper efficiency is a natural concept in vector optimization, various notions of classic proper efficiency have appeared in the literature, such as Geoffrion proper efficiency [1], Benson proper efficiency [2], Borwein proper efficiency [3], Henig proper efficiency [4], super efficiency [5] and strict efficiency [6].