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Using approximate gradients in developing an interactive interior primal-dual multiobjective linear programming algorithm

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1996年第1期89卷 202-211页

作者： Arbel, A Oren, SS UNIV CALIF BERKELEY DEPT IND ENGN & OPERAT RESBERKELEYCA 94720

We present a new interactive multiobjective linear programming algorithm that is based on one variant of Karmarkar's algorithm known as the path-following primal-dual algorithm. The modification of this single-objective linear programming algorithm to the multiobjective case is done by deriving an approximate gradient to the implicitly-known utility function. By interacting with the decision maker, locally-relevant preference information are elicited and the approximated gradient can therefore be continuously updated. The interior step direction is then generated by projecting the approximate gradient and taking an interior step from the current iterate to the new one along this projection.

关键词： Multiobjective Linear Programming (MOLP) multicriteria optimization path-following primal-dual algorithm interior-point algorithms

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An interior multiobjective primal-dual linear programming algorithm using approximated gradients and sequential generation of anchor points

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Optimization 1994年第2期30卷 119-135页

作者： Arbel, A. Industrial Engineering Department Tel-Aviv University Tel-Aviv 69978 Israel

::An algorithm for addressing multiple objective linear programming (MOLP) problems is presented. The algorithm modifies the path-following primal-dual algorithm to MOLP problems by using the singleobjective algorithm to generate interior search directions and later combine them to derive a single direction along which to step to the next iterate. Combining the different interior search directions is done by interacting with a Decision Maker (DM) to obtain locally-relevant preference information for the value vectors along these directions. This preference information is then used to derive an approximation to the gradient of an implicitly-known utility function, and using a projection of this gradient provides a direction vector along which we step to the next iterate. At each iteration the algorithm also generates boundary points that aid in deriving the combined search direction. We refer to these boundary points, generated sequentially during the process, as anchor points that serve as candidate solutions at which to terminate the iterative process. © 1994, Taylor & Francis Group, LLC. All rights reserved.

关键词： Interactive Multiple Criteria Linear Programming (MOLP) path-following primal-dual algorithm

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Using aspiration levels in an interior primal-dual multiobjective linear programming algorithm

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Journal of Multi-Criteria Decision Analysis 1996年第1期5卷 61-71页

作者： Arbel, Ami Korhonen, Pekka Industrial Engineering Department Tel-Aviv University Tel-Aviv 69978 Israel Helsinki School of Economics and Business Administration Helsinki 0100 Runeberginkatu 14-16 Finland

We introduce in this paper a new multiple-objective linear programming (MOLP) algorithm. The algorithm is based on the single-objective path-following primal-dual linear programming algorithm and combines it with aspiration levels and the use of achievement scalarizing functions. The resulting algorithm falls in the class of interactive MOLP algorithms, as it requires interaction with the decision maker (DM) during the iterative process to obtain statements of aspirations for levels of objectives of the MOLP problem. The interior point algorithm is then used to trace a path of interates from a current (interior) solution and approach as closely as desired a non-dominated solution corresponding to the optimum of the achievement scalarizing function. The timing of the interaction with the DM is dependent on the progress of the interior algorithm. It can take place every few, pre-specified, iterations or after the duality gap achieved for the stated aspirations has fallen below a certain threshold. It is expected that an interior algorithm will speed up the overall process of searching and finding the most preferred MOLP solution-especially in large-scale problems-by avoiding the need for numerous pivot operations and their corresponding interactive sessions inherent in simplex-based algorithms. © 1996 John Wiley & Sons, Ltd.

关键词： Aspiration levels Interior point algorithms Multiple-objective linear programming path-following primal-dual algorithm Scalarizing functions

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