scholarly journals Multi-objective dynamic programming with limited precision

2021 ◽  
Author(s):  
L. Mandow ◽  
J. L. Perez-de-la-Cruz ◽  
N. Pozas
Keyword(s):  
Dynamic Programming ◽  
Pareto Front ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Value Iteration ◽  
Multi Objective ◽  
All Solutions ◽  
Markov Decision ◽  
Objective Value ◽  
Limited Precision

AbstractThis paper addresses the problem of approximating the set of all solutions for Multi-objective Markov Decision Processes. We show that in the vast majority of interesting cases, the number of solutions is exponential or even infinite. In order to overcome this difficulty we propose to approximate the set of all solutions by means of a limited precision approach based on White’s multi-objective value-iteration dynamic programming algorithm. We prove that the number of calculated solutions is tractable and show experimentally that the solutions obtained are a good approximation of the true Pareto front.

scholarly journals Unified Polynomial Dynamic Programming Algorithms for P-Center Variants in a 2D Pareto Front

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 453
Author(s):  
Nicolas Dupin ◽  
Frank Nielsen ◽  
El-Ghazali Talbi
Keyword(s):  
Dynamic Programming ◽  
Pareto Front ◽  
Dynamic Programming Algorithm ◽  
Efficient Solutions ◽  
Programming Algorithm ◽  
Planar Case ◽  
Multi Objective ◽  
Isolated Points ◽  
Time Specific ◽  
Pareto Fronts

With many efficient solutions for a multi-objective optimization problem, this paper aims to cluster the Pareto Front in a given number of clusters K and to detect isolated points. K-center problems and variants are investigated with a unified formulation considering the discrete and continuous versions, partial K-center problems, and their min-sum-K-radii variants. In dimension three (or upper), this induces NP-hard complexities. In the planar case, common optimality property is proven: non-nested optimal solutions exist. This induces a common dynamic programming algorithm running in polynomial time. Specific improvements hold for some variants, such as K-center problems and min-sum K-radii on a line. When applied to N points and allowing to uncover M<N points, K-center and min-sum-K-radii variants are, respectively, solvable in O(K(M+1)NlogN) and O(K(M+1)N2) time. Such complexity of results allows an efficient straightforward implementation. Parallel implementations can also be designed for a practical speed-up. Their application inside multi-objective heuristics is discussed to archive partial Pareto fronts, with a special interest in partial clustering variants.

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