# Monotone Approximations of Minimum and Maximum Functions and Multi-objective Problems

Monotone Approximations of Minimum and Maximum Functions and Multi-objective Problems In this paper the problem of accomplishing multiple objectives by a number of agents represented as dynamic systems is considered. Each agent is assumed to have a goal which is to accomplish one or more objectives where each objective is mathematically formulated using an appropriate objective function. Sufficient conditions for accomplishing objectives are derived using particular convergent approximations of minimum and maximum functions depending on the formulation of the goals and objectives. These approximations are differentiable functions and they monotonically converge to the corresponding minimum or maximum function. Finally, an illustrative pursuit-evasion game example with two evaders and two pursuers is provided. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Monotone Approximations of Minimum and Maximum Functions and Multi-objective Problems

, Volume 66 (3) – Dec 1, 2012
19 pages

/lp/springer_journal/monotone-approximations-of-minimum-and-maximum-functions-and-multi-aaqTr0mLG4
Publisher
Springer-Verlag
Subject
Mathematics; Mathematical Methods in Physics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Numerical and Computational Physics; Theoretical, Mathematical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-012-9179-8
Publisher site
See Article on Publisher Site

### Abstract

In this paper the problem of accomplishing multiple objectives by a number of agents represented as dynamic systems is considered. Each agent is assumed to have a goal which is to accomplish one or more objectives where each objective is mathematically formulated using an appropriate objective function. Sufficient conditions for accomplishing objectives are derived using particular convergent approximations of minimum and maximum functions depending on the formulation of the goals and objectives. These approximations are differentiable functions and they monotonically converge to the corresponding minimum or maximum function. Finally, an illustrative pursuit-evasion game example with two evaders and two pursuers is provided.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Dec 1, 2012

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