Applications of Mathematics, Vol. 65, No. 6, pp. 777-783, 2020
Optimization problem under two-sided $(\max,+)/(\min,+)$ inequality constraints
Received January 2, 2020. Published online September 15, 2020.
Abstract: $(\max,+)$-linear functions are functions which can be expressed as the maximum of a finite number of linear functions of one variable having the form $f(x_1, \dots, x_h) = \max_j(a_j+ x_j)$, where $a_j$, $j = 1, \dots, h$, are real numbers. Similarly $(\min,+)$-linear functions are defined. We will consider optimization problems in which the set of feasible solutions is the solution set of a finite inequality system, where the inequalities have $(\max,+)$-linear functions of variables $x$ on one side and $(\min,+)$-linear functions of variables $y$ on the other side. Such systems can be applied e.g. to operations research problems in which we need to coordinate or synchronize release and completion times of operations or departure and arrival times of passengers. A motivation example is presented and the proposed solution method is demonstrated on a small numerical example.
Keywords: nonconvex optimization; $(\max,+)/(\min,+)$-linear functions; OR - arrival-departure coordination
References:  P. Butkovič: Max-Linear Systems: Theory and Algorithms. Springer Monographs in Mathematics. Springer, London (2010). DOI 10.1007/978-1-84996-299-5 | MR 2681232 | Zbl 1202.15032
 R. Cuninghame-Green: Minimax Algebra. Lecture Notes in Economics and Mathematical Systems 166. Springer, Berlin (1979). DOI 10.1007/978-3-642-48708-8 | MR 0580321 | Zbl 0399.90052
 N. K. Krivulin: Methods of Idempotent Algebra in Modelling and Analysis of Complex Systems. Saint Peterburg University Press, St. Petersburg (2009). (In Russian.)
 N. N. Vorobjov: Extremal algebra of positive matrices. Elektron. Inform.-verarb. Kybernetik 3 (1967), 39-70. (In Russian.) MR 0216854 | Zbl 0168.02603
Affiliations: Karel Zimmermann, Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostranské nám. 2/25, 118 00 Praha 1, e-mail: email@example.com