Incorrectly Posed Optimization Problems under Extremally Linear Equation Constraints

In this paper we propose an approach for solving (max, min)-separable linear equation systems. The concept of attainable set for (max, min)-separable linear equation systems will be introduced. Properties of the attainable sets will be studied in detail. The (max, min)-separable linear equation systems, in which the function of unknown variable occur only on one side, will be consider. In this case, attainable set means that "the set of all real vectors on the right hand side of linear separable equation, which make the separable linear equation system solvable". Optimization problem consisting in finding the nearest point of an attainable set to a given fixed point will be considered. An algorithm for solving the optimization problem will be proposed. Motivational example from the area of operations research, which shows possible applications of the optimization problem solved in the paper, will be given. Two numerical examples illustrating the proposed algorithm are included. Hints for further research will be briefly discussed in the conclusions.