Globally Optimal Solution Of The Max-min System Under Some Constraints

A variety of problems can be modeled as max-min systems in computer science,operations research,control theory,etc.Such as digital circuits,computer networks and automated manufacturing plants.For a manufacturing system with input structure,we need to consider the input time of raw materials,the processing time and the order of work of each machine,etc.On the basis of satisfying the system requirements,we want to control the system,and make the system work to achieve the best state.Max-min systems are described by non-linear and non-differentiable max-min functions in which the operations maximum,minimum,and addition appear simultaneously.Max-min systems are non-linear extensions of max systems with only maximum constraints.Relevant literature has given the globally optimal solutions of the max-min function under the constraint condition of 1+ 2+ · · · + 9)= (7;4)? 0,4)= 1,· · ·,9);(7 ? 0.The method of control vector is of great significance to solve the globally optimal solutions of max-min function.The paper studies the globally optimal solutions of max-min systems under more general constraints by using optimal max-only projection set and k-control vector.We define three kinds of globally optimal solutions of max-min system under constraint conditions of 2)(x)= (711+ (722+ · · · + (79)9)-(9 ? 0;5)? 0,5)= 1,· · ·,9);?{2)(x)? 0}?2)4)(x)= (74)11+ (74)22+ · · · + (74)9)9)-?9)5)=1(75)? 0,4)= 1,· · ·,7);0 ? 5)? 1,5)=1,· · ·,9);?4)=1,···,7){2)4)(x)? 0} and 2)4)(x)= (74)11+ (74)22+ · · · + (74)9)9)-(94)? 0 4)=1,· · ·,7);5)? 0,5)= 1,· · ·,9);?4)=1,···,7){2)4)(x)? 0}.We call them the globally optimal solution of the first,second and third kind of max-min system.Firstly,we study three kinds of the global optimal solution of max-only functions.And the necessary and sufficient conditions for the three kinds of the global optimal solution of max-only functions are obtained.Secondly,based on max-only system,the max-min function is transformed into a plurality of max-only functions by using optimal max-only projection.Finally,we study three kinds of the globally optimal solution of max-min function in special conditions,the necessary and sufficient conditions for the three kinds of the global optimal solution of max-min functions are obtained and a polynomial algorithm is obtained.