Solution of Some Max–Separable Optimization Problems with Inequality Constraints

Let R be the set of real numbers, ⊕ is the maximum operation so that α⊕ β = max{α, β} for any α, β ∈ R. We assume further that Rn ≡ R× . . .×R (n-times, the Cartesian product). We shall solve optimization problems consisting in the minimization of function F (f(x), g(y)), where F : R2 → R1 is an isotone function and function f(x), f : Rn → R1 is defined as follows f(x) = f(x1, . . . , xn) = f1(x1)⊕ f2(x2)⊕ . . .⊕ fn(xn) where fj : R 1 → R1 are continuous for all j = 1, . . . , n, and g : Rk → R1 is an isotone function with respect to the relation ≤, i.e. y ′ ≤ y ⇒ g(y ) ≤ g(y′′). This function will be minimized under the constraints max 1≤j≤n rij(xj) = n ⊕ j=1 rij(xj) ≥ bi(y), x ≥ x, y ≥ y for all i = 1, . . . ,m, where rij(xj) are real continuous increasing functions of one variable for all i = 1, . . . ,m and j = 1, . . . , n and bi(y) are isotone in y with respect to ≤ for all i = 1, . . . ,m and x, y are given finite lower bounds. As special examples, so called (⊕,⊗)-linear optimization problems in the so called (⊕,⊗)-algebras with (⊕,⊗) = (max,+) and (⊕,⊗) = (max,min) will be considered (more details about the (⊕,⊗)-algebras can be found e.g. in [3]).