On Minimizing Nonseparable Functions Defined on the Integers with an Inventory Application

The general form of the problem considered is: \[ {Find}\,{the}\,{infimum}\,{of}\,f:X → R, \] where X is a discrete rectangle, that is to say, \[ X = \{ {x:x ∈ I^n ,a_i x_i b_i ,i = 1, ⋯ ,n} \}, \]I is the set of integers \ ⋯ , - 2, - 1,0, + 1, ⋯ \, aᵢ, bᵢ are integers or infinite, and R is the real line. A condition called discrete convexity is developed for functions f which is a sufficient condition for a local optimum of f to be a global optimum of f. A general solution strategy is given, and computational results are presented which show the possibility of solving this problem in cases where n is in the hundreds.