On Some Nonlinear Knapsack Problems

TLDR: In this article, the authors considered the problem of minimizing a separable strictly convex function with nonnegative integer variables when the sum of variables is constrained and provided the condition for the optimum and properties of the optimal solution.

Abstract: Minimization of separable strictly convex function is considered with nonnegative integer variables when the sum of variables is constrained. Theorems concerning the condition for the optimum and properties of the optimal solution are presented. For a few types of functions this problem displays “periodic” properties similar to those in linear integer programming: The difference between the noninteger and integer solution is a function depending solely on the position of the noninteger solution inside a hypercube formed by the neighbouring integer points. Utilization of this property shortens drastically the search for the integer solution, in many cases the problem reduces to nonlinear 0/1 problem.