**A class of nonlinear nonseparable continuous knapsack and multiple-choice knapsack problems**

This paper considers a general class of continuous, nonlinear, and nonseparable knapsack
problems, special cases of which arise in numerous operations and financial contexts.
We develop important properties of optimal solutions for this problem class, based on the
properties of a closely related class of linear programs. Using these properties, we
provide a solution method that runs in polynomial time in the number of decision
variables, while also depending on the time required to solve a particular one-dimensional
optimization problem. Thus, for the many applications in which this one-dimensional
function is reasonably well behaved (e.g., unimodal), the resulting algorithm runs in
polynomial time. We next develop a related solution approach to a class of continuous,
nonlinear, and nonseparable multiple-choice knapsack problems. This algorithm runs in
polynomial time in both the number of variables and the number of variants per item,
while again dependent on the complexity of the same one-dimensional optimization problem
as for the knapsack problem. Computational testing demonstrates the power of the proposed algorithms over a commercial global optimization software package.