The stochastic non-linear fractional knapsack problem is a challenging optimization problem with numerous applications, including resource allocation. The goal is to find the most valuable mix of materials that fits within a knapsack of fixed capacity. When the value functions of the involved materials are fully known and differentiable, the most valuable mixture can be found by direct applicat...

where the pairs (I+$, Xi) 2 0 are assumed to be independent draws from a common . . . . . joint distribution Fwx. If we think of the pairs (w,Xi) as the weights and values, respectively, of a collection of n objects, then this problem can be thought of as finding the collection of objects of maximum value which will fit in a “knapsack” with weight capacity 1. Our main result, Theorem 2.4, compu...

The deterministic knapsack problem is a well known and well studied NP-hard combinatorial optimization problem. It consists in filling a knapsack with items out of a given set such that the weight capacity of the knapsack is respected and the total reward maximized. For a review of references on the stochastic knapsack problem, stochastic gradient algorithms and branch-and-bound methods see [4]...

Motivated by a real world application, we study the multiple knapsack problem with assignment restrictions (MKAR): We are given a set of items N = f1; : : : ; ng and a set of knapsacks M = f1; : : : ;mg. Each item j 2 N has a positive real weight wj and each knapsack i 2 M has a positive real capacity ci associated with it. In addition, for each item j 2 N a set Aj M of knapsacks that can hold ...

In this paper, we present a new methodology to adapt any kind of lattice reduction algorithms to deal with the modular knapsack problem. In general, the modular knapsack problem can be solved using a lattice reduction algorithm, when its density is low. The complexity of lattice reduction algorithms to solve those problems is upper-bounded in the function of the lattice dimension and the maximu...

We study the classical 0-1 knapsack problem with additional restrictions on pairs of items. A con ict constraint states that from a certain pair of items at most one item can be contained in a feasible solution. Reversing this condition, we obtain a forcing constraint stating that at least one of the two items must be included in the knapsack. A natural way for representing these constraints is...

A special class of the knapsack problem is called the separable nonlinear knapsack problem. This problem has received considerable attention recently because of its numerous applications. Dynamic programming is one of the basic approaches for solving this problem. Unfortunately, the size of state-pace will dramatically increase and cause the dimensionality problem. In this paper, an efficient a...