The 0–1 Knapsack Problem: A Continuous Generalized Convex Multiplicative Programming Approach

In this work we propose a continuous approach for solving one of the most studied problems in combinatorial optimization, known as the 0–1 knapsack problem. In the continuous space, the problem is reformulated as a convex generalized multiplicative problem, a special class of nonconvex problems which involves the minimization of a finite sum of products of convex functions over a nonempty convex set. The product of any two convex positive functions is not necessarily convex or quasiconvex, and, therefore, the continuous problem may have local optimal solutions that are not global optimal solutions. In the outcome space, this problem can be solved efficiently by an algorithm which combines a relaxation technique with the procedure branch–and– bound. Some computational experiences are reported.