Hybrid combinatorial optimization: Sample problems and algorithms

We study a class of non-convex optimization problems involving sigmoid functions. We show that sigmoid functions impart a combinatorial element to the optimization variables and make them hybrid of continuous and discrete variables. We formulate versions of the knapsack problem and the bin-packing problem with such hybrid variables. We utilize the approximation algorithms from the combinatorial optimization literature and develop approximation algorithms for these NP-hard hybrid optimization problems.