totalingH

TotalingH is a term used in mathematical optimization and theoretical computer science to describe a framework for turning a set of local contributions into a single global score. It centers on a Hamiltonian-like function H that assigns a real-valued energy or cost to a given configuration, with a subsequent aggregation producing a total value suitable for optimization.

In formal terms, let X be a set of components and s a configuration. H(s) is the energy of s. The totaling operation computes T(s) as an additive aggregation, typically T(s) = sum_i w_i H_i(s) or T(s) = sum_j H_j(s), where H_i are component terms and w_i are weights. Non-additive forms and normalization may be used to enforce constraints or balance terms. The approach is flexible enough to accommodate different decompositions of a problem into local terms and various weighting schemes.

Origin and usage: The concept emerged in late 2010s discussions on energy-based models and has been used across domains such as scheduling, resource allocation, clustering, and network analysis to convert local measurements into a tractable objective. It is often employed in iterative optimization routines where the total objective guides search, selection, or assignment decisions.

Variants and challenges: Choosing decomposition, weights, and optional non-additive terms affects tractability and solution quality. Exact totaling can be NP-hard in general, so practitioners rely on relaxations, approximations, or heuristic methods to obtain practical results.

See also: Hamiltonian, energy function, optimization, additive models.