holdlinearity

Holdlinearity is a proposed property of a multivariate function or optimization problem describing how linear structure in some variables is preserved when other variables are fixed, or held constant. The term is used mainly in theoretical discussions of decomposition, sensitivity analysis, and parametric optimization. It is distinct from ordinary linearity, which requires the entire function to be linear in all arguments.

Definition: Let X = H × R be a product space with variables x = (h, r), where h ∈ H are the held variables and r ∈ R are the remaining variables. A function F: H × R → Y is holdlinear with respect to a family of holds H0 ⊆ H if for every h ∈ H0 the restricted function F_h: R → Y defined by F_h(r) = F(h, r) is linear in r. If H0 = H, then F is linear in r for all fixed h; generally H0 may reflect a subset of admissible holds.

Examples: If F(h, r) = a^T r + g(h) with a fixed vector a and any h, then F_h(r) = a^T r + g(h) is linear in r for every h, so F is holdlinear. If instead F(h, r) = ||r||^2 + h^T r, then F_h is not linear in r for any h ≠ 0, and F is not holdlinear with respect to r for that H0.

Applications: Holdlinearity supports decomposition in optimization and numerical methods, allowing the unfixed subproblem to be solved as a linear program or linear regression when holds are fixed. It also informs sensitivity analysis and model specification in partially linear models. The concept is related to, but distinct from, piecewise linearity and partial linearity.

See also: linearity, partially linear model, parametric programming, decomposition methods.