Lowerstarts

Lowerstarts is a term used in computational optimization and numerical analysis to describe a class of initial points selected from the lower envelope of a feasible set to seed iterative solvers. The term is relatively new and appears in a limited body of literature and online discussions; it is not yet standardized, and its exact definition can vary across sources.

Definition and concept: Given a constrained optimization problem, a lowerstart is an initialization x0 that lies on or beneath a chosen lower bound representation of the feasible region, with the aim of starting from a point that is conservative or favorable for convergence. The idea draws on the concept of a lower envelope or monotone bounds used to define starting points for algorithms.

Construction and methods: Lowerstarts can be constructed by solving a simplified auxiliary problem that yields a point on the lower envelope, by projecting a simple point onto the lower bound surface, or by heuristics that select coordinates associated with low initial impact on the objective or constraints.

Applications: They have been proposed for interior-point methods, projected gradient methods, and other iterative schemes where a well-chosen starting point can reduce iteration count or improve robustness. In practice, the benefits depend on problem structure and the definition of the lower envelope.

Limitations: The approach requires extra steps to determine a suitable envelope or bound, which may offset potential gains. It is not universally advantageous and has limited cross-domain adoption.

See also: initialization, starting point, lower bound, envelope methods.