Issues on a 2–Dimensional Quadratic Sub–Problem and Its Applications in Nonlinear Programming: Trust–Region Methods (TRMs) and Linesearch Based Methods (LBMs)

This paper analyses the solution of a specific quadratic sub-problem, along with its possible applications, within both constrained and unconstrained Nonlinear Programming frameworks. We give evidence that this sub–problem may appear in a number of Linesearch Based Methods (LBM) schemes, and to some extent it reveals a close analogy with the solution of trust–region sub–problems. Namely, we refer to a two-dimensional structured quadratic problem, where five linear inequality constraints are included. Finally, we detail how to compute an exact global solution of our two dimensional quadratic sub-problem, exploiting first order Karush-Khun-Tucker (KKT) conditions.