Mixed integer nonlinear optimization models for the Euclidean Steiner tree problem in $$\mathbb {R}^d$$

Abstract

New mixed integer nonlinear optimization models for the Euclidean Steiner tree problem in d-space (with d ≥ 3) will be presented in this work. Each model features a non smooth objective function but a convex set of feasible solutions. All these models are theoretically equivalent. From these models, six mixed integer linear and nonlinear relaxations will be considered. Each relaxation has the same set of feasible solutions as the model from which it is derived. Finally, preliminary computational results highlighting the main features of the presented relaxations will be discussed.

A new second‐order conic optimization model for the Euclidean Steiner tree problem in Rd$\mathbb {R}^d$

“…These instances are named, respectively, nocta‐x and nsimp‐x, where x indicates the dimension d . All the coordinates of these 10 instances are scaled in order to fulfill Assumption 2 and are the same used in Ouzia and Maculan (2021). We did not consider other Smith's instances because they are too big to be solved exactly by the models we are considering and, to the best of our knowledge, by any other known model for ESTP.…”

“…(2016b), Fampa (2019), D'Ambrosio et al. (2020), and Ouzia and Maculan (2021). The approaches of the second kind are based on a mathematical programming model of the ESTP.…”

“…Two second‐order conic formulations of the ESTP, derived from this model, can be found in Fampa and Maculan (2001) and Fampa and Maculan (2004) (for more details, see Fampa et al., 2016a). In Ouzia and Maculan (2021), several mixed integer nonlinear optimization models are presented for the ESTP featuring a nonsmooth objective function but the continuous relaxations of their set of feasible solutions are convex. From these models, four convex mixed integer linear and nonlinear relaxations are derived and used to compute feasible solutions to the ESTP.…”

The complex evolution of information quality improvement in competitive market

“…Ma found that prices of natural resource products were more sensitive to the impact of Covid-19 than other products [10]. Ouzia and Maculan studied the optimal solution of the Euclidean Steiner tree problem with nonlinear characteristics, which is often used for economic optimization problems in network engineering [11]. For system dynamic and complex risks, Kheir and Mahjoub used RGA graph to analyze system complex risk, risk evolution, and risk early warning [12].…”

“…Liu [8]; Liao [9]; Ma [10]; Ouzia and Maculan [11] Kheir et al [12] Arabadzhyan [13] Uncertainty or nonlinear in system…”