Int. J. Sensor Networks, Vol. 31, No. 2, 2019 65 Network localisation using Lagrangian optimisation Ananya Saha and Buddhadeb Sau* Department of Mathematics, Jadavpur University, Kolkata, 700032, India Email: ananyasaha.rs@jadavpuruniversity.in Email: buddhadeb.sau@gmail.com Email: buddhadeb.sau@jadavpuruniversity.in *Corresponding author Abstract: The network localisation problem with non-convex distance constraints may be modelled as a nonlinear optimisation problem. The existing localisation techniques either eliminate the nonconvex constraints or relax them into convex constraints to employ the traditional convex optimisations like semi-definite programming (SDP), least square approximation, etc. to find the node positions. We propose a method to solve the original network localisation problem with noisy distance measurements without any modification of non-convex constraints. Using the nonlinear Lagrangian technique for non-convex optimisation, we convert the localisation problem to a root finding problem involving single variable. This problem is then solved using bisection method. For computing functional values, it involves finite mini-max problem (FMX). We use sequential quadratic programming (SQP) to fix the FMX problem. Simulations studies show that, the number of iterations in the proposed method is reasonable to achieve any desired label of accuracy in node positions. Keywords: network localisation technique; localisation with noisy distances; Lagrange optimisation in localisation; non-convex optimisation for localisation. Reference to this paper should be made as follows: Saha, A. and Sau, B. (2019) ‘Network localisation using Lagrangian optimisation’, Int. J. Sensor Networks, Vol. 31, No. 2, pp.65–77. Biographical notes: Ananya Saha has received her MSc in Mathematics from Jadavpur University, Kolkata in the year 2011. She has joined in the Stat-Math Unit, Indian Statistical Institute, Kolkata in July, 2012 as research fellow and continued her study for research for one year. She has started her research work for PhD in Science at Jadavpur University under the supervision of Professor Buddhadeb Sau in 2013. Her research interest includes different issues related to ad-hoc networks, graph rigidity theory, etc. She has published research works along with her supervisor in reputed international conference proceedings and international journals. Buddhadeb Sau has received his MSc in Mathematics from Jadavpur University, Kolkata in the year 1995 and M.Teach. degree in Computer Science from Indian Statistical Institute, Kolkata in 2000. He received his PhD in Computer Science in 2012 from Indian Statistical Institute, Kolkata. Currently, he is a Professor in the Department of Mathematics, Jadavpur University. His research interest are related to computer science, wireless sensor network (WSN), localisation of WSN and ad-hoc networks, security in WSN, graph drawing, visibility graph, swarm robots, etc. He has published his research papers at national and international journals, conference proceedings. This paper is a revised and expanded version of a paper entitled ‘Network localization by non-convex optimization’ presented at ACM MobiMWareHN’17, Chennai, India, 10–14 July, 2017. 1 Introduction In recent technological advances, sensor networks are being adopted for collecting data from different hostile environments and monitoring them (Figure 1). A network may consists of sensor nodes, RFID readers, or members in a rescue team in a disaster management system, etc. Air pollution monitoring (Khedo et al., 2010), forest fire detection (Hefeeda and Bagheri, 2007), landslide detection (Ramesh, 2009), water quality detection (Akyildiz et al., 2005), natural disasters Copyright © 2019 Inderscience Enterprises Ltd.