Using Metaheuristic Computations to Find the Minimum-Norm-Residual Solution to Lrnear Systerns of Equations Rodrigo S. Jamisola, Jr.f , Elmer P. Dadiost, and Marcelo H. Ang, Jr.+ Abstract-This work will present metaheuristic computations, namely, probabilistic artificial neural network, simulated anneal- ing, and modified genetic algorithm in finding the minimum- norm-residual solution to linear systems of equations. By demon- strating a set of input parameters, the objective function, and the expectedresults solutions are computed for determined, overde- termined, and underdetermined linear systems. In addition, this work will present a version of genetic algorithm modified in terms of reproduction and mutation. In this modification,every reproduction cycle is performed by matching eachindividual with the rest of the individuals in the population. Further, the offspring chromosomes result from crossover of parent chromosomes without mutation. The selection process only selectsthe best fit individuals in the population. Mutation is only performed when the desired level of fitness cannotbe achieved, and all the possible chromosome combinations were alreadyexhausted. Experimental results for randorrly generated matrices with increasing matrix sizes will be presentedand analyzed. It will be the basis in modeling and identifying the dynamics parameters of a humanoid robot through responseoptimization at excitatory motions. Index Tenns-metaheuristic computation, probabilistic arti- ficial neural network, simulated annealing, modified genetic algorithm, linear systems of equations I. INTRODUCTION Metaheuristic computations are gaining wide acceptance through many fields of specializationbecause of their ability to find near-optimum solutions to Fbemingly unsolved problems using traditional mathematical 'techniques. Because of this, many computational applicatiorn,found workable solutions which would have been almost impossible to find or imprac- tical to implement using non-heuristic approaches. This work will presentthree metaheuristicmethods,namely, probabilistic artificial neural network, simulated annealing, and modified genetic algorithm. All three methods will be used to solve the same set of randomly generated inputs to linear systems of equations and their computational performance are compared. Probabilistic artificial neural network is chosen becauseif its faster computation especially at more complicated systems compared to the other two methods, simulated annealing becauseof its ease of implementation, and genetic algorithm becauseof its high rate of convergence. lDept. of Electronics and Communications Engineering and Dept. of Manufacturing Engineering and Management, De La Salle University - Manila, 24Ol Taft Ave, 1004 Manila, Philippines {j amisolar, dadiose}odlsu. edu. ph. +Dept. of Mechanical Engineering, National University of Singapore, 9 Engineering Dr 1, Singapore I 17576 mpeangh@nus . edu. sg. This work is supported by the University Research Council of De La Salle University - Manila. Artificial neural network (ANN) has addressed a wide rangeof real-world problems including autoregressive moving- averagemodel model parameter estimation [1], identification of linear discrete time systems [2], augmentation.of controller for underwater vehicles [3],2D pattern recognition problems [4], and linear systemsof equations [5] including time-varying systems [6] and linear and quadratic programming [7]. The ANN results presented here varies from [5] in that a prob- abilistic approach is used when ANN does not converge to the desired minimum residual value when a preset number of iterations is reached. Probabilistic neural networks appli- cations include pattern [8] and power quality classification [9], volume segmentation in brain MR images [10], face recognitiorVdecision [11], allocationof power loads [12], and freeway incident detection [13]. Computational applications of genetic algorithm (GA) in- clude traffic engineeringoptimization [14],leaming and struc- turing of afiificial neural network [5], linear array synthesis problem [16], optimizing reactive power planning [17], opti- mizing piecewiselinear function [18], solving linear bilevel programming [19], and automated linear modeling[20]. In this work, when the level of fitness has not reached the desired value but all possible chromosome combinations are exhausted, a probabilistic mutation process is performed" Some applications in probabilistic approach to genetic algo- rithm include alarm processing [21], optimal scheduling [22], tagging model [23], and in embedded systems [24]. Simulated annealing (SA) was first used to optimize NP- complete (nondeterministic polynomial time complete) prob- lems including the physical design of computers such as partitioning, component placement, and wiring of electronic systems [25]. Then simulated annealing found many wide- ranging area of applications that expanded the usefulness of this optimization technique.Among many of theseapplications include transmission system planning [26], combination with genetic algorithm to solve some NP-hard problems [2'll, at- mospheric correction of hyperspectraldata over dark surfaces [28], EEG source localization [29], flow model application [30], and bioclusteringof gene expression.data [31]. Swarm behavior that is hybrid with simulated annealing is shown in 1321. The contribution of this work lies in the analysis of the convergence and computational complexity of each of the methods presented, together with the comparison between these methods based on the analysis presented. A set of inputs are specified together with the desired outputs. Then the Philippine Computing Journal, Vol. 4, No. 2, December 2009, pp. 1-9.