# Chaotic Quantum Double Delta Swarm Algorithm Using Chebyshev Maps: Theoretical Foundations, Performance Analyses and Convergence Issues

Department of Electrical Engineering and Computer Science, Vanderbilt University, 2201 West End Ave, Nashville, TN 37235, USA

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Received: 8 December 2018 / Revised: 31 December 2018 / Accepted: 11 January 2019 / Published: 17 January 2019

(This article belongs to the Special Issue AI and Quantum Computing for Big Data Analytics)

The Quantum Double Delta Swarm (QDDS) Algorithm is a networked, fully-connected novel metaheuristic optimization algorithm inspired by the convergence mechanism to the center of potential generated within a single well of a spatially colocated double–delta well setup. It mimics the wave nature of candidate positions in solution spaces and draws upon quantum mechanical interpretations much like other quantum-inspired computational intelligence paradigms. In this work, we introduce a Chebyshev map driven chaotic perturbation in the optimization phase of the algorithm to diversify weights placed on contemporary and historical, socially-optimal agents’ solutions. We follow this up with a characterization of solution quality on a suite of 23 single–objective functions and carry out a comparative analysis with eight other related nature–inspired approaches. By comparing solution quality and successful runs over dynamic solution ranges, insights about the nature of convergence are obtained. A two-tailed t-test establishes the statistical significance of the solution data whereas Cohen’s d and Hedge’s g values provide a measure of effect sizes. We trace the trajectory of the fittest pseudo-agent over all iterations to comment on the dynamics of the system and prove that the proposed algorithm is theoretically globally convergent under the assumptions adopted for proofs of other closely-related random search algorithms.