Danger theory inspired micro-population immune optimization for probabilistic constrained programming
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ORIGINAL PAPER
Danger theory inspired micro‑population immune optimization for probabilistic constrained programming Zhuhong Zhang1 · Renchong Zhang2 Received: 14 October 2017 / Accepted: 21 January 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract This work solves the problem of a general kind of single-objective probabilistic constrained programming without any a prior stochastic distribution information, after probing into an adaptive sampling-based micro immune optimization approach inspired by the danger theory in immunology. In the whole design of the algorithm, the current population is divided into uninfected, susceptible and infected sub-populations based on the version of individuals’ dominance, relying upon the schemes of sample-dependent constraint handling and objective evaluation. Those uninfected and susceptible sub-populations proliferate their clones and execute adaptive mutation with small variable mutation rates, whereas the infected sub-population directly participates in mutation at a large and variable mutation rate. Two mutation strategies, together with the version of life cycle on individual, are simultaneously designed to evolve those sub-populations along different directions in order to explore those diverse and high-quality solutions. It is shown that the complexity of the algorithm depends mainly on the number of iterations. Experimental results have validated that one such approach is a competitive and potential optimizer because of few parameters, fast convergence and its ability of effective noise suppression. Keywords Probabilistic constrained optimization · Danger theory · Micro immune optimization · Adaptive sampling · Life cycle
1 Introduction In real-world engineering optimization, many design problems, such as mechanical product design, transportation programming, distribution network design, etc. (Liu 2009; Cheng et al. 2015; Pan 2015; Ma et al. 2016), can be formulated by stochastic programming models. It, however, is hard to seek their optimal solutions by conventional approaches, due to their complex uncertain factors and hard constraints. Single-objective chance constrained programming, proposed originally by Charnes and Cooper (1959) is a challenging and valuable branch in the field of uncertain programming. * Zhuhong Zhang [email protected] Renchong Zhang [email protected] 1
College of Big Data and Information Engineering, Guizhou University, Guiyang 550025, Guizhou, People’s Republic of China
Guizhou University of Commerce, Guiyang 551400, Guizhou, People’s Republic of China
2
It includes two types of mathematical programming models, namely E-model and P-model. The former consists of an expected value objective function and at least a probabilistic or chance constraint, whereas the latter, probabilistic constrained programming, is another kind of chance constrained programming model in which the objective function is decided by a probabilistic inequality. The main difficulty of solving P-model involves four points: (1)
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