Αναβαθμισμένη Παραλλαγή της Τεχνικής Σμήνους Σωματιδίων στη Βελτιστοποίηση

Abstract

Στόχο της παρούσης εργασίας αποτέλεσε η ανάπτυξη ενός εναλλακτικού αλγορίθμου Στοχαστικής Βελτιστοποίησης βασιζομένου στην εδραιωμένη μέθοδο Σμήνους Σωματιδίων (Particle Swarm Optimization – PSO). Η προϋπάρχουσα κεντρική ιδέα πλαισιώνεται από δοκιμασμένες στο χώρο της βελτιστοποίησης λύσεις και από ορισμένες προσθήκες του γράφοντος, με στόχο το τελικό αποτέλεσμα να αποτελέσει μία λειτουργική και ανταγωνιστική εναλλακτική λύση, ειδικά όσον αφορά προβλήματα βελτιστοποίησης ως προς περισσότερα του ενός κριτηρίων. Παρουσιάζεται διεξοδικά η πρόοδος του αλγορίθμου με κάθε προσθήκη, ενώ εκτίθενται παράλληλα και διάφορα αντιπροσωπευτικά της αιχμής του δόρατος της Στοχαστικής Βελτιστοποίησης παραδείγματα, για λόγους σύγκρισης αλλά και πληρότητας. Ιδιαίτερη μνεία γίνεται στους δημοφιλείς Εξελικτικούς Αλγορίθμους (ΕΑ), το αντίπαλο δέος, ουσιαστικά, της τεχνικής Σμήνους Σωματιδίων, επί των οποίων το Εργαστήριο Θερμικών Στροβιλομηχανών (ΕΘΣ) έχει να επιδείξει σπουδαία δραστηριότητα και τεχνογνωσία και ο γράφων μία σχετική εμπειρία. Επιχειρείται μία απευθείας αντιπαραβολή, τόσο φιλοσοφίας όσο και πρακτικής, των δύο ιδεών, ενώ είναι σαφής καθ' όλη την έκταση αυτού του εκπονήματος η πρόθεση αντιστοίχισης, τμηματικά, της μίας με την άλλη, ώστε να τονιστεί ο ενιαίος χαρακτήρας του χώρου της Στοχαστικής Βελτιστοποίησης και να ταυτοποιηθούν τα πάγια χαρακτηριστικά των μεθόδων αυτών. Ο ολοκληρωμένος αλγόριθμος δοκιμάζεται, κατόπιν, σε επιλεγμένες εφαρμογές, ακαδημαϊκού και βιομηχανικού ενδιαφέροντος, όλες δύο στόχων: Οι περιπτώσεις των μαθηματικών συναρτήσεων ZDT-1 και ZDT-3 είναι αντιπροσωπευτικά δείγματα προβλημάτων που έχουν αναπτυχθεί από ακαδημαϊκό φορέα ειδικά ως μέσο δοκιμής και σύγκρισης τέτοιων μεθόδων και αποτελούν στην ουσία μοντελοποίηση των δυσχερειών που αναμένεται να συναντήσει ένας αλγόριθμος βελτιστοποίησης σε βιομηχανικές εφαρμογές. Τέλος, δοκιμάζεται και έναντι της, υπό περιορισμούς, αεροδυναμικής βελτιστοποίησης πτερυγίου αεροσυμπιεστή. Τα αποτελέσματα που εκτίθενται προσφέρονται για σύγκριση της παρούσας πρότασης με έναν ενδεικτικό ΕΑ, επιβεβαιώνουν τα όσα είναι γνωστά για τις διαφορές στη συμπεριφορά των δύο τεχνικών, ενώ πιστοποιούν την ανταγωνιστικότητα του παρουσιαζόμενου λογισμικού.

The main objective of this work has been the development and subsequent validation of a complete optimization tool based on the concept of the Particle Swarm. Particle Swarm Optimization (PSO), as which, the entirety of optimization-oriented applications of the Particle Swarm is referred to, is a subcategory of the great family of Swarm Intelligence techniques. As such, it introduces processes inspired from the collective activity of a swarm of insects, flock of birds and school of fish or similar to assess the search for optimal solutions to a variety of problems. The proposed algorithm (PA) borrows the original core idea of PSO, and applies a series of additions and adjustments, some of which original, some inspired from trends in the ongoing advances in the field of optimization. Swarm intelligence itself is a subcategory of the Stochastic Methods, which essentially encapsulate all optimization techniques that rely, to some extent, on randomized search within all specified variable ranges to locate the optima. This thesis extends its perspective beyond Swarm Intelligence and approaches Stochastic Optimization holistically, attempting to outline the common features among its various aspects and extract clues as to how each one can be enhanced. Particular attention is given to the most popular and widely applied Stochastic Methods branch, that of Evolutionary Computation and Evolutionary Algorithms (EA’s). After EA’s and PSO have been introduced and discussed in depth in chapter 2 and 3, a long discussion is conducted to highlight the similarities or equivalences between the two, as far as both their philosophical and mathematical background and their practical application is concerned. The purpose of this is not only to determine the adjacencies between the various components and defining features of these two paradigms, but also to gain insight into possible improvements, either by borrowing principles from each other or by hybridizing. At this point, suffice to say that the prominent product of this analysis is that PSO has a relatively faster rate of progress through the earlier stages of a run, while EA’s in general shine at a later stage, the phase of exploitation, namely the phase when search space has almost been exhausted and the optimizer focuses on refining the located solutions by searching in their immediate vicinity, thus slightly improving the end result. This rough observation greatly impacts this entire work and its efforts in improving the generic PSO optimizer are focused on moderating this fundamental disadvantage. In PSO, the members of the swarm, or particles, are driven by two main forces: the particle’s individual perception of search space, as it is shaped by its own progress thus far (cognitive influence), and its interaction with the rest of the swarm, its awareness of the progress of the swarm as a whole (social influence). The relative effect of these two driving forces is dependent upon a series of tuning parameters. Their choice of value is therefore crucial, especially so since it is understood that cognitively and socially influenced behaviors relate to performance in different stages of the optimization process. In section 3.4, these governing parameters are discussed: Their impact is analyzed, relevant experiments and literature are surveyed and the various existing trends are reviewed. The choice of parameters for the PA is elaborately justified, especially from the perspective of addressing the lacking exploitation capabilities. A scheme that dynamically alters these parameters is adopted, inspired by similar beneficial practices in EA’s. Chapter 4 provides an overview of the entire PA: each section examines a major aspect and its internal processes in depth. A short survey of popular equivalents comes with the introduction of each feature. Unless explicitly stated otherwise, the various processes and features are original. Similarities to existing techniques are present in some cases, while others deviate from common practice. Occasionally, a few alternative approaches to a certain issue will be presented, and their distinctive characteristics will be discussed. The optimizer was generally developed and programmed from scratch. The most notable novelties are the highly directional and strategic social influence structure and the shuffle operator, a scheme designed to intervene late in the algorithm’s progress by appropriately re-positioning the swarm and determining certain directions in which to intensify search, thus maximizing its efficiency. Other main points, like the constraint operator, responsible for administering candidate solutions in breach of any constraints imposed by the problem, and the initialization phase are also worth mention. Emphasis was placed on multi-objective optimization (MOO) problems, namely problems where the optimality of a solution is judged on multiple criteria. As was explained, the multi-objective regime is completely different to the single-objective one and poses additional challenges, some of which are specific to PSO and pertain to the elevated roles of cognitive and social influence. The reader is introduced to the details of MOO and the current trends in dealing with such problems (the Pareto concept, non-dominated solution sorting methods etc.) in chapter 2. In section 4.3 I specifically elaborate on the approaches adopted in the PA to facilitate a successful transition to MOO: A solution selection/sorting procedure determining the best solutions so far, wherein to invest. A solution spacing routine is designed and incorporated to guarantee the sought diversity among the various optimal solutions. The PA is tested against three problems: each of two objectives, with its individual peculiarities. The first two are benchmark mathematical function cases, especially developed by optimization researchers for exactly this purpose: ZDT-1 and ZDT-3. The latter, with its challenging non-contiguous set of optima is a very popular experimental tool. One last test, of a more practical orientation, utilizes the PA for the optimization of a cascade compressor’s stator airfoil, with regard to individual aerodynamic efficiency and good static pressure rise qualities. This case features strict constraints and a higher computational cost per examined candidate solution, thus, a more demanding problem. The PA is subjected to these tests alongside EA-based optimization software of established competitiveness, serving as a point of reference. The demonstrated results showcase the earlier speculated differences in behavior between EA’s and PSO, and how the added features have somewhat bridged this gap. They also grant the PA validation as a fully functional and competent optimizer and a decent foundation for further experimentation. In chapter 6, a few suggestions for future work are laid out; various adjustments to the existing features, possible on a short-term basis, as well as more ambitious enhancements that may be achieved as part of a larger project.