Problemas de otimização em caixas são de grande importância, não só por surgirem naturalmente na formulação de problemas da vida prática, mas também por aparecerem como subproblemas de métodos de penalização ou do tipo Lagrangiano Aumentado para resolução de problemas de programação não-linear. O objetivo do trabalho é estudar um algoritmo de restrições ativas para problemas de otimização em caixas recentemente apresentado chamado ASA e compará-lo à versão mais recente de GENCAN, que é também um método de restrições ativas. Para tanto, foi elaborada uma metodologia de testes robusta e minuciosa, que se propõe a remediar vários dos aspectos comumente criticados em trabalhos anteriores. Com isso, puderam ser extraídas conclusões que levaram à melhoria de GENCAN, conforme ficou posteriormente comprovado por meio da metodologia aqui introduzida. / Box-constrained optimization problems are of great importance not only for naturally arising in several real-life problems formulation, but also for their occurrence as sub-problems in both penalty and Augmented Lagrangian methods for solving nonlinear programming problems. This work aimed at studying a recently introduced active-set method for box-constrained optimization called ASA and comparing it to the latest version of GENCAN, which is also an active-set method. For that purpose, we designed a robust and thorough testing methodology intended to remedy many of the widely criticized aspects of prior works. Thereby, we could draw conclusions leading to GENCAN\'s further development, as it later became evident by means of the same methodology herein proposed.

Problemas de otimização em caixas são de grande importância, não só por surgirem naturalmente na formulação de problemas da vida prática, mas também por aparecerem como subproblemas de métodos de penalização ou do tipo Lagrangiano Aumentado para resolução de problemas de programação não-linear. O objetivo do trabalho é estudar um algoritmo de restrições ativas para problemas de otimização em caixas recentemente apresentado chamado ASA e compará-lo à versão mais recente de GENCAN, que é também um método de restrições ativas. Para tanto, foi elaborada uma metodologia de testes robusta e minuciosa, que se propõe a remediar vários dos aspectos comumente criticados em trabalhos anteriores. Com isso, puderam ser extraídas conclusões que levaram à melhoria de GENCAN, conforme ficou posteriormente comprovado por meio da metodologia aqui introduzida. / Box-constrained optimization problems are of great importance not only for naturally arising in several real-life problems formulation, but also for their occurrence as sub-problems in both penalty and Augmented Lagrangian methods for solving nonlinear programming problems. This work aimed at studying a recently introduced active-set method for box-constrained optimization called ASA and comparing it to the latest version of GENCAN, which is also an active-set method. For that purpose, we designed a robust and thorough testing methodology intended to remedy many of the widely criticized aspects of prior works. Thereby, we could draw conclusions leading to GENCAN\'s further development, as it later became evident by means of the same methodology herein proposed.

Η παρούσα διδακτορική διατριβή πραγματεύεται το θέμα της αποδοτικής και με βεβαιότητα εύρεσης όλων των ριζών της παραμετρικής εξίσωσης f(x;[p]) = 0, μιας συνεχώς διαφορίσιμης συνάρτησης f με [p] ένα διάνυσμα που περιγράφει όλες τις παραμέτρους της παραμετρικής εξίσωσης και τυποποιούνται με τη μορφή διαστημάτων. Για την επίλυση αυτού του προβλήματος χρησιμοποιήθηκαν εργαλεία της Διαστηματικής Ανάλυσης. Το κίνητρο για την ερευνητική ενασχόληση με το παραπάνω πρόβλημα προέκυψε μέσα από ένα κλασικό πρόβλημα αριθμητικής ανάλυσης: την αριθμητική επίλυση συστημάτων πολυωνυμικών εξισώσεων μέσω διαστηματικής ανάλυσης. Πιο συγκεκριμένα, προτάθηκε μια ευρετική τεχνική αναδιάταξης του αρχικού πολυωνυμικού συστήματος που φαίνεται να βελτιώνει σημαντικά, κάθε φορά, τον χρησιμοποιούμενο επιλυτή. Η ανάπτυξη, καθώς και τα αποτελέσματα αυτής της εργασίας αποτυπώνονται στο Κεφάλαιο 2 της παρούσας διατριβής. Στο επόμενο Κεφάλαιο 3, προτείνεται μια μεθοδολογία για την αποδοτική και αξιόπιστη επίλυση μη-γραμμικών εξισώσεων με διαστηματικές παραμέτρους, δηλαδή την αποδοτική και αξιόπιστη επίλυση διαστηματικών εξισώσεων. Πρώτα, δίνεται μια νέα διατύπωση της Διαστηματικής Αριθμητικής και αποδεικνύεται η ισοδυναμία της με τον κλασσικό ορισμό. Στη συνέχεια, χρησιμοποιείται η νέα διατύπωση της Διαστηματικής Αριθμητικής ως θεωρητικό εργαλείο για την ανάπτυξη μιας επέκτασης της διαστηματικής μεθόδου Newton που δύναται να επιλύσει όχι μόνο κλασικές μη-παραμετρικές μη-γραμμικές εξισώσεις, αλλά και παραμετρικές (διαστηματικές) μη-γραμμικές εξισώσεις. Στο Κεφάλαιο 4 προτείνεται μια νέα προσέγγιση για την αριθμητική επίλυση του προβλήματος της Ολικής Βελτιστοποίησης με περιορισμούς διαστήματα, χρησιμοποιώντας τα αποτελέσματα του Κεφαλαίου 3. Το πρόβλημα της ολικής βελτιστοποίησης, ανάγεται σε πρόβλημα επίλυσης διαστηματικών εξισώσεων, και γίνεται εφικτή η επίλυσή του με τη βοήθεια των θεωρητικών αποτελεσμάτων και της αντίστοιχης μεθοδολογίας του Κεφαλαίου 3. Στο τελευταίο Κεφάλαιο δίνεται μια νέα αλγοριθμική προσέγγιση για το πρόβλημα της επίλυσης διαστηματικών πολυωνυμικών εξισώσεων. Η νέα αυτή προσέγγιση, βασίζεται και γενικεύει την εργασία των Hansen και Walster, οι οποίοι πρότειναν μια μέθοδο για την επίλυση διαστηματικών πολυωνυμικών εξισώσεων 2ου βαθμού. / In this dissertation the problem of finding reliably and with certainty all the zeros a pa-rameterized equation f(x;[p]) = 0, of a continuously differentiable function f is considered, where [p] is an interval vector describing all the parameters of the Equation, which are formed with interval numbers. For this kind of problem, methods of Interval Analysis are used. The incentive to this scientific research was emerged from a classic numerical analysis problem: the numerical solution of polynomial systems of equations using interval analysis. In particular, a heuristic reordering technique of the initial polynomial systems of equations is proposed. This approach seems to improve significantly the used solver. The proposed technique, as well as the results of this publication are presented in Chapter 2 of this dissertation. In the next Chapter 3, a methodology is proposed for solving reliably and efficiently parameterized (interval) equations. Firstly, a new formulation of interval arithmetic is given and the equivalence with the classic one is proved. Then, an extension of interval Newton method is proposed and developed, based on the new formulation of interval arithmetic. The new method is able to solve not only classic non-linear equations but, non-linear parameterized (interval) equation too. In Chapter 4 a new approach on solving the Box-Constrained Global Optimization problem is proposed, based on the results of Chapter 3. In details, the Box-Constrained Global Optimization problem is reduced to a problem of solving interval equations. The solution of this reduction is attainable through the methodology developed in Chapter 3. In the last Chapter of this dissertation a new algorithmic approach is given for the problem of solving reliably and with certainty an interval polynomial equation of degree $n$. This approach consists in a generalization of the work of Hansen and Walster. Hansen and Walster proposed a method for solving only quadratic interval polynomial equations