Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-03-16T11:37:46Z No. of bitstreams: 1 rodrigoleppausdearaujo.pdf: 1937685 bytes, checksum: 0f43bd0bab8063bdc6af288e7aa82320 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-03-16T14:17:13Z (GMT) No. of bitstreams: 1 rodrigoleppausdearaujo.pdf: 1937685 bytes, checksum: 0f43bd0bab8063bdc6af288e7aa82320 (MD5) / Made available in DSpace on 2017-03-16T14:17:13Z (GMT). No. of bitstreams: 1 rodrigoleppausdearaujo.pdf: 1937685 bytes, checksum: 0f43bd0bab8063bdc6af288e7aa82320 (MD5) Previous issue date: 2016-11-05 / Meta-heurísticas têm sido frequentemente empregadas na resolução de problemas de otimização. Em particular, pode-se destacar a Evolução Diferencial (DE), que vem sendo aplicada com sucesso em situações onde o espaço de busca é contínuo. Apesar das vantagens dessas técnicas, elas precisam de adequações para tratar as restrições, que comumente limitam o espaço de busca em problemas reais de otimização. Nesse trabalho, uma modificação na DE é proposta a fim de tratar as restrições lineares de igualdade do problema. O método proposto, denotado aqui por DELEqC, gera uma população inicial de soluções candidatas que é factível em relação às restrições lineares de igualdade e gera os novos indivíduos sem utilizar o operador padrão de cruzamento. Com isso, pretende-se gerar novas soluções que também sejam viáveis quanto a esse tipo de restrição. O procedimento proposto de geração de indivíduos e manutenção da factibilidade da população é direto quando restrições lineares de igualdade são consideradas, mas requer o uso de variáveis de folga quando há desigualdades lineares no problema. Caso o problema de otimização envolva restrições não-lineares, o seu tratamento é feito aqui através de uma técnica de penalização adaptativa (APM) ou por meio de um esquema de seleção (DSS). O procedimento proposto é aplicado a problemas disponíveis na literatura e os resultados obtidos são comparados à queles apresentados por outras técnicas de tratamento de restrições. A análise de resultados indica que a proposta apresentada encontrou soluções competitivas em relação às outras técnicas específicas para o tratamento de restrições de igualdade lineares e melhores do que as alcançadas por estratégias comumente adotadas em meta-heurísticas. / Metaheuristics have been used to solve optimization problems. In particular, we can highlight the Differential Evolution(DE),which has been successfully applied insituations where the search space is continuous. Despite the advantages of those techniques, they require adjustments in order to deal with constraints, which commonly restrict the search space in real optimization problems. In this work, a change in the DE is proposed in order to deal with the linear equality constraints of the problem. The proposed method, here denoted by DELEqC, generates an initial population of candidate solutions, which are feasible with respect to the linear equality constraints, and generates new individuals without the standard crossover operation. The idea is to generate new solutions that are also feasible with respect to this kind of constraint. The proposed procedure for generating individuals and maintaining the feasibility of the population is straightforward when linear equality constraints are considered, but requires the use of slack variables when linear inequalities are present. If the optimization problem involves nonlinear constraints, their treatment is done here using an adaptive penalty method (APM), or by means of a selection scheme (DSS). The proposed procedure is applied to problems available in the literature and the results obtained are compared to those presented by other constraint handling techniques. The analysis of results indicates that the presented proposal found competitive solutions in relation to other specific techniques for the treatment of linear equality constraints and better than those achieved by strategies commonly adopted in metaheuristics.

Since augmented Lagrangian methods were introduced by Powell and Hestenes, this class of methods has been investigated very intensively. While the finite dimensional case has been treated in a satisfactory manner, the infinite dimensional case is studied much less. The general approach to solve an infinite dimensional optimization problem subject to equality constraints is as follows: First one proves convergence for a basic algorithm in the Hilbert space setting. Then one discretizes the given spaces and operators in order to make numerical computations possible. Finally, one constructs a discretized version of the infinite dimensional method and tries to transfer the convergence results to the finite dimensional version of the basic algorithm. In this thesis we discuss a globally convergent augmented Lagrangian algorithm and discretize it in terms of functional analytic restriction operators. Given this setting, we prove global convergence of the discretized version of this algorithm to a stationary point of the infinite dimensional optimization problem. The proposed algorithm includes an explicit rule of how to update the discretization level and the penalty parameter from one iteration to the next one - questions that had been unanswered so far. In particular the latter update rule guarantees that the penalty parameters stay bounded away from zero which prevents the Hessian of the discretized augmented Lagrangian functional from becoming more and more ill conditioned. / Master of Science

<p>U disertaciji je razmatran problem stohastičkog programiranja s ograničenjima tipa jednakosti, odnosno problem minimizacije s ograničenjima koja su u obliku matematičkog očekivanja. Za re&scaron;avanje posmatranog problema kreirana su dva iterativna postupka u kojima se u svakoj iteraciji računa s uzoračkim očekivanjem kao aproksimacijom matematičkog očekivanja. Oba postupka koriste prednosti postupaka s promenljivom veličinom uzorka zasnovanih na adaptivnom ažuriranju veličine uzorka. To znači da se veličina uzorka određuje na osnovu informacija u tekućoj iteraciji. Konkretno, tekuće informacije o preciznosti aproksimacije očekivanja i tačnosti aproksimacije re&scaron;enja problema defini&scaron;u veličinu uzorka za narednu iteraciju. Oba iterativna postupka su zasnovana na linijskom pretraživanju, a kako je u pitanju problem s ograničenjima, i na kvadratnom kaznenom postupku prilagođenom stohastičkom okruženju. Postupci su zasnovani na istim idejama, ali s različitim pristupom.<br />Po prvom pristupu postupak je kreiran za re&scaron;avanje SAA reformulacije problema stohastičkog programiranja, dakle za re&scaron;avanje aproksimacije originalnog problema. To znači da je uzorak definisan pre iterativnog postupka, pa je analiza konvergencije algoritma deterministička. Pokazano je da se, pod standardnim pretpostavkama, navedenim algoritmom dobija podniz iteracija čija je tačka nagomilavanja KKT tačka SAA reformulacije.<br />Po drugom pristupu je formiran algoritam za re&scaron;avanje samog problema<br />stohastičkog programiranja, te je analiza konvergencije stohastička. Predstavljenim algoritmom se generi&scaron;e podniz iteracija čija je tačka nagomilavanja, pod standardnim pretpostavkama za stohastičku optimizaciju, skoro sigurno<br />KKT tačka originalnog problema.<br />Predloženi algoritmi su implementirani na istim test problemima. Rezultati numeričkog testiranja prikazuju njihovu efikasnost u re&scaron;avanju posmatranih problema u poređenju s postupcima u kojima je ažuriranje veličine uzorka<br />zasnovano na unapred definisanoj &scaron;emi. Za meru efikasnosti je upotrebljen<br />broj izračunavanja funkcija. Dakle, na osnovu rezultata dobijenih na skupu<br />testiranih problema može se zaključiti da se adaptivnim ažuriranjem veličine<br />uzorka može u&scaron;tedeti u broju evaluacija funkcija kada su u pitanju i problemi s<br />ograničenjima.<br />Kako je posmatrani problem deterministički, a formulisani postupci su stohastički, prva tri poglavlja disertacije sadrže osnovne pojmove determinističke<br />i stohastiˇcke optimizacije, ali i kratak pregled definicija i teorema iz drugih<br />oblasti potrebnih za lak&scaron;e praćenje analize originalnih rezultata. Nastavak disertacije čini prikaz formiranih algoritama, analiza njihove konvergencije i numerička implementacija.<br />&nbsp;</p> / <p>Stochastic programming problem with equality constraints is considered within thesis. More precisely, the problem is minimization problem with constraints in the form of mathematical expectation. We proposed two iterative methods for solving considered problem. Both procedures, in each iteration, use a sample average function instead of the mathematical expectation function, and employ the advantages of the variable sample size method based on adaptive updating the sample size. That means, the sample size is determined at every iteration using information from the current iteration. Concretely, the current precision of the approximation of expectation and the quality of the approximation of solution determine the sample size for the next iteration. Both iterative procedures are based on the line search technique as well as on the quadratic penalty method adapted to stochastic environment, since the considered problem has constraints. Procedures relies on same ideas, but the approach is different.<br />By first approach, the algorithm is created for solving an SAA reformulation of the stochastic programming problem, i.e., for solving the approximation of the original problem. That means the sample size is determined before the iterative procedure, so the convergence analyses is deterministic. We show that, under the standard assumptions, the proposed algorithm generates a subsequence which accumulation point is the KKT point of the SAA problem. Algorithm formed by the second approach is for solving the stochastic programming problem, and therefore the convergence analyses is stochastic. It generates a subsequence with&nbsp; accumulation point that is almost surely the KKT point of the original problem, under the standard assumptions for stochastic optimization.for sample size. The number of function evaluations is used as measure of efficiency. Results of the set of tested problems suggest that it is possible to make smaller number of function evaluations by adaptive sample size scheduling in the case of constrained problems, too.<br />Since the considered problem is deterministic, but the formed procedures are stochastic, the first three chapters of thesis contain basic notations of deterministic and stochastic optimization, as well as a short sight of definitions and theorems from another fields necessary for easier tracking the original results analysis. The rest of thesis consists of the presented algorithms, their convergence analysis and numerical implementation.</p>

This research presents a Hybrid Particle Swarm Optimization with Bat Algorithm (HPSOBA) based approach to solve Optimal Reactive Power Dispatch (ORPD) problem. The primary objective of this project is minimization of the active power transmission losses by optimally setting the control variables within their limits and at the same time making sure that the equality and inequality constraints are not violated. Particle Swarm Optimization (PSO) and Bat Algorithm (BA) algorithms which are nature-inspired algorithms have become potential options to solving very difficult optimization problems like ORPD. Although PSO requires high computational time, it converges quickly; while BA requires less computational time and has the ability of switching automatically from exploration to exploitation when the optimality is imminent. This research integrated the respective advantages of PSO and BA algorithms to form a hybrid tool denoted as HPSOBA algorithm. HPSOBA combines the fast convergence ability of PSO with the less computation time ability of BA algorithm to get a better optimal solution by incorporating the BA’s frequency into the PSO velocity equation in order to control the pace. The HPSOBA, PSO and BA algorithms were implemented using MATLAB programming language and tested on three (3) benchmark test functions (Griewank, Rastrigin and Schwefel) and on IEEE 30- and 118-bus test systems to solve for ORPD without DG unit. A modified IEEE 30-bus test system was further used to validate the proposed hybrid algorithm to solve for optimal placement of DG unit for active power transmission line loss minimization. By comparison, HPSOBA algorithm results proved to be superior to those of the PSO and BA methods. In order to check if there will be a further improvement on the performance of the HPSOBA, the HPSOBA was further modified by embedding three new modifications to form a modified Hybrid approach denoted as MHPSOBA. This MHPSOBA was validated using IEEE 30-bus test system to solve ORPD problem and the results show that the HPSOBA algorithm outperforms the modified version (MHPSOBA). / Electrical and Mining Engineering / M. Tech. (Electrical Engineering)