<p>Posmatra se problem minimizacije bez ograničenja. U determinističkom&nbsp;slučaju ti problemi se uspe&scaron;no re&scaron;avaju iterativnim Kvazi Njutnovim postupcima.&nbsp;Ovde se istražuje &nbsp;stohastički slučaj, kada su poznate vrednosti funkcije cilja i njenog gradijenta na koje je uticao &scaron;um. Koristi se novi način određivanja dužina koraka, koji kombinuje metod linijskog pretraživanja i metod stohastičke aproksimacije tako da zadrži dobre osobine oba pristupa i obezbedi veću efikasnost postupka. Metod je testiran u kombinaciji sa vi&scaron;e načina izbora pravca u iterativnom postupku. Dokazana je konvergencija novog postupka i testiranjem na velikom broju standardnih test problema pokazana njegova efikasnost. Takođe se za re&scaron;avanje problema ekvilibriuma u Neoklasičnoj ekonomiji predlaže i dokazuje konvergencija jednog Fiksnog Njutnovog postupka. U zadatku nalaženja re&scaron;enja za niz problema kojima se preciznije modelira slučajni sistem, ovaj Fiksni Njutnov postupak ostvaruje veliku u&scaron;tedu CPU vremena u odnosu na Njutnov metod. U prvom delu teze je dat op&scaron;ti teoretski uvod. U drugom delu je dat pregled relevantnih rezultata iz posmatranih oblasti zajedno sa dva originalna rezultata. U trećem &nbsp;delu su dati rezultati numeričkih testova.</p> / <p>The problem under consideration is unconstrained minimization pro-blem. The problem in deterministic case is often solved with Quasi Newton met-hods. In noisy environment, which is considered, new approach for step length along descent direction is used. The new approach combines line search and stoc-hastic&nbsp; approximation method using good characteristics of both enabling better efficiency. The convergence is proved. New step length is tested with three de-scent directions. Many standard test problems show the efficiency of the met-hod. Also, a new, affordable procedure based on application of the fixed Newton method for a sequence of equilibrium problems generated by simulation is intro-duced. The convergence conditions of the method are derived. The numerical results show a clear difference in the quality of information obtained by solving a sequence of problems if compared with the single equilibrium problem. In the first part general theoretical introduction is given. In the second part a survey of results from scientific community is given together with original results. The third part contains many numerical tests of new methods that show its efficiency.</p>

<p>There are many real models in which unbounded solution to conservation law system occur. Most often we have some kind of delta function in the solution as a result of the accumulation of mass or some other variable. There is no general method of approaching<br />such problems with nonlinearities. This dissertation provides solutions to conservation law systems that contain division by a dependent variable, which is a problematic part when working with measures. For example, a basic model of chromatography and similar chemical processes has a division with a variable that is unbounded in some cases. The denition of the split delta shock and the general method of using it in such systems is given. Finally, the solution for the singular chromatography model is given.<br />&nbsp;</p> / <p>Postoji mnogo realnih modela u kojima se javljaju neoranicena resenja zakona odrzanja. Najcesce imamo neku vrstu delta funkcije u resenju kao posledicu nagomilavanja mase ili neke druge velicine. Ne postoji opsti metod prilaza takvim problemima sa nelinearnostima. U ovoj disertaciji su data resenja problema zakona odrzanja koja sadrze delenje zavisnom promenljivom, sto je problematican deo kod rada sa merama. Na primer, osnovni model hromatograje i slicnih hemijskih procesa ima delenje promenljivom koja je neogranicena u nekim slucajevima. Data je denicija inverza delenjog delta udarnog talasa i opsti metod primene u takvim sistemima. Na kraju je dato resenje kod modela singularne hromatograje.<br />&nbsp;</p>

<p>The problem under consideration is an unconstrained optimization&nbsp;problem with the objective function in the form of mathematical ex-pectation. The expectation is with respect to the random variable that represents the uncertainty. Therefore, the objective &nbsp;function is in fact deterministic. However, nding the analytical form of that objective function can be very dicult or even impossible. This is the reason why the sample average approximation is often used. In order to obtain reasonable good approximation of the objective function, we have to use relatively large sample size. We assume that the sample is generated at the beginning of the optimization process and therefore we can consider this sample average objective function as the deterministic one. However, applying some deterministic method on that sample average function from the start can be very costly. The number of evaluations of the function under expectation is a common way of measuring the cost of an algorithm. Therefore, methods that vary the sample size throughout the optimization process are developed. Most of them are trying to determine the optimal dynamics of increasing the sample size.</p><p>The main goal of this thesis is to develop the clas of methods that&nbsp;can decrease the cost of an algorithm by decreasing the number of&nbsp;function evaluations. The idea is to decrease the sample size whenever&nbsp;it seems to be reasonable - roughly speaking, we do not want to impose&nbsp;a large precision, i.e. a large sample size when we are far away from the&nbsp;solution we search for. The detailed description of the new methods&nbsp;<br />is presented in Chapter 4 together with the convergence analysis. It&nbsp;is shown that the approximate solution is of the same quality as the&nbsp;one obtained by dealing with the full sample from the start.</p><p>Another important characteristic of the methods that are proposed&nbsp;here is the line search technique which is used for obtaining the sub-sequent iterates. The idea is to nd a suitable direction and to search&nbsp;along it until we obtain a sucient decrease in the &nbsp;function value. The&nbsp;sucient decrease is determined throughout the line search rule. In&nbsp;Chapter 4, that rule is supposed to be monotone, i.e. we are imposing&nbsp;strict decrease of the function value. In order to decrease the cost of&nbsp;the algorithm even more and to enlarge the set of suitable search directions, we use nonmonotone line search rules in Chapter 5. Within that chapter, these rules are modied to t the variable sample size framework. Moreover, the conditions for the global convergence and the R-linear rate are presented.&nbsp;</p><p>In Chapter 6, numerical results are presented. The test problems&nbsp;are various - some of them are academic and some of them are real&nbsp;world problems. The academic problems are here to give us more&nbsp;insight into the behavior of the algorithms. On the other hand, data&nbsp;that comes from the real world problems are here to test the real&nbsp;applicability of the proposed algorithms. In the rst part of that&nbsp;chapter, the focus is on the variable sample size techniques. Different&nbsp;implementations of the proposed algorithm are compared to each other&nbsp;and to the other sample schemes as well. The second part is mostly&nbsp;devoted to the comparison of the various line search rules combined&nbsp;with dierent search directions in the variable sample size framework.&nbsp;The overall numerical results show that using the variable sample size&nbsp;can improve the performance of the algorithms signicantly, especially&nbsp;when the nonmonotone line search rules are used.</p><p>The rst chapter of this thesis provides the background material&nbsp;for the subsequent chapters. In Chapter 2, basics of the nonlinear&nbsp;optimization are presented and the focus is on the line search, while&nbsp;Chapter 3 deals with the stochastic framework. These chapters are&nbsp;here to provide the review of the relevant known results, while the&nbsp;rest of the thesis represents the original contribution.&nbsp;</p> / <p>U okviru ove teze posmatra se problem optimizacije bez ograničenja pri čcemu je funkcija cilja u formi matematičkog očekivanja. Očekivanje se odnosi na slučajnu promenljivu koja predstavlja neizvesnost. Zbog toga je funkcija cilja, u stvari, deterministička veličina. Ipak, odredjivanje analitičkog oblika te funkcije cilja može biti vrlo komplikovano pa čak i nemoguće. Zbog toga se za aproksimaciju često koristi uzoračko očcekivanje. Da bi se postigla dobra aproksimacija, obično je neophodan obiman uzorak. Ako pretpostavimo da se uzorak realizuje pre početka procesa optimizacije, možemo posmatrati uzoračko očekivanje kao determinističku funkciju. Medjutim, primena nekog od determinističkih metoda direktno na tu funkciju&nbsp; moze biti veoma skupa jer evaluacija funkcije pod ocekivanjem često predstavlja veliki tro&scaron;ak i uobičajeno je da se ukupan tro&scaron;ak optimizacije meri po broju izračcunavanja funkcije pod očekivanjem. Zbog toga su razvijeni metodi sa promenljivom veličinom uzorka. Većcina njih je bazirana na odredjivanju optimalne dinamike uvećanja uzorka.</p><p>Glavni cilj ove teze je razvoj algoritma koji, kroz smanjenje broja izračcunavanja funkcije, smanjuje ukupne tro&scaron;skove optimizacije. Ideja je da se veličina uzorka smanji kad god je to moguće. Grubo rečeno, izbegava se koriscenje velike preciznosti&nbsp; (velikog uzorka) kada smo daleko od re&scaron;senja. U čcetvrtom poglavlju ove teze opisana je nova klasa metoda i predstavljena je analiza konvergencije. Dokazano je da je aproksimacija re&scaron;enja koju dobijamo bar toliko dobra koliko i za metod koji radi sa celim uzorkom sve vreme.</p><p>Jo&scaron; jedna bitna karakteristika metoda koji su ovde razmatrani je primena linijskog pretražzivanja u cilju odredjivanja naredne iteracije. Osnovna ideja je da se nadje odgovarajući pravac i da se duž njega vr&scaron;si pretraga za dužzinom koraka koja će dovoljno smanjiti vrednost funkcije. Dovoljno smanjenje je odredjeno pravilom linijskog pretraživanja. U čcetvrtom poglavlju to pravilo je monotono &scaron;to znači da zahtevamo striktno smanjenje vrednosti funkcije. U cilju jos većeg smanjenja tro&scaron;kova optimizacije kao i pro&scaron;irenja skupa pogodnih pravaca, u petom poglavlju koristimo nemonotona pravila linijskog pretraživanja koja su modifikovana zbog promenljive velicine uzorka. Takodje, razmatrani su uslovi za globalnu konvergenciju i R-linearnu brzinu konvergencije.</p><p>Numerički rezultati su predstavljeni u &scaron;estom poglavlju. Test problemi su razliciti - neki od njih su akademski, a neki su realni. Akademski problemi su tu da nam daju bolji uvid u pona&scaron;anje algoritama. Sa druge strane, podaci koji poticu od stvarnih problema služe kao pravi test za primenljivost pomenutih algoritama. U prvom delu tog poglavlja akcenat je na načinu ažuriranja veličine uzorka. Različite varijante metoda koji su ovde predloženi porede se medjusobno kao i sa drugim &scaron;emama za ažuriranje veličine uzorka. Drugi deo poglavlja pretežno je posvećen poredjenju različitih pravila linijskog pretraživanja sa različitim pravcima pretraživanja u okviru promenljive veličine uzorka. Uzimajuci sve postignute rezultate u obzir dolazi se do zaključcka da variranje veličine uzorka može značajno popraviti učinak algoritma, posebno ako se koriste nemonotone metode linijskog pretraživanja.</p><p>U prvom poglavlju ove teze opisana je motivacija kao i osnovni pojmovi potrebni za praćenje preostalih poglavlja. U drugom poglavlju je iznet pregled osnova nelinearne optimizacije sa akcentom na metode linijskog pretraživanja, dok su u trećem poglavlju predstavljene osnove stohastičke optimizacije. Pomenuta poglavlja su tu radi pregleda dosada&scaron;njih relevantnih rezultata dok je originalni doprinos ove teze predstavljen u poglavljima 4-6.</p>

<p>Posmatra se problem optimizacije bez ograničenja. Za re&scaron;avanje<br />problema&nbsp; optimizacije bez ograničenja postoji mno&scaron;tvo raznovrsnih<br />metoda. Istraživanje ovde motivisano je potrebom za metodama koje<br />će brzo konvergirati.<br />Cilj je sistematizacija poznatih rezultata, kao i teorijska i numerička<br />analiza mogućnosti uvođenja parametra u gradijentne metode.<br />Najpre se razmatra problem minimizacije konveksne funkcije vi&scaron;e<br />promenljivih.<br />Problem minimizacije konveksne funkcije vi&scaron;e promenljivih ovde se<br />re&scaron;ava bez izračunavanja matrice hesijana, &scaron;to je naročito aktuelno za<br />sisteme velikih dimenzija, kao i za probleme optimizacije kod kojih<br />ne raspolažemo ni tačnom vredno&scaron;ću funkcije cilja, ni tačnom<br />vredno&scaron;ću gradijenta. Deo motivacije za istraživanjem ovde leži i u<br />postojanju problema kod kojih je funkcija cilja rezultat simulacija.<br />Numerički rezultati, predstavljeni u Glavi 6, pokazuju da uvođenje<br />izvesnog parametra može biti korisno, odnosno, dovodi do ubrzanja<br />određenog metoda optimizacije.<br />Takođe se predstavlja jedan novi hibridni metod konjugovanog<br />gradijenta, kod koga je parametar konjugovanog gradijenta<br />konveksna kombinacija dva poznata parametra konjugovanog<br />gradijenta.<br />U prvoj glavi opisuje se motivacija kao i osnovni pojmovi potrebni za<br />praćenje preostalih glava.<br />U drugoj glavi daje se pregled nekih gradijentnih metoda prvog i<br />drugog reda.<br />Četvrta glava sadrži pregled osnovnih pojmova i nekih rezultata<br />vezanih za metode konjugovanih gradijenata.<br />Pomenute glave su tu radi pregleda nekih poznatih rezultata, dok se<br />originalni doprinos predstavlja u trećoj, petoj i &scaron;estoj glavi.<br />U trećoj glavi se opisuje izvesna modifikacija određenog metoda u<br />kome se koristi multiplikativni parametar, izabran na slučajan način.<br />Dokazuje se linearna konvergencija tako formiranog novog metoda.<br />Peta glava sadrži originalne rezultate koji se odnose na metode<br />konjugovanih gradijenata. Naime, u ovoj glavi predstavlja se novi<br />hibridni metod konjugovanih gradijenata, koji je konveksna<br />kombinacija dva poznata metoda konjugovanih gradijenata.<br />U &scaron;estoj glavi se daju rezultati numeričkih eksperimenata, izvr&scaron;enih<br />na&nbsp; izvesnom skupu test funkcija, koji se odnose na metode iz treće i<br />pete glave. Implementacija svih razmatranih algoritama rađena je u<br />paketu MATHEMATICA. Kriterijum upoređivanja je vreme rada<br />centralne procesorske jedinice.6</p> / <p>The problem under consideration is an unconstrained optimization<br />problem. There are many different methods made in aim to solve the<br />optimization problems.&nbsp; The investigation made here is motivated by<br />the fact that the methods which converge fast are necessary.<br />The main goal is the systematization of some known results and also<br />theoretical and numerical analysis of the possibilities to int roduce<br />some parameters within gradient methods.<br />Firstly, the minimization problem is considered, where the objective<br />function is a convex, multivar iable function. This problem is solved<br />here without the calculation of Hessian, and such solution is very<br />important, for example, when the&nbsp; big dimension systems are solved,<br />and also for solving optimization problems with unknown values of<br />the objective function and its gradient. Partially, this investigation is<br />motivated by the existence of problems where the objective function<br />is the result of simulations.<br />Numerical results, presented in&nbsp; Chapter&nbsp; 6, show that the introduction<br />of a parameter is useful, i.e., such introduction results by the<br />acceleration of the known optimization method.<br />Further, one new hybrid conjugate gradient method is presented, in<br />which the conjugate gradient parameter is a convex combination of<br />two known conjugate gradient parameters.<br />In the first chapter, there is motivation and also the basic co ncepts<br />which are necessary for the other chapters.<br />The second chapter contains the survey of some first order and<br />second order gradient methods.<br />The fourth chapter contains the survey of some basic concepts and<br />results corresponding to conjugate gradient methods.<br />The first, the second and the fourth&nbsp; chapters are here to help in<br />considering of some known results, and the original results are<br />presented in the chapters 3,5 and 6.<br />In the third chapter, a modification of one unco nstrained optimization<br />method is presented, in which the randomly chosen multiplicative<br />parameter is used. Also, the linear convergence of such modification<br />is proved.<br />The fifth chapter contains the original results, corresponding to<br />conjugate gradient methods. Namely, one new hybrid conjugate<br />gradient method is presented, and this&nbsp; method is the convex<br />combination of two known conjugate gradient methods.<br />The sixth chapter consists of the numerical results, performed on a set<br />of test functions, corresponding to methods in the chapters 3 and 5.<br />Implementation of all considered algorithms is made in Mathematica.<br />The comparison criterion is CPU time.</p> / <p>The problem under consideration is an unconstrained optimization<br />problem. There are many different methods made in aim to solve the<br />optimization problems.&nbsp; The investigation made here is motivated by<br />the fact that the methods which converge fast are necessary.<br />The main goal is the systematization of some known results and also<br />theoretical and numerical analysis of the possibilities to int roduce<br />some parameters within gradient methods.<br />Firstly, the minimization problem is considered, where the objective<br />function is a convex, multivar iable function. This problem is solved<br />here without the calculation of Hessian, and such solution is very<br />important, for example, when the&nbsp; big dimension systems are solved,<br />and also for solving optimization problems with unknown values of<br />the objective function and its gradient. Partially, this investigation is<br />motivated by the existence of problems where the objective function<br />is the result of simulations.<br />Numerical results, presented in&nbsp; Chapter&nbsp; 6, show that the introduction<br />of a parameter is useful, i.e., such introduction results by the<br />acceleration of the known optimization method.<br />Further, one new hybrid conjugate gradient method is presented, in<br />which the conjugate gradient parameter is a convex combination of<br />two known conjugate gradient parameters.<br />In the first chapter, there is motivation and also the basic co ncepts<br />which are necessary for the other chapters.<br />Key&nbsp; Words Documentation&nbsp; 97<br />The second chapter contains the survey of some first order and<br />second order gradient methods.<br />The fourth chapter contains the survey of some basic concepts and<br />results corresponding to conjugate gradient methods.<br />The first, the second and the fourth&nbsp; chapters are here to help in<br />considering of some known results, and the original results are<br />presented in the chapters 3,5 and 6.<br />In the third chapter, a modification of one unco nstrained optimization<br />method is presented, in which the randomly chosen multiplicative<br />parameter is used. Also, the linear convergence of such modification<br />is proved.<br />The fifth chapter contains the original results, corresponding to<br />conjugate gradient methods. Namely, one new hybrid conjugate<br />gradient method is presented, and this&nbsp; method is the convex<br />combination of two known conjugate gradient methods.<br />The sixth chapter consists of the numerical results, performed on a set<br />of test functions, corresponding to methods in the chapters 3 and 5.<br />Implementation of all considered algorithms is made in Mathematica.<br />The comparison criterion is CPU time</p>

<p>The problem under consideration is an unconstrained mini-mization problem in noisy environment. The common approach for solving the problem is Stochastic Approximation (SA) algorithm. We propose a class of adaptive step size schemes for the SA algorithm. The step size selection in the proposed schemes is based on the objective functionvalues. At each iterate, interval estimates of the optimal function&nbsp; value are constructed using the xed number of previously observed function values.&nbsp;If the observed function value in the current iterate is larger than the upper bound of the interval, we reject the current iterate. If the observed function value in the current iterate is smaller than the lower bound of the interval, we suggest a larger step size in the next iterate. Otherwise, if the function value lies in the interval, we propose a small safe step size in the next iterate. In this manner, a faster progress of the algorithm is ensured&nbsp;when it is expected that larger steps will improve the performance of the algorithm. We propose two main schemes which dier in the intervals that we construct at each iterate. In the rst scheme, we construct a symmetrical interval that can be viewed as a condence-like interval for the optimal function value. The bounds of the interval are shifted means of the xed number of previously observed function values. The generalization&nbsp;of this scheme using a convex combination instead of the mean is also presented. In the second scheme, we use the minimum and the maximum of previous noisy function values as the lower and upper bounds of the interval, respectively. The step size sequences generated by the proposed schemes satisfy the step size convergence conditions for the SA algorithm almost surely. Performance of SA algorithms with the new step size schemes is tested on a set of standard test problems. Numerical results&nbsp;support theoretical expectations and verify eciency of the algorithms in comparison to other relevant modications of SA algorithms. Application of the algorithms in LASSO regression models is also considered. The algorithms are applied for estimation of the regression parameters where the objective function contains L₁ penalty.</p> / <p>Predmet istraživanja doktorske disertacije su numerički postupci za re&scaron;avanje problema stohastičke optimizacije. Najpoznatiji numerički postupak za re&scaron;avanje pomenutog problema je algoritam stohastičke aproksimacije (SA). U disertaciji se&nbsp;&nbsp; predlaže nova klasa &scaron;ema za prilagođavanje dužina koraka u svakoj iteraciji. Odabir dužina koraka u predloženim &scaron;emama se zasniva na vrednostima funkcije cilja. U svakoj iteraciji formira se intervalna ocena optimalne vrednosti funkcije cilja koristeći samo registrovane vrednosti funkcije cilja iz ksnog broja prethodnih iteracija. Ukoliko je vrednost funkcije cilja u trenutnoj iteraciji veća od gornje granice intervala, iteracija se odbacuje. Korak dužine 0 se koristi u narednoj iteraciji. Ako je trenutna vrednost funkcije cilja manja od donje granice intervala, predlaže se duži korak u narednoj iteraciji. Ukoliko vrednost funkcije leži u intervalu, u narednoj iteraciji se koristi korak dobijen harmonijskim pravilom. Na ovaj način se obezbeđuje brzi progres algoritma i&nbsp; izbegavaju mali koraci posebno kada se povećava broj iteracija.&nbsp; &Scaron;eme izbegavaju korake proporcionalne sa 1/k kada se očekuje da ce duži koraci pobolj&scaron;ati proces optimizacije. Predložene &scaron;eme se razlikuju u intervalima koji se formiraju u svakoj iteraciji. U prvoj predloženoj &scaron;emi se formira ve&scaron;tački interval poverenja za ocenu optimalne vrednosti funkcije cilja u svakoj iteraciji. Granice tog intervala se uzimaju za&nbsp; kriterijume dovoljnog smanjenja ili rasta funkcije cilja. Predlaže se i uop&scaron;tenje ove &scaron;eme tako &scaron;to se umesto srednje vrednosti koristi konveksna kombinacija prethodnih vrednosti funkcije cilja. U drugoj &scaron;emi, kriterijum po kom se prilagođavaju dužine koraka su minimum i maksimum prethodnih registrovanih vrednosti funkcije cilja. Nizovi koji se formiranju predloženim &scaron;emama zadovoljavaju uslove potrebne za konvergenciju SA algoritma skoro sigurno. SA algoritmi sa novim &scaron;emama za prilagođavanje dužina koraka su testirani na standardnim test&nbsp; problemima i upoređ eni sa SA algoritmom i njegovim postojećim modikacijama. Rezultati pokazuju napredak u odnosu na klasičan algoritam stohastičke aproksimacije sa determinističkim nizom dužine koraka kao i postojećim adaptivnim algoritmima. Takođe se razmatra primena novih algoritama na LASSO regresijske modele. Algoritmi su primenjeni za ocenjivanje parametara modela.</p>

<p>Algorithmic trading is an automated process of order execution on electronic stock markets. It can be applied to a broad range of financial instruments, and it is&nbsp; characterized by a signicant investors&#39; control over the execution of his/her orders, with the principal goal of finding the right balance between costs and risk of not (fully) executing an order. As the measurement of execution performance gives information whether best execution is achieved, a signicant number of diffeerent benchmarks is&nbsp; used in practice. The most frequently used are price benchmarks, where some of them are determined before trading (Pre-trade benchmarks), some during the trading&nbsp; day (In-traday benchmarks), and some are determined after the trade (Post-trade benchmarks). The two most dominant are VWAP and Arrival Price, which is along with other pre-trade price benchmarks known as the Implementation Shortfall (IS).</p><p>We introduce Negative Selection as a posteriori measure of the execution algorithm performance. It is based on the concept of Optimal Placement, which represents the ideal order that could be executed in a given time win-dow, where the notion of ideal means that it is an order with the best execution price considering&nbsp; market &nbsp;conditions&nbsp; during the time window. Negative Selection is dened as a difference between vectors of optimal and executed orders, with vectors dened as a quantity of shares at specied price positionsin the order book. It is equal to zero when the order is optimally executed; negative if the order is not (completely) filled, and positive if the order is executed but at an unfavorable price.</p><p>Negative Selection is based on the idea to offer a new, alternative performance measure, which will enable us to find the&nbsp; optimal trajectories and construct optimal execution of an order.</p><p>The first chapter of the thesis includes a list of notation and an overview of denitions and theorems that will be used further in the thesis. Chapters 2 and 3 follow with a&nbsp; theoretical overview of concepts related to market microstructure, basic information regarding benchmarks, and theoretical background of algorithmic trading. Original results are presented in chapters 4 and 5. Chapter 4 includes a construction of optimal placement, definition and properties of Negative Selection. The results regarding the properties of a Negative Selection are given in [35]. Chapter 5 contains the theoretical background for stochastic optimization, a model of the optimal execution formulated as a stochastic optimization problem with regard to Negative Selection, as well as original work on nonmonotone line search method [31], while numerical results are in the last, 6th chapter.</p> / <p>Algoritamsko trgovanje je automatizovani proces izvr&scaron;avanja naloga na elektronskim berzama. Može se primeniti na &scaron;irok spektar nansijskih instrumenata kojima se trguje na berzi i karakteri&scaron;e ga značajna kontrola investitora nad izvr&scaron;avanjem njegovih naloga, pri čemu se teži nalaženju pravog balansa izmedu tro&scaron;ka i rizika u vezi sa izvr&scaron;enjem naloga. S ozirom da se merenjem performasi izvr&scaron;enja naloga određuje da li je postignuto najbolje izvr&scaron;enje, u praksi postoji značajan broj različitih pokazatelja. Najče&scaron;će su to pokazatelji cena, neki od njih se određuju pre trgovanja (eng. Pre-trade), neki u toku trgovanja (eng. Intraday), a neki nakon trgovanja (eng. Post-trade). Dva najdominantnija pokazatelja cena su VWAP i Arrival Price koji je zajedno sa ostalim &quot;pre-trade&quot; pokazateljima cena poznat kao Implementation shortfall (IS).</p><p>Pojam negative selekcije se uvodi kao &quot;post-trade&quot; mera performansi algoritama izvr&scaron;enja, polazeći od pojma optimalnog naloga, koji predstavlja idealni nalog koji se&nbsp; mogao izvrsiti u datom vremenskom intervalu, pri ćemu se pod pojmom &quot;idealni&quot; podrazumeva nalog kojim se postiže najbolja cena u trži&scaron;nim uslovima koji su vladali&nbsp; u toku tog vremenskog intervala. Negativna selekcija se defini&scaron;e kao razlika vektora optimalnog i izvr&scaron;enog naloga, pri čemu su vektori naloga defisani kao količine akcija na odgovarajućim pozicijama cena knjige naloga. Ona je jednaka nuli kada je nalog optimalno izvr&scaron;en; negativna, ako nalog nije (u potpunosti) izvr&scaron;en, a pozitivna ako je nalog izvr&scaron;en, ali po nepovoljnoj ceni.</p><p>Uvođenje mere negativne selekcije zasnovano je na ideji da se ponudi nova, alternativna, mera performansi i da se u odnosu na nju nađe optimalna trajektorija i konstrui&scaron;e optimalno izvr&scaron;enje naloga.</p><p>U prvom poglavlju teze dati su lista notacija kao i pregled definicija i teorema&nbsp; neophodnih za izlaganje materije. Poglavlja 2 i 3 bave se teorijskim pregledom pojmova i literature u vezi sa mikrostrukturom trži&scaron;ta, pokazateljima trgovanja i algoritamskim trgovanjem. Originalni rezultati su predstavljeni u 4. i 5. poglavlju. Poglavlje 4 sadrži konstrukciju optimalnog naloga, definiciju i osobine negativne selekcije. Teorijski i praktični rezultati u vezi sa osobinama negativna selekcije dati su u [35]. Poglavlje 5 sadrži teorijske osnove stohastičke optimizacije, definiciju modela za optimalno izvr&scaron;enje, kao i originalni rad u vezi sa metodom nemonotonog linijskog pretraživanja [31], dok 6. poglavlje sadrži empirijske rezultate.</p>

<p>U&nbsp; disertaciji&nbsp; je&nbsp; sprovedeno&nbsp; teorijsko&nbsp; istraživanje&nbsp; fizičko-hemijskih&nbsp; svojstava&nbsp;elektronskog podsistema molekula sumanena. Proračuni su sprovedeni u okvirima DFT i TDDFT&nbsp; teorije&nbsp; sa&nbsp; B3LYP&nbsp; funkcionalom&nbsp; i&nbsp; 6-31Gd,&nbsp; 6-31Gdp i&nbsp; 6-31++Gdp&nbsp; bazisom.&nbsp; Za&nbsp; sve&nbsp;proračune&nbsp; u&nbsp; okvirima&nbsp; navedenog&nbsp; nivoa&nbsp; teorije,&nbsp; kori&scaron;ćen&nbsp; je&nbsp; softverski&nbsp; paket&nbsp; Gaussian03.&nbsp;Ispitana&nbsp; su:&nbsp; strukturna&nbsp; i&nbsp; optička&nbsp; (zajedno&nbsp; sa&nbsp; nelinearnim)&nbsp; svojstva&nbsp; sumanena&nbsp; i&nbsp; njegovih&nbsp;derivata dobijenih modifikovanjem sa atomima bora iazota, aromatična i inverziona svojstva&nbsp;svojstva&nbsp; sumanena&nbsp; i&nbsp; njegovih&nbsp; derivata,&nbsp; adsorpciona&nbsp; svojstva&nbsp; sumanena&nbsp; prema&nbsp; molekulima&nbsp;H₂, CO, CO_2&nbsp;i NH_3&nbsp;i uticaj spolja&scaron;njeg električnog polja na svojstva sumanena.</p><p>Prema&nbsp; dobijenim&nbsp; rezultatima,&nbsp; fundamentalna&nbsp; fizičko-hemijska&nbsp; svojstva&nbsp; molekula&nbsp;<br />sumanena mogu se efikasno i fino pode&scaron;avati supstitucijom i disupstitucijom hetero-atomima&nbsp;bora&nbsp; i&nbsp; azota.&nbsp; Dubina&nbsp; i&nbsp; inverziona&nbsp; barijera&nbsp; sumanena se&nbsp; na&nbsp; ovaj&nbsp; način&nbsp; može&nbsp; menjati&nbsp; u&nbsp; oba&nbsp;smera.&nbsp; Pokazalo&nbsp; se&nbsp; da&nbsp; su&nbsp; ova&nbsp; dva&nbsp; parametra&nbsp; u&nbsp; korelaciji,&nbsp; s&nbsp; obzirom&nbsp; da inverziona&nbsp; barijera&nbsp;skoro potpuno zavisi od dubine sumanena. Slično je i sa aromatičnim svojstvima prstenova,&nbsp;koja se u zavisnosti od broja uvedenih hetero-atomamenja od aromatične preko nearomatične&nbsp;do antiaromatične prirode. Optička svojstva su, pomenutim modifikacijama, takođe značajno&nbsp;pobolj&scaron;ana: apsorpcioni pikovi kod UV/Vis spektra se pomeraju ka vidljivoj oblasti, dok je&nbsp;hiperpolarizibilnost&nbsp; sumanena&nbsp; i&nbsp; derivata&nbsp; sa&nbsp; jednim&nbsp; atomom&nbsp; bora&nbsp; vi&scaron;a&nbsp; od&nbsp; referentnog&nbsp;molekula uree, respektivno, 9 i 49 puta.</p><p>Prema dobijenim rezultatima, adsorpciona svojstva sumanena su veoma kompetitivna&nbsp;sa&nbsp; ostalim&nbsp; organskim&nbsp; molekulima&nbsp; kao&nbsp; &scaron;to&nbsp; su&nbsp; ugljenične&nbsp; nanotube&nbsp; i&nbsp; fuleren&nbsp; C₆₀,&nbsp; usled&nbsp;postojanja&nbsp; značajnog&nbsp; dipolnog&nbsp; momenta&nbsp; zbog&nbsp; specifične&nbsp; geometrije.&nbsp; Posebno&nbsp; se&nbsp; ističu&nbsp;pozitivna adsorpciona svojstva prema molekulima H_2&nbsp;<br />i CO. Dok je prvo pomenuti bitan sa&nbsp;energetskog&nbsp; aspekta,&nbsp; specifičnost&nbsp; drugog&nbsp; molekula&nbsp; se&nbsp; ogleda&nbsp; u&nbsp; činjenici&nbsp; da&nbsp; ne&nbsp; može&nbsp; biti&nbsp;adsorbovan od strane ugljeničnih nanotuba.&nbsp;</p><p>Potencijal sumanena, posebno u oblasti organske elektronike, dodatno ističu i rezultati&nbsp;ispitivanja&nbsp; uticaja&nbsp; spolja&scaron;njeg&nbsp; električnog&nbsp; polja.&nbsp; Naime,&nbsp; dobijeni&nbsp; rezultati&nbsp; ukazuju&nbsp; da&nbsp; se&nbsp;specifična&nbsp; (posebno&nbsp; adsorpciona)&nbsp; svojstva&nbsp; sumanena&nbsp; mogu&nbsp; dodatno&nbsp; pobolj&scaron;ati&nbsp; primenom&nbsp;slabog, i eksperimentalno lako ostvarljivog, električnog &nbsp;polja, jer se intenzivira razdvajanje&nbsp;naelektrisanja.</p> / <p>Theoretical&nbsp; investigation&nbsp; of&nbsp; physicochemical&nbsp; properties&nbsp; of&nbsp; electron&nbsp; subsystem&nbsp; of&nbsp;sumanene&nbsp; molecule&nbsp; has&nbsp; been&nbsp; conducted&nbsp; in&nbsp; this&nbsp; doctoral&nbsp; work.&nbsp; Calculations &nbsp;are&nbsp; performed&nbsp;within DFT and TD-DFT with B3LYP functional and 6-31Gd, 6-31Gdp i &nbsp; 6-31++Gdp basis&nbsp;sets. For all calculations, on the mentioned level of theory, Gaussian03software package was&nbsp;used.&nbsp; Investigation&nbsp; encompassed:&nbsp; structural &nbsp; and&nbsp; optical&nbsp; (including&nbsp; nonlinear)&nbsp; properties&nbsp; of&nbsp;sumanene&nbsp; and&nbsp; its&nbsp; derivatives&nbsp; obtained&nbsp; by&nbsp; modification&nbsp; with&nbsp; boron&nbsp; and&nbsp; nitrogen&nbsp; atoms,&nbsp;aromatic and bowl to bowl inversion properties of sumanene and its derivatives, adsorption&nbsp;properties of sumanene towards H₂, CO, CO_2&nbsp;i NH_3&nbsp;molecules and the influence of external&nbsp;electric field to the properties of sumanene.</p><p>According to obtained results, fundamental physicochemical properties of sumanene&nbsp;molecule could be efficiently and finely adjusted with the monosubstitution and disubstitution&nbsp;with&nbsp; hetero-atoms&nbsp; of&nbsp; boron&nbsp; and&nbsp; nitrogen.&nbsp; Bowl&nbsp; depth and&nbsp; bowl&nbsp; to&nbsp; bowl&nbsp; inversion&nbsp; barrier&nbsp;could&nbsp; be&nbsp; tuned&nbsp; in&nbsp; both&nbsp; directions.&nbsp; It&nbsp; was&nbsp; demonstrated&nbsp; that&nbsp; these&nbsp; two&nbsp; parameters&nbsp; are&nbsp;correlated, since bowl to bowl inversion barrier principally depends on the fourth power of&nbsp;bowl&nbsp; depth.&nbsp; Similar&nbsp; situation&nbsp; is&nbsp; with&nbsp; aromatic&nbsp; nature&nbsp; of&nbsp; sumanene&nbsp; rings,&nbsp; which&nbsp; change,&nbsp;depending&nbsp; on&nbsp; the&nbsp; number&nbsp; of&nbsp; introduced&nbsp; boron&nbsp; and&nbsp; nitrogen&nbsp; atoms,&nbsp; from&nbsp; aromatic&nbsp; through&nbsp;non-aromatic&nbsp; towards&nbsp; anti-aromatic&nbsp; nature.&nbsp; Thanks&nbsp; to&nbsp; mentioned&nbsp; modifications,&nbsp; optical&nbsp;properties&nbsp; are&nbsp; improved&nbsp; as&nbsp; well:&nbsp; absorption&nbsp; peaks&nbsp; in&nbsp; UV/Vis&nbsp; spectra&nbsp; shift&nbsp; towards&nbsp; visible&nbsp;area,&nbsp; while&nbsp; hyperpolarizabilities&nbsp; of&nbsp; sumanene&nbsp; and&nbsp; its&nbsp; derivative&nbsp; containing&nbsp; one&nbsp; boron&nbsp; atom&nbsp;are higher than hyperpolarizability of referent molecule of urea, respectively, 9 and 49 times.&nbsp;</p><p>Thanks&nbsp; to&nbsp; significant&nbsp; dipole&nbsp; moment&nbsp; due&nbsp; to&nbsp; the&nbsp; specific&nbsp; geometry, &nbsp;nvestigated &nbsp;adsorption properties of sumanene are very competitive to the other &nbsp;organic molecules such&nbsp;as carbon nanotubes and fullerene C₆₀. Positive adsorption properties of sumanene towards H_2&nbsp;and CO molecules are distinguished. While the H_2&nbsp;molecule is important from the energetic&nbsp;aspects, specificity of CO molecules is that it can&rsquo;t be adsorbed by carbon nanotubes.</p><p>Potential&nbsp; of&nbsp; sumanene,&nbsp; especially&nbsp; in&nbsp; the&nbsp; field&nbsp; of&nbsp; organic&nbsp; electronics,&nbsp; is &nbsp;emphasized&nbsp;through the obtained results related to the investigation of the influence of external electric&nbsp;field.&nbsp; Obtained&nbsp; results&nbsp; indicate&nbsp; that&nbsp; specific&nbsp; (especially&nbsp; adsorption)&nbsp; properties&nbsp; of&nbsp; sumanene&nbsp;could be additionally improved by application of &nbsp;weak, and experimentally easily achievable,&nbsp;external electric field, due to the higher charge separation.&nbsp;</p>