A globally convergent primal-dual interior-point filter method for nonlinear programming

In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primal-dual interior-point algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters. The new algorithm decomposes the primal-dual step obtained from the perturbed first-order necessary conditions into a normal and a tangential step, whose sizes are controlled by a trust-region type parameter. Each entry in the filter is a pair of coordinates: one resulting from feasibility and centrality, and associated with the normal step; the other resulting from optimality (complementarity and duality), and related with the tangential step. Global convergence to first-order critical points is proved for the new primal-dual interior-point filter algorithm

Hybrid Filter Methods for Nonlinear Optimization

Globalization strategies used by algorithms to solve nonlinear constrained optimization problems must balance the oftentimes conflicting goals of reducing the objective function and satisfying the constraints. The use of merit functions and filters are two such popular strategies, both of which have their strengths and weaknesses. In particular, traditional filter methods require the use of a restoration phase that is designed to reduce infeasibility while ignoring the objective function. For this reason, there is often a significant decrease in performance when restoration is triggered. In Chapter 3, we present a new filter method that addresses this main weakness of traditional filter methods. Specifically, we present a hybrid filter method that avoids a traditional restoration phase and instead employs a penalty mode that is built upon the l-1 penalty function; the penalty mode is entered when an iterate decreases both the penalty function and the constraint violation. Moreover, the algorithm uses the same search direction computation procedure during every iteration and uses local feasibility estimates that emerge during this procedure to define a new, improved, and adaptive margin (envelope) of the filter. Since we use the penalty function (a combination of the objective function and constraint violation) to define the search direction, our algorithm never ignores the objective function, a property that is not shared by traditional filter methods. Our algorithm thusly draws upon the strengths of both filter and penalty methods to form a novel hybrid approach that is robust and efficient. In particular, under common assumptions, we prove global convergence of our algorithm. In Chapter 4, we present a nonmonotonic variant of the algorithm in Chapter 3. For this version of our method, we prove that it generates iterates that converge to a first-order solution from an arbitrary starting point, with a superlinear rate of convergence. We also present numerical results that validate the efficiency of our method. Finally, in Chapter 5, we present a numerical study on the application of a recently developed bound-constrained quadratic optimization algorithm on the dual formulation of sparse large-scale strictly convex quadratic problems. Such problems are of particular interest since they arise as subproblems during every iteration of our new filter methods

Modifikacije metoda NJutnovog tipa za rešavanje semi-glatkih problema stohastičke optimizacije

 In numerous optimization problems originating from real-world and scientific applications, we often face nonsmoothness. A large number of problems belong to this class, from models of natural phenomena that exhibit sudden changes, shape optimization, to hinge loss functions in machine learning and deep neural networks. In practice, solving a on smooth convex problem tends to be more challenging, usually more difficult and costly than a smooth one. The aim of this thesis is the formulation and theoretical analysis of Newton-type algorithms for solving nonsmooth convex stochastic optimization problems. The optimization problems with the objective function given in the form of a mathematical expectation without differentiability assumption of the function are considered. The Sample Average Approximation (SAA) is used to estimate the objective function. As the accuracy of the SAA objective functions and its derivatives is naturally proportional to the computational costs – higher precision implies larger costs in general, it is important to design an efficient balance between accuracy and costs. Therefore, the main focus of this thesis is the development of adaptive sample size control algorithms in a nonsmooth environment, with particular attention given to the control of the accuracy and selection of search directions. Several options are investigated for the search direction, while the accuracy control involves cheaper objective function approximations (with looser accuracy) during the initial stages of the process to save computational effort. This approach aims to conserve computational resources, reserving the deployment of high-accuracy objective function approximations for the final stages of the optimization process. A detailed description of the proposed methods is presented in Chapter 5 and 6. Also, the theoretical properties of the numerical procedures are analyzed, i.e., their convergence is proved, and the complexity of the developed methods is studied. In addition to the theoretical framework, the successful practical implementation of the given algorithms is presented. It is shown that the proposed methods are more efficient in practical application compared to the existing methods from the literature. Chapter 1 of this thesis serves as a foundation for the subsequent chapters by providing the necessary background information. Chapter 2 covers the fundamentals of nonlinear optimization, with a particular emphasis on line search techniques. In Chapter 3, the focus shifts to the nonsmooth framework. This chapter serves the purpose of reviewing the existing knowledge and established results in the field. The remaining sections of the thesis, starting from Chapter 4, where the framework for the subject of this thesis (the minimization of the expected value function) is introduced, onwards, represent the original contribution made by the author.У бројним проблемима оптимизације који потичу из стварних и научних примена, често се суочавамо са недиференцијабилношћу. У ову класу спада велики број проблема, од модела природних феномена који показују нагле промене, оптимизације облика, до функције циља у машинском учењу и дубоким неуронским мрежама. У пракси, решавање семи-глатких конвексних проблема обично је изазовније и захтева веће рачунске трошкове у односу на глатке проблеме. Циљ ове тезе је формулација и теоријска анализа метода Њутновог типа за решавање семи-глатких конвексних стохастичких проблема оптимизације. Разматрани су проблеми оптимизације са функцијом циља датом у облику математичког очекивања без претпоставке о диференцијабилности функције. Како је врло тешко, па некад чак и немогуће одредити аналитички облик математичког очекивања, функција циља се апроксимира узорачким очекивањем. Имајући у виду да је тачност апроксимације функције циља и њених извода пропорционална рачунским трошковима – већа прецизност подразумева веће трошкове у општем случају, важно је дизајнирати ефикасан баланс између тачности и трошкова. Стога, главни фокус ове тезе је развојалгоритама базираних на одређивању оптималне динамике увећања узорка у семи-глатком окружењу, са посебном пажњом на контроли тачности и одабиру праваца претраге. По питању одабира правца, размотрено је неколико опција, док контрола тачности укључује јефтиније апроксимације функције циља (са мањом прецизношћу) током почетних фаза процеса да би се уштедели рачунски напори. Овај приступ има за циљ очување рачунских ресурса, резервишући примену апроксимација функције циља високе тачности за завршне фазе процеса оптимизације. Детаљан опис предложених метода представљен је у поглављима 5 и 6, где су анализиране и теоријске особине нумеричких поступака, тј. доказана је њихова конвергенција и приказана сложеност развијених метода. Поред теоријског оквира, потврђена је успешна практична имплементација датих алгоритама. Показано је да су предложене методе ефикасније у практичној примени у односу на постојеће методе из литературе. Поглавље 1 ове тезе служи као основа за праћење наредних поглавља пружајући преглед основних појмова. Поглавље 2 се односи на нелинеарну оптимизацију, при чему је посебан акценат стављен на технике линијског претраживања. У поглављу 3 фокус се помера на семи-глатке проблеме оптимизације и методе за њихово решавање и служи као преглед постојећих резултата из ове области. Преостали делови тезе, почевши од поглавља 4, где се уводи проблем изучавања ове тезе (минимизација функције дате у облику очекиване вредности), па надаље, представљају оригинални допринос аутора.U brojnim problemima optimizacije koji potiču iz stvarnih i naučnih primena, često se suočavamo sa nediferencijabilnošću. U ovu klasu spada veliki broj problema, od modela prirodnih fenomena koji pokazuju nagle promene, optimizacije oblika, do funkcije cilja u mašinskom učenju i dubokim neuronskim mrežama. U praksi, rešavanje semi-glatkih konveksnih problema obično je izazovnije i zahteva veće računske troškove u odnosu na glatke probleme. Cilj ove teze je formulacija i teorijska analiza metoda NJutnovog tipa za rešavanje semi-glatkih konveksnih stohastičkih problema optimizacije. Razmatrani su problemi optimizacije sa funkcijom cilja datom u obliku matematičkog očekivanja bez pretpostavke o diferencijabilnosti funkcije. Kako je vrlo teško, pa nekad čak i nemoguće odrediti analitički oblik matematičkog očekivanja, funkcija cilja se aproksimira uzoračkim očekivanjem. Imajući u vidu da je tačnost aproksimacije funkcije cilja i njenih izvoda proporcionalna računskim troškovima – veća preciznost podrazumeva veće troškove u opštem slučaju, važno je dizajnirati efikasan balans između tačnosti i troškova. Stoga, glavni fokus ove teze je razvojalgoritama baziranih na određivanju optimalne dinamike uvećanja uzorka u semi-glatkom okruženju, sa posebnom pažnjom na kontroli tačnosti i odabiru pravaca pretrage. Po pitanju odabira pravca, razmotreno je nekoliko opcija, dok kontrola tačnosti uključuje jeftinije aproksimacije funkcije cilja (sa manjom preciznošću) tokom početnih faza procesa da bi se uštedeli računski napori. Ovaj pristup ima za cilj očuvanje računskih resursa, rezervišući primenu aproksimacija funkcije cilja visoke tačnosti za završne faze procesa optimizacije. Detaljan opis predloženih metoda predstavljen je u poglavljima 5 i 6, gde su analizirane i teorijske osobine numeričkih postupaka, tj. dokazana je njihova konvergencija i prikazana složenost razvijenih metoda. Pored teorijskog okvira, potvrđena je uspešna praktična implementacija datih algoritama. Pokazano je da su predložene metode efikasnije u praktičnoj primeni u odnosu na postojeće metode iz literature. Poglavlje 1 ove teze služi kao osnova za praćenje narednih poglavlja pružajući pregled osnovnih pojmova. Poglavlje 2 se odnosi na nelinearnu optimizaciju, pri čemu je poseban akcenat stavljen na tehnike linijskog pretraživanja. U poglavlju 3 fokus se pomera na semi-glatke probleme optimizacije i metode za njihovo rešavanje i služi kao pregled postojećih rezultata iz ove oblasti. Preostali delovi teze, počevši od poglavlja 4, gde se uvodi problem izučavanja ove teze (minimizacija funkcije date u obliku očekivane vrednosti), pa nadalje, predstavljaju originalni doprinos autora

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Standard Bundle Methods: Untrusted Models and Duality

We review the basic ideas underlying the vast family of algorithms for nonsmooth convex optimization known as "bundle methods|. In a nutshell, these approaches are based on constructing models of the function, but lack of continuity of first-order information implies that these models cannot be trusted, not even close to an optimum. Therefore, many different forms of stabilization have been proposed to try to avoid being led to areas where the model is so inaccurate as to result in almost useless steps. In the development of these methods, duality arguments are useful, if not outright necessary, to better analyze the behaviour of the algorithms. Also, in many relevant applications the function at hand is itself a dual one, so that duality allows to map back algorithmic concepts and results into a "primal space" where they can be exploited; in turn, structure in that space can be exploited to improve the algorithms' behaviour, e.g. by developing better models. We present an updated picture of the many developments around the basic idea along at least three different axes: form of the stabilization, form of the model, and approximate evaluation of the function

A vision-based optical character recognition system for real-time identification of tractors in a port container terminal

Automation has been seen as a promising solution to increase the productivity of modern sea port container terminals. The potential of increase in throughput, work efficiency and reduction of labor cost have lured stick holders to strive for the introduction of automation in the overall terminal operation. A specific container handling process that is readily amenable to automation is the deployment and control of gantry cranes in the container yard of a container terminal where typical operations of truck identification, loading and unloading containers, and job management are primarily performed manually in a typical terminal. To facilitate the overall automation of the gantry crane operation, we devised an approach for the real-time identification of tractors through the recognition of the corresponding number plates that are located on top of the tractor cabin. With this crucial piece of information, remote or automated yard operations can then be performed. A machine vision-based system is introduced whereby these number plates are read and identified in real-time while the tractors are operating in the terminal. In this paper, we present the design and implementation of the system and highlight the major difficulties encountered including the recognition of character information printed on the number plates due to poor image integrity. Working solutions are proposed to address these problems which are incorporated in the overall identification system.postprin

Job shop scheduling with artificial immune systems

The job shop scheduling is complex due to the dynamic environment. When the information of the jobs and machines are pre-defined and no unexpected events occur, the job shop is static. However, the real scheduling environment is always dynamic due to the constantly changing information and different uncertainties. This study discusses this complex job shop scheduling environment, and applies the AIS theory and switching strategy that changes the sequencing approach to the dispatching approach by taking into account the system status to solve this problem. AIS is a biological inspired computational paradigm that simulates the mechanisms of the biological immune system. Therefore, AIS presents appealing features of immune system that make AIS unique from other evolutionary intelligent algorithm, such as self-learning, long-lasting memory, cross reactive response, discrimination of self from non-self, fault tolerance, and strong adaptability to the environment. These features of AIS are successfully used in this study to solve the job shop scheduling problem. When the job shop environment is static, sequencing approach based on the clonal selection theory and immune network theory of AIS is applied. This approach achieves great performance, especially for small size problems in terms of computation time. The feature of long-lasting memory is demonstrated to be able to accelerate the convergence rate of the algorithm and reduce the computation time. When some unexpected events occasionally arrive at the job shop and disrupt the static environment, an extended deterministic dendritic cell algorithm (DCA) based on the DCA theory of AIS is proposed to arrange the rescheduling process to balance the efficiency and stability of the system. When the disturbances continuously occur, such as the continuous jobs arrival, the sequencing approach is changed to the dispatching approach that involves the priority dispatching rules (PDRs). The immune network theory of AIS is applied to propose an idiotypic network model of PDRs to arrange the application of various dispatching rules. The experiments show that the proposed network model presents strong adaptability to the dynamic job shop scheduling environment.postprin