DC Algorithm for Sample Average Approximation of Chance Constrained Programming: Convergence and Numerical Results

Chance constrained programming refers to an optimization problem with uncertain constraints that must be satisfied with at least a prescribed probability level. In this work, we study a class of structured chance constrained programs in the data-driven setting, where the objective function is a difference-of-convex (DC) function and the functions in the chance constraint are all convex. By exploiting the structure, we reformulate it into a DC constrained DC program. Then, we propose a proximal DC algorithm for solving the reformulation. Moreover, we prove the convergence of the proposed algorithm based on the Kurdyka-\L ojasiewicz property and derive the iteration complexity for finding an approximate KKT point. We point out that the proposed pDCA and its associated analysis apply to general DC constrained DC programs, which may be of independent interests. To support and complement our theoretical development, we show via numerical experiments that our proposed approach is competitive with a host of existing approaches.Comment: 31 pages, 3 table

Solving And Applications Of Multi-Facility Location Problems

This thesis is devoted towards the study and solving of a new class of multi-facility location problems. This class is of a great theoretical interest both in variational analysis and optimization while being of high importance to a variety of practical applications. Optimization problems of this type cannot be reduced to convex programming like, the much more investigated facility location problems with only one center. In contrast, such classes of multi-facility location problems can be described by using DC (difference of convex) programming, which are significantly more involved from both theoretical and numerical viewpoints.In this thesis, we present a new approach to solve multi-facility location problems, which is based on mixed integer programming and algorithms for minimizing differences of convex (DC) functions. We then computationally implement the proposed algorithm on both artificial and real data sets and provide many numerical examples. Finally, some directions and insights for future work are detailed

Applications of non-convex optimization in portfolio selection

Die vorgelegte Arbeit befasst sich mit nicht-konvexer Optimierung in dem Gebiet der Portfolio Selection. Thematisch lässt sich die Arbeit in zwei Teilgebiete strukturieren: (1) Das Lösen von Mean-Risk Problemen mit Value-at-Risk als Risikomaß: Es werden Methoden zum Auffinden von effizienten Portfolios für den Fall von diskret verteilten Asset Returns vorgestellt. Die behandelten Probleme sind (wegen der Nicht-Konvexität des Value-at-Risk) nicht konvex und lassen sich als Differenz von konvexen Funktionen darstellen. Es werden sowohl Branch-and-Bound als auch approximative Lösungsverfahren angewandt. Die globalen Lösungen des Branch-and-Bound werden mit den Lösungen der approximativen Verfahren verglichen. (2) Robustifizierung von Portfolio-Selection Problemen: In den letzten Jahren gibt es in der Literatur verstärkt Bemühungen Optimierungsprobleme bezüglich Unsicherheiten in den Parametern zu robustifizieren. Robustifizierte Lösungen haben die Eigenschaft, dass moderate Variationen von Parametern nicht zu dramatischen Verschlechterungen der Lösungen führen. Im Rahmen der robusten Portfolio Optimierung geht es hauptsächlich darum, Lösungen in Bezug auf Abweichungen in den Verteilungen der Gewinne der verwendeten Finanzinstrumente zu kontrollieren. In der gegenständlichen Arbeit werden mit Hilfe von Wahrscheinlichkeitsmetriken sogenannte Ambiguity Mengen definiert, welche alle Verteilungen enthalten, die aufgrund der Datenlage als mögliche Verteilungen in Frage kommen. Die verwendete Metrik, die sogenannte Kantorovich (Wasserstein) Metrik, ermöglicht es mittels Ergebnissen der nichtparametrischen Statistik, die Ambiguity Mengen als Konfidenzmengen um die empirischen Verteilungschätzer zu interpretieren. Mittels der beschriebenen Methoden werden Mean-Risk Probleme robustifiziert. Diese Probleme sind zunächst infinit und werden in einem weiteren Schritt zu nicht konvexen semi-definiten Problemen umformuliert. Die Lösung dieser Probleme basiert einerseits auf einem Algortihmus zum Lösen von semi-definiten Problemen mit unendlich vielen Nebenbedingungen und andererseits auf Methoden zum approximativen Lösen von nicht konvexen Problemen (dem sogenannten Difference of Convex Algorithm).The thesis is concerned with application of non-convex programming to problems of portfolio optimization in a single stage stochastic optimization framework. In particular two different classes of portfolio selection problems are investigated. In both the problems a scenario based approach to modeling uncertainty is pursued, i.e. the randomness in the models is always described by finitely many joint realizations of the asset returns. The thesis is structured into three chapters briefly outlined below: (1) A D.C. Formulation of Value-at-Risk constrained Optimization: In this Chapter the aim is to solve mean risk models with the Value-at-Risk as a risk measure. In the case of finitely supported return distributions, it is shown that the Value-at-Risk can be written as a D.C. function and the mentioned mean risk problem therefore corresponds to a D.C. problem. The non-convex problem of optimizing the Value at Risk is rather extensively treated in the literature and there are various approximative solution techniques as well as some approaches to solve the problem globally. The reformulation as D.C. problem provides an insight into the structure of the problem, which can be exploited to devise a Branch-and-Bound algorithm for finding global solutions for small to medium sized instances. The possibility of refining epsilon-optimal solutions obtained from the Branch-and-Bound framework via local search heuristics is also discussed in this Chapter. (2) Value-at-Risk constrained optimization using the DCA: In this part of the thesis the Value-at-Risk problem is once again investigated with the aim of solving problems of realistic sizes in relatively short time. Since the Value at Risk optimization can be shown to be a NP hard problem, this can only be achieved by sacrificing on the guaranteed globality of the solutions. Therefore a local solution technique for unconstrained D.C. problems called Difference of Convex Algorithm (DCA) is employed. To solve the problem a new variant of the DCA the so called 'hybrid DCA' is proposed, which preserves the favorable convergence properties of the computationally hard 'complete DCA' as well as the computational tractability of the so called 'simple DCA'. The results are tested for small problems and the solutions are shown to actually coincide with the global optima obtained with the Branch-and-Bound algorithm in most of the cases. For realistic problem sizes the proposed method is shown to consistently outperform known heuristic approximations implemented in commercial software. (3) A Framework for Optimization under Ambiguity: The last part of the thesis is devoted to a different topic which received much attention in the recent stochastic programming literature: the topic of robust optimization. More specifically the aim is to robustify single stage stochastic optimization models with respect to uncertainty about the distributions of the random variables involved in the formulation of the stochastic program. The aim is to explore ways of explicitly taking into account ambiguity about the distributions when finding a decision while imposing only very weak restrictions on possible probability models that are taken into consideration. Ambiguity is defined as possible deviation from a discrete reference measure Q (in this work the empirical measure). To this end a so called ambiguity set B, that contains all the measures that can reasonably be assumed to be the real measure P given the available data, is defined. Since the idea is to devise a general approach not restricted by assuming P to be an element of any specific parametric family, we define our ambiguity sets by the use of general probability metrics. Relative to these measures a worst case approach is adopted to robustify the problem with respect to B. The resulting optimization problems turn out to be infinite and are reduced to non-convex semi-definite problems. In the last part of the paper we show how to solve these problems numerically for the example of a mean risk portfolio selection problem with Expected Shortfall under a Threshold as the risk measure. The DCA in combination with an iterative algorithm to approximate the infinite set of constraints by finitely many ones is used to obtain numerical solutions to the problem

Joint resource allocation for full-duplex ambient backscatter communication: A difference convex algorithm

Nowadays, Ambient Backscatter Communication (AmBC) systems have emerged as a green communication technology to enable massive self-sustainable wireless networks by leveraging Radio Frequency (RF) Energy Harvesting (EH) capability. A Full-duplex Ambient Backscatter Communication (FAmBC) network with a Full-duplex Access Point (AP), a dedicated Legacy User (LU), and several Backscatter Devices (BDs) is considered in this study. The AP with two antennas transfers downlink Orthogonal Frequency Division Multiplexing (OFDM) information and energy to the dedicated LU and several BDs, respectively, while receiving uplink backscattered information from BDs at the same time. One of the key aims in AmBC networks is to ensure fairness among BDs. To address this, we propose the Multi-objective Lexicographical Optimization Problem (MLOP), which aims to maximize the minimum BD’s

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

On multiobjective optimization from the nonsmooth perspective

Practical applications usually have multiobjective nature rather than having only one objective to optimize. A multiobjective problem cannot be solved with a single-objective solver as such. On the other hand, optimization of only one objective may lead to an arbitrary bad solutions with respect to other objectives. Therefore, special techniques for multiobjective optimization are vital. In addition to multiobjective nature, many real-life problems have nonsmooth (i.e. not continuously differentiable) structure. Unfortunately, many smooth (i.e. continuously differentiable) methods adopt gradient-based information which cannot be used for nonsmooth problems. Since both of these characteristics are relevant for applications, we focus here on nonsmooth multiobjective optimization. As a research topic, nonsmooth multiobjective optimization has gained only limited attraction while the fields of nonsmooth single-objective and smooth multiobjective optimization distinctively have attained greater interest. This dissertation covers parts of nonsmooth multiobjective optimization in terms of theory, methodology and application. Bundle methods are widely considered as effective and reliable solvers for single-objective nonsmooth optimization. Therefore, we investigate the use of the bundle idea in the multiobjective framework with three different methods. The first one generalizes the single-objective proximal bundle method for the nonconvex multiobjective constrained problem. The second method adopts the ideas from the classical steepest descent method into the convex unconstrained multiobjective case. The third method is designed for multiobjective problems with constraints where both the objectives and constraints can be represented as a difference of convex (DC) functions. Beside the bundle idea, all three methods are descent, meaning that they produce better values for each objective at each iteration. Furthermore, all of them utilize the improvement function either directly or indirectly. A notable fact is that none of these methods use scalarization in the traditional sense. With the scalarization we refer to the techniques transforming a multiobjective problem into the single-objective one. As the scalarization plays an important role in multiobjective optimization, we present one special family of achievement scalarizing functions as a representative of this category. In general, the achievement scalarizing functions suit well in the interactive framework. Thus, we propose the interactive method using our special family of achievement scalarizing functions. In addition, this method utilizes the above mentioned descent methods as tools to illustrate the range of optimal solutions. Finally, this interactive method is used to solve the practical case studies of the scheduling the final disposal of the spent nuclear fuel in Finland.Käytännön optimointisovellukset ovat usein luonteeltaan ennemmin moni- kuin yksitavoitteisia. Erityisesti monitavoitteisille tehtäville suunnitellut menetelmät ovat tarpeen, sillä monitavoitteista optimointitehtävää ei sellaisenaan pysty ratkaisemaan yksitavoitteisilla menetelmillä eikä vain yhden tavoitteen optimointi välttämättä tuota mielekästä ratkaisua muiden tavoitteiden suhteen. Monitavoitteisuuden lisäksi useat käytännön tehtävät ovat myös epäsileitä siten, etteivät niissä esiintyvät kohde- ja rajoitefunktiot välttämättä ole kaikkialla jatkuvasti differentioituvia. Kuitenkin monet optimointimenetelmät hyödyntävät gradienttiin pohjautuvaa tietoa, jota ei epäsileille funktioille ole saatavissa. Näiden molempien ominaisuuksien ollessa keskeisiä sovelluksia ajatellen, keskitytään tässä työssä epäsileään monitavoiteoptimointiin. Tutkimusalana epäsileä monitavoiteoptimointi on saanut vain vähän huomiota osakseen, vaikka sekä sileä monitavoiteoptimointi että yksitavoitteinen epäsileä optimointi erikseen ovat aktiivisia tutkimusaloja. Tässä työssä epäsileää monitavoiteoptimointia on käsitelty niin teorian, menetelmien kuin käytännön sovelluksien kannalta. Kimppumenetelmiä pidetään yleisesti tehokkaina ja luotettavina menetelminä epäsileän optimointitehtävän ratkaisemiseen ja siksi tätä ajatusta hyödynnetään myös tässä väitöskirjassa kolmessa eri menetelmässä. Ensimmäinen näistä yleistää yksitavoitteisen proksimaalisen kimppumenetelmän epäkonveksille monitavoitteiselle rajoitteiselle tehtävälle sopivaksi. Toinen menetelmä hyödyntää klassisen nopeimman laskeutumisen menetelmän ideaa konveksille rajoitteettomalle tehtävälle. Kolmas menetelmä on suunniteltu erityisesti monitavoitteisille rajoitteisille tehtäville, joiden kohde- ja rajoitefunktiot voidaan ilmaista kahden konveksin funktion erotuksena. Kimppuajatuksen lisäksi kaikki kolme menetelmää ovat laskevia eli ne tuottavat joka kierroksella paremman arvon jokaiselle tavoitteelle. Yhteistä on myös se, että nämä kaikki hyödyntävät parannusfunktiota joko suoraan sellaisenaan tai epäsuorasti. Huomattavaa on, ettei yksikään näistä menetelmistä hyödynnä skalarisointia perinteisessä merkityksessään. Skalarisoinnilla viitataan menetelmiin, joissa usean tavoitteen tehtävä on muutettu sopivaksi yksitavoitteiseksi tehtäväksi. Monitavoiteoptimointimenetelmien joukossa skalarisoinnilla on vankka jalansija. Esimerkkinä skalarisoinnista tässä työssä esitellään yksi saavuttavien skalarisointifunktioiden perhe. Yleisesti saavuttavat skalarisointifunktiot soveltuvat hyvin interaktiivisten menetelmien rakennuspalikoiksi. Täten kuvaillaan myös esiteltyä skalarisointifunktioiden perhettä hyödyntävä interaktiivinen menetelmä, joka lisäksi hyödyntää laskevia menetelmiä optimaalisten ratkaisujen havainnollistamisen apuna. Lopuksi tätä interaktiivista menetelmää käytetään aikatauluttamaan käytetyn ydinpolttoaineen loppusijoitusta Suomessa

Integrated Development and Parallelization of Automated Dicentric Chromosome Identification Software to Expedite Biodosimetry Analysis

Manual cytogenetic biodosimetry lacks the ability to handle mass casualty events. We present an automated dicentric chromosome identification (ADCI) software utilizing parallel computing technology. A parallelization strategy combining data and task parallelism, as well as optimization of I/O operations, has been designed, implemented, and incorporated in ADCI. Experiments on an eight-core desktop show that our algorithm can expedite the process of ADCI by at least four folds. Experiments on Symmetric Computing, SHARCNET, Blue Gene/Q multi-processor computers demonstrate the capability of parallelized ADCI to process thousands of samples for cytogenetic biodosimetry in a few hours. This increase in speed underscores the effectiveness of parallelization in accelerating ADCI. Our software will be an important tool to handle the magnitude of mass casualty ionizing radiation events by expediting accurate detection of dicentric chromosomes