[pt] O Partial Digest Problem (problema de digestão parcial), também conhecido como o Turnpike Problem, consiste na construção de um conjunto de pontos na reta real dadas as distâncias não designadas entre todos os pares de pontos. Uma variante deste problema, chamada Min Distance Superset Problem (problema de superset de distância mínimo), lida com entradas incompletas em que algumas distâncias podem estar faltando. O objetivo deste problema é encontrar um conjunto mínimo de pontos na reta real, tal que as distâncias entre cada par de pontos contenham todas as distâncias de entrada. As principais contribuições deste trabalho são duas formulações de programação matemática diferentes para o Min Distance Superset Problem: uma formulação de programação quadrática e uma formulação de programação inteira. Mostramos como aplicar um método de cálculo direto de limites de valores de variáveis através de uma relaxação Lagrangeana da formulação quadrática. Também introduzimos duas abordagens diferentes para resolver a formulação inteira, ambas baseadas em buscas binárias na cardinalidade de uma solução ótima. A primeira baseia-se num subconjunto de variáveis de decisão, na tentativa de lidar com um problema de viabilidade mais simples, e o segundo é baseado na distribuição de distâncias entre possíveis pontos disponíveis. / [en] The Partial Digest Problem, also known as the Turnpike Problem, consists of building a set of points on the real line given their unlabeled pairwise distances. A variant of this problem, named Min Distance Superset Problem, deals with incomplete input in which distances may be missing. The goal is to find a minimal set of points on the real line such that the multiset of their pairwise distances is a superset of the input. The main contributions of this work are two different mathematical programming formulations for the Min Distance Superset Problem: a quadratic programming formulation and an integer programming formulation.We show how to apply direct computation methods for variable bounds on top of a Lagrangian relaxation of the quadratic formulation. We also introduce two approaches to solve the integer programming formulation, both based on binary searches on the cardinality of an optimal solution. One is based on a subset of decision variables, in an attempt to deal with a simpler feasibility problem, and the other is based on distributing available distances between possible points.

Massive proliferation of wireless devices throughout world in the past decade comes with a host of tough and demanding design problems. Noise at receivers and wireless interference are the two major issues which severely limits the received signal quality and the quantity of users that can be simultaneously served. Traditional approaches to this problems are known as Power Control (PC), SINR Balancing (SINRB) and User Selection (US) in Wireless Networks respectively. Interestingly, for a large class of wireless system models, both this problems have a generic form. Thus any approach to this generic optimization problem benefits the transceiver design of all the underlying wireless models. In this thesis, we propose an Eigen approach based on the Joint Numerical Range (JNR) of hermitian matrices for PC, SINRB and US problems for a class of wireless models. In the beginning of the thesis, we address the PC and SINRB problems. PC problems can be expressed as Homogeneous Quadratic Constrained Quadratic Optimization Problems (HQCQP) which are known to be NP-Hard in general. Leveraging their connection to JNR, we show that when the constraints are fewer, HQCQP problems admit iterative schemes which are considerably fast compared to the state of the art and have guarantees of global convergence. In the general case for any number of constraints, we show that the true solution can be bounded above and below by two convex optimization problems. Our numerical simulations suggested that the bounds are tight in almost all scenarios suggesting the achievement of true solution. Further, the SINRB problems are shown to be intimately related to PC problems, and thus share the same approach. We then proceed on to comment on the convexity of PC problems and SINRB problems in the general case of any number of constraints. We show that they are intimately related to the convexity of joint numerical range. Based on this connection, we derive results on the attainability of solution and comment on the same about the state-of-the-art technique Semi-De nite Relaxation (SDR). In the subsequent part of the thesis, we address the US problem. We show that the US problem can be formulated as a combinatorial problem of selecting a feasible subset of quadratic constraints. We propose two approaches to the US problem. The first approach is based on the JNR view point which allows us to propose a heuristic approach. The heuristic approach is then shown to be equivalent to a convex optimization problem. In the second approach, we show that the US is equivalent to another non-convex optimization problem. We then propose a convex approximation approach to the latter. Both the approaches are shown to have near optimal performance in simulations. We conclude the thesis with a discussion on applicability and extension to other class of optimization problems and some open problems which has come out of this work.

Confidentiel / Confidential

This thesis evaluates the mean-variance analysis framework by comparing the performance of an optimized portfolio consisting of stocks from the Dow Jones Industrial Average to the performance of the Dow Jones Industrial Average index itself. The results show that the optimized portfolio performs better than the corresponding index when evaluated on the period between 2015 and 2019. However, the variance of the returns are high and therefore it is difficult to determine if mean-variance analysis performs better than its corresponding index in the general case. Furthermore, it is shown that individual stocks can still influence the movement of an optimized portfolio significantly, even though the model is supposed to diversify firm-specific risk. Thus, the authors recommend modifying the model by restricting the amount that is allowed to be invested in a single stock, if one wishes to apply mean-variance analysis in reality. To be able to draw further conclusions, more practical research within the subject needs to be done. / Denna uppsats utvärderar ramverket ”mean-variance analysis” genom att jämföra prestandan av en optimerad portfölj bestående av aktier från Dow Jones Industrial Average med prestandan av indexet Dow Jones Industrial Average självt. Resultaten visar att att den optimerade portföljen presterar bättre än motsvarande index när de utvärderas på perioden 2015 till 2019. Dock är variansen av avkastningen hög och det är därför svårt att bedöma om mean-variance analysis generellt sett presterar bättre än sitt motsvarande index. Vidare visas det att individuella aktier fortfarande kan påverka den optimerade portföljens rörelser, fastän modellen antas diversifiera företagsspecifik risk. På grund av detta rekommenderar författarna att modifiera modellen genom att begränsa mängden som kan investeras i en individuell aktie, om man önskar att tillämpa mean-variance analysis i verkligheten. För att kunna dra vidare slutsatser så krävs mer praktisk forskning inom området.

This project aims to analyze Nordic companies active in the sector of Information and Communications Technology (ICT), and does this in two parts. Part I entails analyzing public companies to construct a valuation model aimed at predicting the enterprise value of private companies. Part II deals with analyzing private companies to determine if there are opportunities providing excess returns as compared to investments in public companies. In part I, a multiple regression approach is utilized to identify suitable valuation models. In doing so, it is revealed that 1-factor models provide best statistical results in terms of significance and prediction error. In descending order, in terms of prediction accuracy, these are (1) total assets, (2) turnover, (3) EBITDA, and (4) cash flow. Part II uses model (1) and finds that Nordic ICT private equity does provide opportunities for positive alphas, and that it is possible to construct portfolio strategies that increase this alpha. However, with regards to previous research, it seems as though the returns offered by the private equity market analyzed does not adequately compensate investors for the additional risks related to investing in private equity. / Det här projektet analyserar nordiska bolag aktiva inom Informations- och Kommunikationsteknologi (ICT) i två delar. Del I behandlar analys av publika bolag för att konstruera en värderingsmodell avsedd att förutsäga privata bolags enterprise value. Del II analyserar privata bolag för att undersöka huruvida det finns möjligheter att uppnå överavkastning jämfört med investeringar i publika bolag. I del I utnyttjas multipel regressionsanalys för att identifiera tillämpliga värderingsmodeller. I den processen påvisas att modeller med enbart en faktor ger bäst statistiska resultat i fråga om signifikans och förutsägelsefel. I fallande ordning, med avseende på precision i förutsägelser, är dessa modeller (1) totala tillgångar, (2) omsättning, (3) EBITDA, och (4) kassaflöde. Del II använder modell (1) och finner att den nordiska marknaden för privata ICT-bolag erbjuder möjligheter för överavkastning jämfört med motsvarande publika marknad, samt att det är möjligt att konstruera portföljstrategier som ökar avkastningen ytterligare. Dock, med hänsyn till tidigare forskning, verkar det som att de möjligheter för avkastning som går att finna på marknaden av privata bolag som undersökts inte kompenserar investerare tillräckligt för de ytterligare risker som är relaterade till investeringar i privata bolag.