A Bundle Approach for SDPs with Exact Subgraph Constraints

The 'exact subgraph' approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational challenge because of the potentially large number of violated subgraph constraints. We introduce a computational framework for these relaxations designed to cope with these difficulties. We suggest a partial Lagrangian dual, and exploit the fact that its evaluation decomposes into two independent subproblems. This opens the way to use the bundle method from non-smooth optimization to minimize the dual function. Computational experiments on the Max-Cut, stable set and coloring problem show the efficiency of this approach

The merger of populations, the incidence of marriages, and aggregate unhappiness

Let a society’s unhappiness be measured by the aggregate of the levels of relative deprivation of its members. When two societies of equal size, F and M, merge, unhappiness in the merged society is shown to be higher than the sum of the levels of unhappiness in the constituent societies when apart; merger alone increases unhappiness. But when societies F and M merge and marriages are formed such that the number of households in the merged society is equal to the number of individuals in one of the constituent societies, unhappiness in the merged society is shown to be lower than the aggregate unhappiness in the two constituent societies when apart. This result obtains regardless of which individuals from one society form households with which individuals from the other, and even when the marriages have not (or not yet) led to income gains to the married couples from increased efficiency, scale economies, and the like. While there are various psychological reasons for people to become happier when they get married as opposed to staying single, the very formation of households reduces social distress even before any other happiness-generating factors kick in.Merger of populations, Integration of societies, Unhappiness, Marriages, Relative deprivation, Institutional and Behavioral Economics, D0, D10, D31, D63,

Using a Factored Dual in Augmented Lagrangian Methods for Semidefinite Programming

In the context of augmented Lagrangian approaches for solving semidefinite programming problems, we investigate the possibility of eliminating the positive semidefinite constraint on the dual matrix by employing a factorization. Hints on how to deal with the resulting unconstrained maximization of the augmented Lagrangian are given. We further use the approximate maximum of the augmented Lagrangian with the aim of improving the convergence rate of alternating direction augmented Lagrangian frameworks. Numerical results are reported, showing the benefits of the approach.Comment: 7 page

The merger of populations, the incidence of marriages, and aggregate unhappiness

Let a society's unhappiness be measured by the aggregate of the levels of relative deprivation of its members. When two societies of equal size, F and M, merge, unhappiness in the merged society is shown to be higher than the sum of the levels of unhappiness in the constituent societies when apart; merger alone increases unhappiness. But when societies F and M merge and marriages are formed such that the number of households in the merged society is equal to the number of individuals in one of the constituent societies, unhappiness in the merged society is shown to be lower than the aggregate unhappiness in the two constituent societies when apart. This result obtains regardless of which individuals from one society form households with which individuals from the other, and even when the marriages have not (or not yet) led to income gains to the married couples from increased efficiency, scale economies, and the like. While there are various psychological reasons for people to become happier when they get married as opposed to staying single, the very formation of households reduces social distress even before any other happiness-generating factors kick in. --Merger of populations,Integration of societies,Unhappiness,Marriages,Relative Deprivation

Assessing the financial situation of construction company

Bakalářská práce se v první části zaměřuje na teoretické vysvětlení principu finanční analýzy. Dále popisu zdrojů pro finanční analýzu, základních i odborných termínů nutných k pochopení principů, jenž se ve finanční analýze vyskytují. Ve druhé části se teoretické poznatky aplikovaly na konkrétní stavební podnik. Dostupná data byla zpracována dle zmíněných prvků finanční analýzy. Následně bylo vyhodnoceno finanční zdraví podniku a vytvořeny doporučení pro zlepšení finanční situace podniku. Inspiraci z této práce mohou čerpat další stavební podniky o stejné nebo podobné velikosti a technické vybavenosti jako je společnost podrobená analýze, kdy na základě jejího vypracování může též zjistit, v kterém oboru, nebo v které části podnikání má mezery, nebo nedostatky.In the first part the thesis focuses on an theoretical explanation of financial analysis. Further it deals with description of sources for financial analysis and with fundamental and technical terms necessary for understanding the principles, which can be found in the financial analysis. In the second part of the thesis I applied the theoretical knowledge on a particular construction company. Available data were processed according to the aforementioned elements of financial analysis. Subsequently I evaluated a financial health of the company and made several recommendations on how their financial situation could be improved. This work could be beneficial to other construction companies with the same or similar size and technical features as the company that is a subject of the analysis. Based on its elaboration they may also determine in which field or in which part of the business they have gaps or shortcomings

Exact Solution Methods for the kk-item Quadratic Knapsack Problem

The purpose of this paper is to solve the 0-1 $k$-item quadratic knapsack problem $(kQKP)$, a problem of maximizing a quadratic function subject to two linear constraints. We propose an exact method based on semidefinite optimization. The semidefinite relaxation used in our approach includes simple rank one constraints, which can be handled efficiently by interior point methods. Furthermore, we strengthen the relaxation by polyhedral constraints and obtain approximate solutions to this semidefinite problem by applying a bundle method. We review other exact solution methods and compare all these approaches by experimenting with instances of various sizes and densities.Comment: 12 page