An Extension of the Fundamental Theorem of Linear Programming

In 1947 George Dantzig developed the Simplex Algorithm for linear programming, and in doing so became known as The Father of Linear Programming. The invention of the Simplex Algorithm has been called one of the most important discoveries of the 20th century, and linear programming techniques have proven useful in numerous fields of study. As such, topics in linear optimization are taught in a variety of disciplines. The finite convergence of the simplex algorithm hinges on a result stating that every linear program with an optimal solution has a basic optimal solution; a result known as the Fundamental Theorem of Linear Programming. We develop an analog to the fundamental theorem, and the perspective from which we view the problem allows a much greater class of functions. Indeed, not only do we relinquish the assumption of linearity, but we also do not assume the functions under consideration are continuous. Our new result implies the Fundamental Theorem of Linear Programming

Portfolio selection problems in practice: a comparison between linear and quadratic optimization models

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model, based on reformulation as a Standard Quadratic Program and on some recent theoretical results. With this approach we obtain optimal solutions both for some well-known financial data sets used by several other authors, and for some unsolved large size portfolio problems. We also test our method on five new data sets involving real-world capital market indices from major stock markets. Our computational experience shows that, rather unexpectedly, it is easier to solve the quadratic LAM model with our algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of the best commercial codes for mixed integer linear programming (MILP) problems. Finally, on the new data sets we have also compared, using out-of-sample analysis, the performance of the portfolios obtained by the Limited Asset models with the performance provided by the unconstrained models and with that of the official capital market indices

Portfolio selection problems in practice: a comparison between linear and quadratic optimization models

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model, based on reformulation as a Standard Quadratic Program and on some recent theoretical results. With this approach we obtain optimal solutions both for some well-known financial data sets used by several other authors, and for some unsolved large size portfolio problems. We also test our method on five new data sets involving real-world capital market indices from major stock markets. Our computational experience shows that, rather unexpectedly, it is easier to solve the quadratic LAM model with our algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of the best commercial codes for mixed integer linear programming (MILP) problems. Finally, on the new data sets we have also compared, using out-of-sample analysis, the performance of the portfolios obtained by the Limited Asset models with the performance provided by the unconstrained models and with that of the official capital market indices

Fundamentos teórico-metodológicos sobre fluxo de potência ótimo e aplicações com o Power World Simulator

The present study focuses on economic dispatch, optimal power flow, and optimal power flow with security constraints in Power World Simulator, using the Seven Bus Brazilian System Model Equivalent. The objective of this research was to assess the feasibility of applying these techniques in power systems to optimize generation dispatch and ensure operational security. The results demonstrated that the use of optimal power flow with security constraints provided a significant improvement in system performance, including cost reduction and enhancement in energy supply quality. Furthermore, the effectiveness of the employed model for simulating the studied power system was verifiedO trabalho em questão tem como tema o despacho econômico, o fluxo de potência ótimo e o fluxo de potência ótimo com restrições de segurança no Power World Simulator, utilizando o sistema Seven Bus Brazilian System Model Equivalent. O objetivo do estudo foi avaliar a viabilidade de aplicação dessas técnicas em sistemas elétricos de potência, visando otimizar o despacho de geração e garantir a segurança operacional. Os resultados mostraram que o uso do fluxo de potência ótimo com restrições de segurança proporcionou uma melhoria significativa no desempenho do sistema, em termos de redução de custos e melhoria na qualidade do fornecimento de energia. Além disso, foi possível verificar a eficácia do modelo utilizado para simulação do sistema elétrico de potência estudad

Investigación en matemáticas, economía y ciencias sociales

El resultado de este libro que reune inquietudes académicas en torno a temas tan estudiados como los que están alrededor del maíz, del frijol o del café; y tan contemporáneos como las aplicaciones concretas de las ciencias ya citadas, al estudio de la adopción del comercio electrónico en empresas del sector agroindustrial o, el caso de la generación de biogas o energía eléctrica por medio de biodigestores. Al editar este texto e incorporarlo a la bibliografía de los temas de referencia, se enriquecen opciones de consulta para los estudiosos de esos temas en general; pero también para interesados en aspectos tan específicos como la cadena de suministro del mercado hortofrutícola en Texcoco

Connections between continuous and combinatorial optimization problems through an extension of the fundamental theorem of Linear Programming

We describe a common extension of the fundamental theorem of Linear Programming on the existence of a global minimum in a vertex for lower bounded linear programs, and of the Frank-Wolfe theorem on the existence of the minimum of a lower bounded quadratic function on a polyhedron. We then show that several known results providing continuous formulations for discrete optimization problems can be easily derived and generalized with our result. These include the Quadratic Programming formulation of the maximum clique problem by Motzkin and Straus and its weighted extension by Gibbons et al., the equivalence between the minimization of a multilinear function on the continuous and discrete unit hypercube by Rosenberg, and a recent continuous polynomial formulation of the maximum independent set problem by Abello et al. Furthermore, we use our extension of the fundamental theorem of Linear Programming to obtain combinatorial formulations and polynomiality results for some nonlinear problems with simple polyhedral constraints. © 2004