Constrained Optimization Problems in Network Models

Operations Research is the field of mathematics that deals with solving various application problems. Constrained optimization problems are one of the most important and useful fields of mathematics, particularly in Operations Research. In this thesis, we focus our attention on some mathematical models that are decision problems and which are all based on networks and applied to different real situations. We analyze different thematic areas such as Cloud Computing, Financial Market, Business Management and Cybersecurity and for each of them we formulate the associated linear or nonlinear constrained problems which allows us to solve the decision problems related to the specific applications. The purpose of one of our mathematical models, in this thesis, is to represent a cloud environment. This mathematical model could allows us to identify a rational strategy for reaching a final goal, which is to maximize the Iaas provider's profit. We get a mixed-Integer nonlinear programming problem, which can be solved through the proposed computational algorithm. A second step is the linearization of the problem. The effectiveness of the model and of the algorithm is tested, by comparing the final data with the results obtained by solving the linearized problem through an existing software. Another topic we have dealt with in depth in this thesis is the financial market. We studied some optimization models based on networks which allow us to formulate two new multi-period portfolio selection problems as Markowitz mean-variance optimization problems with intermediaries, and therefore with transaction costs, the addition of capital gains tax, but also with short selling and transfer of securities. We proposed two constrained Integer nonlinear programming problems with which it is possible to establish if and when it is suitable to buy and to sell financial securities, not only while maximizing the profits, but also while minimizing the risk (through the use of a weight). We applied the Lagrange theory and analyzed the variational inequality studying an optimization model for business management and cybersecurity investments.