MATRIX REDUCTION IN NUMERICAL OPTIMIZATION

Matrix reduction by eliminating some terms in the expansion of a matrix has been applied to a variety of numerical problems in many different areas. Since matrix reduction has different purposes for particular problems, the reduced matrices also have different meanings. In regression problems in statistics, the reduced parts of the matrix are considered to be noise or observation error, so the given raw data are purified by the matrix reduction. In factor analysis and principal component analysis (PCA), the reduced parts are regarded as idiosyncratic (unsystematic) factors, which are not shared by multiple variables in common. In solving constrained convex optimization problems, the reduced terms correspond to unnecessary (inactive) constraints which do not help in the search for an optimal solution. In using matrix reduction, it is both critical and difficult to determine how and how much we will reduce the matrix. This decision is very important since it determines the quality of the reduced matrix and the final solution. If we reduce too much, fundamental properties will be lost. On the other hand, if we reduce too little, we cannot expect enough benefit from the reduction. It is also a difficult decision because the criteria for the reduction must be based on the particular type of problem. In this study, we investigatematrix reduction for three numerical optimization problems. First, the total least squares problem uses matrix reduction to remove noise in observed data which follow an underlying linear model. We propose a new method to make the matrix reduction successful under relaxed noise assumptions. Second, we apply matrix reduction to the problem of estimating a covariance matrix of stock returns, used in financial portfolio optimization problem. We summarize all the previously proposed estimation methods in a common framework and present a new and effective Tikhonov method. Third, we present a new algorithm to solve semidefinite programming problems, adaptively reducing inactive constraints. In the constraint reduction, the Schur complement matrix for the Newton equations is the object of the matrix reduction. For all three problems, we propose appropriate criteria to determine the intensity of the matrix reduction. In addition, we verify the correctness of our criteria by experimental results and mathematical proof

Thucydides on the Fate of the Democratic Empire

This paper investigates Thucydides instruction on the problematic concept of the democratic empire. Although the term democratic empire, that is, the combination of democracy and empire, is often justified in the modern context, it appeared to the ancient to be very problematic because of its inherent contradiction. The paper examines how Thucydides dealt with the Athenian Empire as an exemplary case of democratic empire. More specifically it examines how Thucydides related the rise and fall of the Athenian Empire to its characteristics of democratic empire. Many scholars attributed the collapse of the Empire to the excessive desire of the demos. I do not deny this traditional reading. Yet I argue that the Athenian demos was fully aware of what it was doing: it preferred democracy to empire when it had to choose either. By reading closely Thucydides I try to show how the Athenian demos constantly maintained democracy even when its preference for democracy could endanger its empire. Based on this reading of Thucydides, I conclude that Thucydides instructs both democratic citizens and imperialist elites that they cannot maintain democratic empire in the long run: they should choose democracy or imperialism at a certain point