Polynomial Optimization Methods

This thesis is an exposition of ideas and methods that help un- derstanding the problem of minimizing a polynomial over a basic closed semi-algebraic set. After the introduction of some the- ory on mathematical tools such as sums of squares, nonnegative polynomials and moment matrices, several Positivstellensa ̈tze are considered. Positivstellens ̈atze provide sums of squares represen- tations of polynomials, positive on basic closed semi-algebraic sets. Subsequently, semi-definite programming methods, in par- ticular based on Putinar’s Postivstellensatz, are considered. In order to use semi-definite programming, certain degree bounds are set. These bounds give rise to a hierarchy of approximations of the minimum of a polynomial, which will also be discussed. Finally, some new results are given that are obtained by looking at sums of squares representations of a positive polynomial when minimizing over the unit hypercube

Macroeconomics without laws : methodological and theoretical aspects

This study develops an economic methodology in which,behavioural laws (in the sense of necessary connections between cause and effect) play no essential role. Hayek and Menger are important sources of inspiration. Economic behaviour is explained by way of tendencies rather than laws and insight into economic phenomena is gained by laying bare their "action structure" in which behavioural explanation and behavioural laws play no role. This methodology is applied to the explanation of macroeconomic coordination. The appropriate equilibrium conditions are developed and the relevant tendencies away from or towards equilibrium are identified. The institutions responsible for these tendencies are identified and anarysed. In the light of these findings, pre-Keynesian macroeconomics, the macroeoconomics of Walrasian theory, as well as Keynes's General Theory itself are critically assessed.Economics and Management SciencesD. Comm. (Economics