Computing global minima to polynomial optimization problems using Gröbner bases

The local optimality conditions to polynomial optimization problems are a set of polynomial equations (plus some inequality conditions). With the recent techniques of Gröbner bases one can find all solutions to such systems, and hence also find global optima. We give a short survey of these methods. We also apply them to a set of problems termed with exact solutions unknown in the problem sets of Hock and Schittkowski. To these problems we give exact solutions.     

Magnetic field value in the weak energy release region of the solar corona

Spectrographic data from observations of the polarized weak steplike (with relatively sharp growth t 3 sec and slow decay t 10² sec) solar continuum radioemission are analyzed below. The main results are found using a sweeping spectrograph in the 8–12 GHz frequency range (sweep rate 1 sample per second and frequency resolution 120 MHz) and using the RT-22 radio telescope at the Crimean Astrophysical Observatory (angular resolution 4–6). In a series of events at the leading edge of the continuum a narrow-band frequency drifting emission termed blip-type is found. The degree of continuum circular polarization (p<10%) is low compared to that of the narrow-band component (p20%). Both components of the aforementioned events should be radiating on the second harmonic of the local plasma frequency in the context of a plasma generating mechanism. Based on the model proposed and using the observed degree of circular polarization of each component, we can estimate the strength of the magnetic field within the area of the source and the width of the angular spectrum from the radiating plasma turbulence. The results of analysis of the real microwave events are given below.Radiophysical Research Institute, Nizhnii Novgorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 7, pp. 874–882, July, 1994.