Global extremum

This section presents (i) the definitions and properties of convex and concave functions, (ii) the definitions of continuity, semicontinuity and subgradients, (iii) the definitions and properties of differentiable convex and concave functions, and (iv) the definitions and properties of local and global extremum points. 

In Figure 17.12 we see four NMF thickness maps and spectra found in a highly weathered soil from a forest site in Kenya. While especially the regions found in the first shown component seem to be of special interest, the spectra need to be evaluated with caution. It is often unclear if the NMF algorithm found the global extremum of the cost function or did not move beyond a local extremum. The spectra found may not be close to reality. While the results can be better in many other cases, it is always worthwhile to compare them with the results of other methods or verify against additional information one might have. 

Mathematical optimization is often used in various fields of applied science. Despite the significant progress made over a century of research, there is still no universal way to treat all variety of problems one can encounter. One faces a typical problem of finding global extremum... 

Taking into account all assumptions mentioned above, we ve developed an algorithm, which was proven to give global extremums. First, the whole domain is divided into smaller... 

The extremal scales for the quantity —dnS (a.T) correspond to the extremal subtotal wavelet contribution (summing over all translation index values) contributing to the wavefunction, at point b. The global extremum for this set is defined by a, and satisfies dlS au,T) = 0. 

Irreversible processes may promote disorder at near equilibrium, and promote order at far from equilibrium known as the nonlinear region. For systems at far from global equilibrium, flows are no longer linear functions of the forces, and there are no general extremum principles to predict the final state. Chemical reactions may reach the nonlinear region easily, since the affinities of such systems are in the range of 10-100 kJ/mol. However, transport processes mainly take place in the linear region of the thermodynamic branch. 

However, a local extremum does not necessarily occur at every point that satisfies (11) that is, (11) is not a sufficient condition for optimality. In practice, necessary conditions are used to identify stationary points, which are candidate extrema, whereas sufficient conditions are used to classify the stationary points as local maxima, local minima, or saddle points (inflection points in Ej). Once all local extrema are found, the global extrema can be found by selecting the absolute maximum or minimum. The necessary and sufficient conditions for determining and classifying the stationary points of a function of one variable are summarized in Table 1. These conditions are easily derived using a Taylor series expansion. 

A stationary point for a general Lagrangian function may or may not be a loctil extremum. If, as described in Section 2, suitable convexity conditions hold, then the method of Lagrange multipliers will yield a global minimum. 

Due to the Lagrangian of the functional (99) is the sum of the dissipation potentials, which is equal to the entropy production in case of every real steady-state physical processes, this extremum theorem involves the minimum principle of global entropy production (MPGEP). The physical meaning of MPGEP needs a clarification. Consider the variations of the fluxes and of the intensive parameters as fluctuations of the system around their stationer state values. When these fluctuations are small, the fluctuation of the global entropy production of the system is equal to its first approximations and it has a form... 

The function provides necessary and sufficient conditions of global stability it has an extremum at each stationary state... 

Thermodynamically, only global minima in the free-energy landscape represent stable equilibrium states. The extremum condition (dF/dE)E=E n — yields... 

Several methods have been used to locate a global energy minimum in the 5-dimensional subspace. Quadratic interpolation and steepest descent were used to locate approximate minima. A hybrid first- and second-derivative method as adapted by Zemer was used to refine these estimates.In this approach the Hessian matrix is generated by successive approximation. An extremum was adjudged to have been located when the sum of the squares of the components of the gradient divided by the number of components was less than lO"" a.u. 

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