Optimum saddle point

Many real problems do not satisfy these convexity assumptions. In chemical engineering applications, equality constraints often consist of input-output relations of process units that are often nonlinear. Convexity of the feasible region can only be guaranteed if these constraints are all linear. Also, it is often difficult to tell if an inequality constraint or objective function is convex or not. Hence it is often uncertain if a point satisfying the KTC is a local or global optimum, or even a saddle point. For problems with a few variables we can sometimes find all KTC solutions analytically and pick the one with the best objective function value. Otherwise, most numerical algorithms terminate when the KTC are satisfied to within some tolerance. The user usually specifies two separate tolerances a feasibility tolerance Sjr and an optimality tolerance s0. A point x is feasible to within if... 

The potential (6.37) corresponds with the previously discussed projection of the three-dimensional PES V(p,p2,p3) onto the proton coordinate plane (pi,p3), shown in Figure 6.20b. As pointed out by Miller [1983], the bifurcation of reaction path and resulting existence of more than one transition state is a rather common event. This implies that at least one transverse vibration, q in the case at hand, turns into a double-well potential. The instanton analysis of the PES (6.37) was carried out by Benderskii et al. [1991b], The existence of the onedimensional optimum trajectory with q = 0, corresponding to the concerted transfer, is evident. On the other hand, it is clear that in the classical regime, T > Tcl (Tc] is the crossover temperature for stepwise transfer), the transition should be stepwise and occur through one of the saddle points. Therefore, there may exist another characteristic temperature, Tc2, above which there exists two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points at T = Tcl. The existence of the second crossover temperature Tc2 for two-proton transfer was noted by Dakhnovskii and Semenov [1989]. 

Stationary points can be a (1) local maximum, (2) local minimum, or (3) saddle point. The existence of a stationary point is a necessary condition for an optimum. 

Martin and Davidson ( 3) have examined the structure of the sodium trimer at the SCF-CI level and find results that are very similar to those which have been obtained for Lis. Nas has an optimum C2V geometry with 82 symmetry, a Ai configuration corresponding to a saddle point lying only 0.6 kcal/mol higher in energy. The linear form lies only 3 kcal/mol above the minimum 82 geometry. The Nas molecule is bound with respect to the dissociation limit Na2 + Na by 8.5 kcal/mol. 

Now we can visualize evolutionary optimization as a hill-climbing process on a landscape that is given by an extremely simple potential [Eqn. (11.15)]. This potential, an ( — 1 )-dimensional hyperplane in n-dimensional space, seems to be a trivial function at first glance. It is linear and hence has no maxima, minima, or saddle points. However, as with every chemical reaction, evolutionary optimization is confined to the cone of nonnegative concentration restricts the physically accessible domain of relative concentrations to the unit simplex (xj > 0, X2 > 0,..., x > 0 Z x = 1). The unit simplex intersects the (n — 1 )-dimensional hyperplane of the potential on a simplex (a three-dimensional example is shown in Figure 4). Selection in the error-free scenario approaches a corner of this simplex, and the stationary state corresponds to a corner equilibrium, as such an optimum on the intersection of a restricted domain with a potential surface is commonly called in theoretical economics. 

Description of the paths is given in ref. 49. Figure 14 illustrates the vibrational character of reactant motion under H transfer. The barrier height is 0.65 eV. Due to intermolecular vibrations the optimum distance between reactants turns out to be close to the saddle point [R R = 0.2-0.3 A) instead of 0.9 A at equilibrium configuration. 

A stationary point does not always correspond to the optimum conditions. It can be a saddle-point (minimax) at which the surface passes through a maximum in certain directions, and through a minimum in other directions. Saddle points are rather conunon. To improve (increase) the response, the directions in which the response surface increases should be explored. 

When the improvement procedure is repeated n times, the local optimum is expected to be found with an acceptable degree of accuracy. The greater the values in x the more inaccurate the result. The lower the values in x the higher the possibilities of being trapped in a saddle point, being affected by rounding errors (note that at every point... 

Examples illustrating what can go wrong if the constraint gradients are dependent at x can be found in Luenberger (1984). It is important to remember that all local maxima and minima of an NLP satisfy the first-order necessary conditions if the constraint gradients at each such optimum are independent. Also, because these conditions are necessary but not, in general, sufficient, a solution of Equations (8.17)-(8.18) need not be a minimum or a maximum at all. It can be a saddle or inflection point. This is exactly what happens in the unconstrained case, where there are no constraint functions hj = 0. Then conditions (8.17)-(8.18) become... 

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