Stationary point constrained

Now consider the imposition of inequality [g(x) < 0] and equality constraints 7i(x) = 0] in Fig. 3-55. Continuing the kinematic interpretation, the inequality constraints g(x) < 0 act as fences in the valley, and equality constraints h(x) = 0 act as "rails. Consider now a ball, constrained on a rail and within fences, to roll to its lowest point. This stationary point occurs when the normal forces exerted by the fences [- Vg(x )] and rails [- V/i(x )] on the ball are balanced by the force of gravity [— Vfix )]. This condition can be stated by the following Karush-Kuhn-Tucker (KKT) necessary conditions for constrained optimality ... 

A similar constrained optimization problem has been solved in Section 2.5.4 by the method of Lagrange multipliers. Using the same method we look for the stationary point of the Lagrange function... 

A key idea in developing necessary and sufficient optimality conditions for nonlinear constrained optimization problems is to transform them into unconstrained problems and apply the optimality conditions discussed in Section 3.1 for the determination of the stationary points of the unconstrained function. One such transformation involves the introduction of an auxiliary function, called the Lagrange function L(x,A, p), defined as... 

Remark 1 The implications of transforming the constrained problem (3.3) into finding the stationary points of the Lagrange function are two-fold (i) the number of variables has increased from n (i.e. the x variables) to n + m + p (i.e. the jc, A and /z variables) and (ii) we need to establish the relation between problem (3.3) and the minimization of the Lagrange function with respect to x for fixed values of the lagrange multipliers. This will be discussed in the duality theory chapter. Note also that we need to identify which of the stationary points of the Lagrange function correspond to the minimum of (3.3). 

Fig. 19 Main plot SAXS intensity (I) vs momentum transfer for a solution of 51 in acetonitrile (5.1 g L 1). The symbols and the solid line correspond to the experimental data points and the numerical fit using GNOM/DAMMIN simulated annealing, constraining the symmetry to the point group P432 (% = 1.397). Inset reconstructed low resolution particle shape for 51 obtained by the GNOM/DAMMIN fit (semitransparent spheres) superimposed onto the PM3 stationary point (space-filling model, iso-butyl groups substituted by methyl groups)...

For non-identity reactions it is often useful to start a search for stationary points by minimizing high symmetry geometries. A subsequent frequency calculation on the symmetry constrained (and minimized) structure will reveal the nature of the stationary... 

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