Local minimum problem

Conventional energy minimization approaches all suffer from the local minimum problem, since all known nonlinear minimization methods are... 

For some cases, it is possible to use refinement techniques (6) to obtain very accurate structural data. This is especially true for the well understood diffraction techniques and, therefore, for characterizations of polycrystalline material. With an increasing level of physical understanding the number of techniques which are suitable for use in refinement is bound to grow. Because of the local minimum problem, the starting model should already be close to reality for single-technique refinements. We expect that the... 

As a newly developed method for chemical data processing, SVM has following obvious advantages in comparison with classical chemometrical methods (1) It can treat both linear and nonlinear data sets so the trouble of underfitting can be depressed or controlled in some problems (2) It is so designed that the overfitting can be depressed or controlled by the capacity control of the indicator functions used so that the prediction results are often more reliable (3) As compared with ANN, SVM has no local minimum problem and the solution is unique. As... 

Local Minimum Point for Unconstrained Problems Consider the following unconstrained optimization problem ... 

Most strategies hmit themselves to finding a local minimum point in the vicinity of the starting point for the search. Such a strategy will find the global optimum only if the problem has a single minimum point or a set of connected minimum points. A convex problem has only a global optimum. 

In the previous section we discussed how a Hopfield net can sometimes converge to a local minimum that docs not correspond to any of the desired stored patterns. The problem is that while the dynamics embodied by equation 10.7 steadily decreases the net s energy (equation 10.9), because of the general bumpiness of the energy landscape (see figure 10.5), whether or not such a steady decrease eventually lands the system at one of the desired minima depends entirely on where the system begins its descent, or on its initial state. There is certainly no general assurance that the system will evolve towards the desired minimum. 

Instead of a formal development of conditions that define a local optimum, we present a more intuitive kinematic illustration. Consider the contour plot of the objective function fix), given in Fig. 3-54, as a smooth valley in space of the variables X and x2. For the contour plot of this unconstrained problem Min/(x), consider a ball rolling in this valley to the lowest point offix), denoted by x. This point is at least a local minimum and is defined by a point with a zero gradient and at least nonnegative curvature in all (nonzero) directions p. We use the first-derivative (gradient) vector Vf(x) and second-derivative (Hessian) matrix V /(x) to state the necessary first- and second-order conditions for unconstrained optimality ... 

Neither of the problems illustrated in Figures 4.5 and 4.6 had more than one optimum. It is easy, however, to construct nonlinear programs in which local optima occur. For example, if the objective function / had two minima and at least one was interior to the feasible region, then the constrained problem would have two local minima. Contours of such a function are shown in Figure 4.7. Note that the minimum at the boundary point x1 = 3, x2 = 2 is the global minimum at / = 3 the feasible local minimum in the interior of the constraints is at / = 4. 

For example, it is usually impossible to prove that a given algorithm will find the global minimum of a nonlinear programming problem unless the problem is convex. For nonconvex problems, however, many such algorithms find at least a local minimum. Convexity thus plays a role much like that of linearity in the study of dynamic systems. For example, many results derived from linear theory are used in the design of nonlinear control systems. 

Let x be a local minimum or maximum for the problem (8.15), and assume that the constraint gradients Vhj(x ),j — 1,m, are linearly independent. Then there exists a vector of Lagrange multipliers A = (Af,..., A ) such that (x A ) satisfies the first-order necessary conditions (8.17)-(8.18). 

The KTC comprise both the necessary and sufficient conditions for optimality for smooth convex problems. In the problem (8.25)-(8.26), if the objective fix) and inequality constraint functions gj are convex, and the equality constraint functions hj are linear, then the feasible region of the problem is convex, and any local minimum is a global minimum. Further, if x is a feasible solution, if all the problem functions have continuous first derivatives at x, and if the gradients of the active constraints at x are independent, then x is optimal if and only if the KTC are satisfied at x. ... 

The Kuhn-Tucker necessary conditions are satisfied at any local minimum or maximum and at saddle points. If (x, A, u ) is a Kuhn-Tucker point for the problem (8.25)-(8.26), and the second-order sufficiency conditions are satisfied at that point, optimality is guaranteed. The second order optimality conditions involve the matrix of second partial derivatives with respect to x (the Hessian matrix of the... 

If the penalty weight w is larger than the maximum of the absolute multiplier values for the problem, then minimizing P(x, w) subject to 1 x < u is equivalent to minimizing/in the original problem. Often, such a threshold is known in advance, say from the solution of a closely related problem. If w is too small, PSLP will usually converge to an infeasible local minimum of P, and w can then be increased. Infeasibility in the original NLP is detected if several increases of w fail to yield a... 

Determine whether the point jc=[0 0 0]risa local minimum of the problem ... 

An indication of the severity of this problem is illustrated by some data for four-configuration wavefunctions of He. Table 1 shows two such wavefunctions and the energy obtained from each. The first of these is near a local minimum the second can be further refined to move toward what we believe to be the global minimum, ultimately yielding a wavefunction of energy —2.903688260 hartree, and with the parameters shown in Table 5. 

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