Functional, convex

This criterion resumes all the a priori knowledge that we are able to convey concerning the physical aspect of the flawed region. Unfortunately, neither the weak membrane model (U2 (f)) nor the Beta law Ui (f)) energies are convex functions. Consequently, we need to implement a global optimization technique to reach the solution. Simulated annealing (SA) cannot be used here because it leads to a prohibitive cost for calculations [9]. We have adopted a continuation method like the GNC [2]. 

The previous considerations can be specified. Namely, if J is a strictly convex functional then... 

Convex functionals have a convenient description in terms of their derivatives. We briefly discuss this question. 

Here i —> i is a continuous convex function describing the plastic yield condition. The equations (5.7) provide a decomposition of the strain tensor Sij u) into a sum of an elastic part aijuicru and a plastic part ij, and (5.6) are the equilibrium equations. 

One simple but useful way to demonstrate the convexity upward (downward) of a function is to show that it is the sum of convex upward (downward) functions. The proof of this property follows immediately from the definition of convexity. For functions of one variable convexity upward (downward) can also be demonstrated by showing that the second derivative is negative (positive) or zero over the interval of interest. Much of the usefulness of convex functions, for our purposes, stems from the following theorem ... 

All of the interpretations of Theorem 4-11 given in Section 4.7 carry over immediately to the continuous output channel The set of Eqs. (4-104) and (4-105), for finding the optimum input probabilities for a given p become virtually useless, but. is still a convex function of p, sb that p can be optimized numerically. 

The concept of convexity is useful both in the theory and applications of optimization. We first define a convex set, then a convex function, and lastly look at the role played by convexity in optimization. 

Next, let us examine the matter of a convex function. The concept of a convex function is illustrated in Figure 4.10 for a function of one variable. Also shown is a concave function, the negative of a convex function. (If /(x) is convex, -/(x) is concave.) A function /(x) defined on a convex set F is said to be a convex function if the following relation holds... 

Illustration of a convex set formed by a plane /(x) = k cutting a convex function. 

The definitions of convexity and a convex function are not directly useful in establishing whether a region or a function is convex because the relations must be applied to an unbounded set of points. The following is a helpful property arising from the concept of a convex set of points. A set of points x satisfying the relation... 

Kahler structures are easy to construct and flexible. For example, any complex submanifold of a Kahler manifold is again Kahler, and a Kahler metric is locally given by a Kahler potential, i.e. uj = / ddu for a strictly pseudo convex function u. However, hyper-Kahler structures are neither easy to construct nor flexible (even locally). A hypercomplex submanifold of a hyper-Kahler manifold must be totally geodesic, and there is no good notion of hyper-Kahler potential. The following quotient construction, which was introduced by Hitchin et al.[39] as an analogue of Marsden-Weinstein quotients for symplectic manifolds, is one of the most powerful tool for constructing new hyper-Kahler manifolds. 

Beale, E. (1955) On minimizing a convex function subject to linear inequalities. Journal of the Royal Statistical Society, Series B (Methodological), 17, 173. 

All linear functions are simply alike, and all nonlinear functions are different in numerous ways. Perhaps the next level of complication is monotonic increasing functions, where the sign of the first derivative dy/dx is always positive, so the function is always rising in value. An example is y = x" where n > 0. It would be useful here to distinguish between the convex functions, where the second derivative d y/dx is negative (when n < ) and concave functions, where the second derivative is positive (when n > 1). The exponential function y = e" where n > 0 is concave the logarithm function y = log(x) is convex. 

The monotonic decreasing functions have a first derivative that is always negative, so that the function is always decreasing with x. Examples of the monotonic increasing functions that are concave include y = ax", where n is negative, and y = e", where n is negative an example of a monotonic decreasing convex function is y = (a - x) where 0 < n < 1. 

It is advantageous to do mixing when the property is a convex function of the composition, so that the mixture property is higher than the arithmetic average such functions have a second derivative d y/dx < 0, such as the function y =... 

Notice that the expression (8) for the free energy of the step network, irrespectively of the sign of s, is a non-convex function ofp and q. This result implies that the network is always unstable some surface orientations disappear from the equilibrium shape of the crystal and are replaced by sharp edges. 

The central concept of the DA algorithm is based on developing a homotopy from an appropriate convex function to the nonconvex cost function the local minima of cost function at every... 

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