Convex region

A convex hull is a molecular surface that is determined by running a planar probe over a molecule. This gives the smallest convex region containing the molecule. It also serves as the maximum volume a molecule can be expected to reach. 

We have seen that the output neuron in a binary-threshold perceptron without hidden layers can only specify on which side of a particular hyperplane the input lies. Its decision region consists simply of a half-plane bounded by a hyperplane. If one hidden layer is added, however, the neurons in the hidden layer effectively take an intersection (i.e. a Boolean AND operation) of the half-planes formed by the input neurons and can thus form arbitrary (possible unbounded) convex regions. ... 

The number of sides of the convex regions is equal to the miinber of half-planes whose intersection formed the decision region, and is thus bounded by the number of input neurons. 

Theorem 4-6. Let /(p) be a convex upward (downward) function over a convex region of space, R, and let/(p) have a stationary point at qx with respect to variations in R. (That is, for any p in R,... 

Note that the theorem does not rule out the possibility of a maximum persisting over a convex region on which the function is constant. 

It should be noted that it is not sufficient to simply have a convex region in order to ensure that a search can locate the global optimum. The objective function must also be convex if it is to be minimized or concave if it is to be maximized. 

Does the following set of constraints that form a closed region form a convex region ... 

Solution. See Figure E4.9 for the region delineated by the inequality constraints. By visual inspection, the region is convex. This set of linear inequality constraints forms a convex region because all the constraints are concave. In this case the convex region is closed. 

For well-posed quadratic objective functions the contours always form a convex region for more general nonlinear functions, they do not (see tlje next section for an example). It is helpful to construct contour plots to assist in analyzing the performance of multivariable optimization techniques when applied to problems of two or three dimensions. Most computer libraries have contour plotting routines to generate the desired figures. 

Does the constraint set form a convex region Is it closed (Hint A plot will help you decide.)... 

In problems in which there are n variables and m equality constraints, we could attempt to eliminate m variables by direct substitution. If all equality constraints can be removed, and there are no inequality constraints, the objective function can then be differentiated with respect to each of the remaining (n — m) variables and the derivatives set equal to zero. Alternatively, a computer code for unconstrained optimization can be employed to obtain x. If the objective function is convex (as in the preceding example) and the constraints form a convex region, then any stationary point is a global minimum. Unfortunately, very few problems in practice assume this simple form or even permit the elimination of all equality constraints. 

Figure 7 Application of the dimer method to a two-dimensional test problem. Three different starting points are generated in the reactant region by taking extrema along a high temperature dynamical trajectory. From each one of these, the dimer isjirst translated only in the direction of the lowest mode, but once the dimer is out of the convex region a full optimization of the effective force is carried oat at each step (thus the kink in two of the paths). Each one of the three starting p>oints leads to a different saddle point in this case.

Linear, No-Threshold Model. This simplest model is based on the assumption that risk is directly proportional to the dose P(d) = ad. When it is assumed that the true dose-response curve is convex, linear extrapolation in the low-dose region may overestimate the true risk. However, it is not known if the experimental dose is in the convex region of the curve. 

Deep within the crystal, fiy = 0 and fiA = fi°A, and therefore < AL = n°A. The diffusion potential at the convex region of the surface is greater than that at the concave region, and atoms therefore diffuse to smooth the surface as indicated in Fig. 3.7. 

The feasible region defined by constraints (a)-(c) is the convex region below the dashed line in Fig. 9. All other feasibility constraints for the HEN (for the heating only case) lie outside this region. 

These constraints are assumed to define nonempty (convex) regions containing the point (0, 0). 

The kinetics at longer times were quite different. As shown in Figure 6-10, the initial period of gelation was characterized by concave cure curves, after which they became convex and then almost linear. At 5°C, the convex region was not perceptible. This different behavior, and the effect of temperature on the rate of aging of the HM pectin gels, could be clearly illustrated when the SDRs were analyzed at different aging temperatures. 

Property 1. If the projected (onto R ) PFR trajectories are such that X, i G 7i, 73 encloses a convex region with respect to any element of Xa - a E. I and no two PFR projections meet within the bounds of possible concentrations, then none of these projected (two dimensional) spaces can be extended under the operations of mixing, additional PFRs/CSTRs/RRs, or any combination of these. This convex region implies that the projected curve is a concave function, d%ldXj < 0. 

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