Concave function

The same relations (11) and (12) hold for the Gibbs free energy in the (N, p,T) ensemble. Equation (11) is also valid for a quanmm mechanical system. Note that for a linear coupling scheme such as Eq. (10), the first term on the right of Eq. (12) is zero the matrix of second derivatives can then be shown to be definite negative, so that the free energy is a concave function of the Xi. 

Minimum number of mass exchanger units. Combinatorics determines the minimum number of mass exchanger units required in the network. This objective attempts to minimize indirectly the fixed cost of the network, since the cost of each mass exchanger is usually a concave function of the unit size. FuithetTnore, in a practical context it is desirable to minimize the number of separators so as to reduce pipework, foundations, maintenance, and instrumentation. Normally, the minimum number of units is related to the total number of streams by the following expression (El-Halwagi and Manousiouthakis, 1989). 

If f(x) and g(x) are concave functions, then a solution to the above maximum problem is equivalent to the solution of the saddle value problem that requires finding x° and A°, which satisfy... 

In this case,. 4(A) is a concave function which can be very useful to check the accuracy of the calculation. This is proved by computing the second derivative of... 

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... 

That the enclosed region is convex can be demonstrated by showing that both gj(x) and g2(x) are concave functions ... 

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. 

This is a case of Jensen s inequality, which states that, for a convex function, a chord between any two points is always below the curve that is, the property of the average is always higher than the average of the two properties. On the other hand, it is not advantageous to do mixing when the property is a concave function of the composition, so that the mixture property is less than the arithmetic average. 

Based on the X-ray data (see above), the accessibility of the concave functional group was studied. By computer modelling (Connolly-routine [34]), spheres of varying sizes were rolled over the van der Waals-surfaces of the concave reagents which were calculated from the X-ray data, and the resulting contact surface was monitored. 

This chapter discusses the elements of convex analysis which are very important in the study of optimization problems. In section 2.1 the fundamentals of convex sets are discussed. In section 2.2 the subject of convex and concave functions is presented, while in section 2.3 generalizations of convex and concave functions are outlined. 

This section presents (i) the definitions and properties of convex and concave functions, (ii) the definitions of continuity, semicontinuity and subgradients, (iii) the definitions and properties of differentiable convex and concave functions, and (iv) the definitions and properties of local and global extremum points. 

Remark 2 The function /(x) is concave on S if and only if -/(x) is convex on S. Then, the results obtained for convex functions can be modified into results for concave functions by multiplication by -1 and vice versa. 

Illustration 2.2.2 Figure 2.8 shows the epigraph and hypograph of a convex and concave function. Theorem 2.2.1... 

Remark 1 The epigraph of a convex function and the hypograph of a concave function are convex sets. 

Remark 3 Convex and concave functions may not be continuous everywhere but the points of discontinuity have to be on the boundary of S. 

Remark 5 The above two theorems can be directly extended to strictly convex and strictly concave functions by replacing the inequalities > and < with strict inequalities > and <. 

Illustration 2.2.5 Figure 2.11 shows a differentiable convex and concave function, as well as their linearizations around a point x1. Note that the linearization always underestimates the convex function and always overestimates the concave function. 

This section presents the definitions, properties and relationships of quasi-convex, quasi-concave, pseudo-convex and pseudo-concave functions. 

Definition 2.3.2 (Quasi-concave function) /(jc) is quasi-concave if... 

Illustration 2.3.1 Figure 2.12 shows a quasi-convex and quasi-concave function. 

Similarly a concave function is also quasi-concave since... 

Figure 2.12 Quasi-convex and quasi-concave functions...

Definition 2.3.4 (Strictly quasi-concave function) f(x) is strictly quasi-concave if... 

Illustration 2.3.2 Figure 2.13 shows a strictly quasi-convex and strictly quasi-concave function. Theorem 2.3.2... 

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