Generalizations of convex and concave functions

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

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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. 

Section 2.3 focuses on the generalizations of convex and concave functions and treats the quasi-convex, quasi-concave, pseudo-convex and pseudo-concave functions, and their properties. Further reading in this subject is the excellent book of Avriel et al. (1988). 

Part 1, comprised of three chapters, focuses on the fundamentals of convex analysis and nonlinear optimization. Chapter 2 discusses the key elements of convex analysis (i.e., convex sets, convex and concave functions, and generalizations of convex and concave functions), which are very important in the study of nonlinear optimization problems. Chapter 3 presents the first and second order optimality conditions for unconstrained and constrained nonlinear optimization. Chapter 4 introduces the basics of duality theory (i.e., the primal problem, the perturbation function, and the dual problem) and presents the weak and strong duality theorem along with the duality gap. Part 1 outlines the basic notions of nonlinear optimization and prepares the reader for Part 2. 

To. .generalize the results of Examples 8.2 and 83, we define convex and concave functions. As presented in the introductory calculus course, the simplest version pertains to functions having at least two derivatives. In that case, a function is convex (concave upward) if its second derivative is everywhere greater than or equal to zero. A function is concave (concave downward) if its second derivative is everywhere less than or equal to zero, as shown in Figure 8.24... 

This simple linear variation of the vapom pressure is found very rarely in practice. In general the components influence one another in such a manner that the physical properties of the solution are no longer additive functions of the properties of the pure components. The vapour pressure curve (at constant temperature) may then be either concave or convex to the axis of abscissae, and may even have maxima or minima. 

Nonlinear costs could arise when the production cost function exhibits economies of scale and is concave. Another example of nonlinear flow costs is in the area of transportation and distribution planning. Models for analyzing the traffic congestion on a road network usually have arc travel costs that are convex functions of the arc flow in order to represent congestion effects on the network. These models also generally have multiple commodities to represent the various origin-destination characteristics of the network users. 

In general, a locally convex domain D2(b),j(a> Fj) of a functional group F, relative to a reference curvature b, shows local shape complementarity with a locally concave domain Do( b),j(a, F2) of a complementing functional group F2, relative to a reference curvature of -b. The threshold values a and a are also likely to complement each other the shape complementarity between the higher electron density contours of one functional group and the lower electron density contours of the other functional group is relevant. 

Figure 5 illustrates more generally various cases that can occur for simple quadratic functions of form q x) — JxTHx, for n = 2, where H is a constant matrix. The contour plots display different characteristics when H is (a) positive-definite (elliptical contours with lowest function value at the center) and q is said to be a convex quadratic, (b) positive-semidefinite, (c) indefinite, or (d) negative-definite (elliptical contours with highest function value at the center), and q is a concave quadratic. For this figure, the following matrices are used for those different functions ... 

Figure 1.42. Schlegel diagram and individual hemispheres of a hexakis-adduct of Cgo with a pseudooctahedral general functionalization pattern which is noninherently chiral due to two different sets of three addends arranged in a relative e,e,e-fashion. It should be noted that the front hemisphere is viewed from the convex side (extra-cage view) and the rear hemisphere from the concave side (intra-cage view).

Plots of Z vs. t for various soil reactions and other solid-fluid processt generally behave in a more complex fashion. They are S-shaped convex i small t, concave at large t, and linear at some intermediate range of t. Fc example, the plot of Z against t for the sorption of phosphate by a Typ Dystrochrept soil depicted in Fig. 1-6 illustrates this S-shaped behavior. Th property of the Z (Z) plot suggests that the kinetics of soil reactions ofte obey some complex function that can be approximated by Eq. [12] at sma t, by Eq. [13] at an intermediate t, and by Eq. [14] at large t. 

Due to the pyramidalization of the C-atoms and the rigid cage structure of Cgo the outer convex surface is very reactive towards addition reactions but at the same time the inner concave surface is inert (chemical Faraday cage). This allows the encapsulation, the observation, and the tuning of the wave function of extremely reactive species. This pronounced shape dependence of reactivity which was revealed for the first time with Cgg as ambido-shaped system is predicted to be a general topicity principle. 

Since the conditions in general are difficult to verify, Thowsen discusses special cases under which the (y,p) policy is optimal for the problem considered. For instance, under a linear expected demand curve, linear stockout costs, convex holding costs, and a demand distribution that is a PF2 distribution, the iViP) policy is optimal. Furthermore, if excess demand is backlogged, the demand curve is concave and the revenue is collected a fixed number of periods after the time orders are placed, then no assumptions are needed on the cost and demand distribution for optimality of the critical number policy, and for this case the decision on price and quantity decisions can be made separately. Thowsen also shows that if negative demand is disallowed, the optimal price will be a decreasing function of increasing initial inventory. 

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