Generalized convexity

The first iteration in a CG method is the same as in SD, with a step along the current negative gradient vector. Successive directions are constructed differently so that they form a set of mutually conjugate vectors with respect to the (positive-definite) Hessian A of a general convex quadratic function. 

A more general family of methods for absolute shape domain subdivision of molecular surfaces with reference to regular standard objects, such as plane, spheres, and ellipsoids, can be described within the common framework of generalized convexity [199]. These techniques are applicable for smooth (differentiable) molecular surfaces. 

The average dimension of the interparticle pores in a packed bed is of the order of a fraction of the particle diameter. As the particles are generally convex and the packed bed is not consolidated (f.e., the particles are not fused but remain independent), the structure of the extrapartide space is relatively simple, the porosity distribution is rather narrow, and the channel anastomosis that is illustrated in Figure 5.3 does not leave any significant part of the bed isolated over more than a few particle diameters [75]. 

In general, the task of proving that an arbitrary subset of is convex can be quite difficult. However, the feasible region of problem (P) will be a convex set if each function g,(x) is a convex function and each function hj x) is a linear function. Actually, these conditions on the functions g ix), hj(X) can be relaxed somewhat using the concepts of generalized convexity. A nice summary of generalized convexity is provided by Avriel (1976). 

Anticline A fold of layered, sedimentary rocks whose core contains strati-graphically older rocks the shape of the fold is generally convex upward. 

Note that as reported in the literature ([20], [24]), the utilization of the linear-cuts results in faster convergence since the maximum separation is always greater for the general convex function L(x). Decreasing the tolerance to lOE-6 results in the same solutions although it requires a larger number of iterations, 923 compared to 834, and additional computational time (a total of 35.1 CPU sec). 

The end portion is then contoured into a generally convex curvature which... 

Unique chemistry is associated with the cyclopentenone all five carbon atoms can be functionalized, and the endo-methyl groups of the acetonide assure clean stereoselective addition of the alkenylcopper reagent from the convex side. The use of the acetonide group to control enolate regioselectivity and to mask alcohols should be generally applicable. 

This assumption does not restrict the generality. We introduce the closed and convex set... 

McCabe-Thiele diagrams for nonlinear and more practical systems with pertinent inequaUty constraints are illustrated in Figures 11 and 12. The convex isotherms are generally observed for 2eohtic adsorbents, particularly in hydrocarbon separation systems, whereas the concave isotherms are observed for ion-exchange resins used in sugar separations. 

Le Chatelier (1888) has discussed the general form of the solubility curve in the light of equation (5). If dA/dT is negative (which is usually the case) the curve begins asymptotically to the T axis, and is convex to it. It then passes through a point of inflexion, and is concave up to the maximum where A = 0, df/dT = 0. If A then becomes negative, the solubility... 

A second approach considers that the regions of equivalent parameter values must enclose parameters for which the loss function is nearly the same or at any rate less different than some threshold. In other words, the equivalence regions should take the form 015(0) < c 5(6) for some appropriate constant of. Note that in this case the shape of the regions would not necessarily be ellipsoidal, or even convex In fact, we might postulate in general the existence of multiple minima surrounded by disjoint equivalence neigh-... 

It can be shown that this can be generalized to the case of more than two variables. The standard solution of a linear programming problem is then to define the comer points of the convex set and to select the one that yields the best value for the objective function. This is called the Simplex method. 

Some of the common terms used for defining or describing particle shapes in a qualitative way are presented in Table 2.1. It is by now quite clear that particle shape cannot be very precisely defined. However, it is fortunate that mineral particles occur in a variety of generally simple shapes some are acicular, several are plate-like, most are convex, others are mildly concave, and in this manner a phraseological descriptive list is built for the different shapes that are formed or produced or generally encountered of powders of fragments of... 

Completely closed, convex, single-shell clusters are called closo clusters their atoms form a polyhedron. If the polyhedron has only triangular faces, it is also called a delta-hedron. Depending on the number of available electrons, we can distinguish four general bonding types for closo clusters ... 

However, bubble nonhomogeneous distribution exists in two-phase shear flow. As yet, the following general trends in void fraction radial profiles are being identified for bubbly upward flow (Zun, 1990) concave profiles (Serizawa et al., 1975) convex profiles (Sekoguchi et al., 1981), and intermediate profiles (Sekoguchi et al., 1981 Zun, 1988). Two theories are currently dominant ... 

As shown in Fig. 3-53, optimization problems that arise in chemical engineering can be classified in terms of continuous and discrete variables. For the former, nonlinear programming (NLP) problems form the most general case, and widely applied specializations include linear programming (LP) and quadratic programming (QP). An important distinction for NLP is whether the optimization problem is convex or nonconvex. The latter NLP problem may have multiple local optima, and an important question is whether a global solution is required for the NLP. Another important distinction is whether the problem is assumed to be differentiable or not. 

When the second stage decisions are real-valued variables, the value function Qu(x) is piecewise-linear and convex in x. However, when some of the second stage variables are integer-valued, the convexity property is lost. The value function Qafx) is in general non-convex and non-differentiable in x. The latter property prohibits the use of gradient-based search methods for solving (MASTER). 

In general, the first derivative of the Gibbs energy is sufficient to determine the conditions of equilibrium. To examine the stability of a chemical equilibrium, such as the one described above, higher order derivatives of G are needed. We will see in the following that the Gibbs energy versus the potential variable must be upwards convex for a stable equilibrium. Unstable equilibria, on the other hand, are... 

In summary, the optimum of a nonlinear programming problem is, in general, not at an extreme point of the feasible region and may not even be on the boundary. Also, the problem may have local optima distinct from the global optimum. These properties are direct consequences of nonlinearity. A class of nonlinear problems can be defined, however, that are guaranteed to be free of distinct local optima. They are called convex programming problems and are considered in the following section. 

---