Convex cone

The first-order necessary conditions for problems with inequality constraints are called the Kuhn-Tucker conditions (also called Karush-Kuhn-Tucker conditions). The idea of a cone aids the understanding of the Kuhn-Tucker conditions (KTC). A cone is a set of points R such that, if x is in R, Tx is also in R for X 0. A convex cone is a cone that is a convex set. An example of a convex cone in two dimensions is shown in Figure 8.2. In two and three dimensions, the definition of a convex cone coincides with the usual meaning of the word. 

It can be shown from the preceding definitions that the set of all nonnegative linear combinations of a finite set of vectors is a convex cone, that is, that the set... 

Because of the choice of enumeration, the vectors of logarithms of reaction rate constants form a convex cone in which is described by the system of inequalities lnfc2i> lnfc,y, (/,/)t (2,1). For each of the possible auxiliary systems (Figure 4) additional inequalities between constants should be valid, and we get four correspondent cones in These cones form a partitions of the initial one (we neglect intersections of faces which have zero measure). Let us discuss the typical behavior of systems from these cones separately. (Let us remind that if in a cone for some values of coefficients dp then,... 

Illustration 2.1.6 Consider the cases shown in Figure 2.5(a,b). Let us denote the columns of A4 as flj, aa, and a3. System 1 has a solution if the closed convex cone defined by Ax < 0 and the open half-space defined by c x > 0 have a nonempty intersection. System 2 has a solution if c lies within the convex cone generated by alra3, and a3. 

Each compartment is associated with a convex cone QA generated by vectors k3 , ys[Pg.138]

If a flux vector J is thermodynamically feasible, then there exist concentrations Ci that satisfy the above inequality. In fact, Equation (9.27) defines a feasible space for the metabolites concentrations as a convex cone in the log-concentration space. 

It is often possible to show that either the union of these cones or some subset of them form a trapping region. That is, there are cones Ta for the A cycle, 7g for the B cycle, and so on, such that the image of T = 17 7 (where k ranges over the letters of the possible cycles) is in T. We define a (convex) cone ... 

Yu, P. L. (1974), Cone Convexity, Cone Extreme Points and nondominated Solutions in Decision Problems with Multiobjectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 

Silhav, M. On measures, convex cones, and foundations of thermodynamics, I systems with vector valued actions. Czech. J. Phys. B30(8), 841-861 (1980)... 

It can be constructed the convex cone of vectors A Q. in H, but if Mis not abelian, this cone not necessarily has the property of self-duality. Under these conditions, the associated state... 

The set of all metric spaces satisfying the distance limits, being the intersection of the box defined by the limits and the convex cone defined by the triangle inequalities, is itself a convex set. Thus every value of every distance between its lower and upper limits is attained in some metric space consistent with all the limits. 

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