Convex functions definition

One simple but useful way to demonstrate the convexity upward (downward) of a function is to show that it is the sum of convex upward (downward) functions. The proof of this property follows immediately from the definition of convexity. For functions of one variable convexity upward (downward) can also be demonstrated by showing that the second derivative is negative (positive) or zero over the interval of interest. Much of the usefulness of convex functions, for our purposes, stems from the following theorem ... 

The definitions of convexity and a convex function are not directly useful in establishing whether a region or a function is convex because the relations must be applied to an unbounded set of points. The following is a helpful property arising from the concept of a convex set of points. A set of points x satisfying the relation... 

Definition 2.3.1 (Quasi-convex function) f(x) is quasi-convex if... 

Definition 2.3.5 (Pseudo-convex function) /(x) is pseudo-convex if for every xi,x2 S,... 

The condition for stability of the ground state is that E[p] be a convex functional and therefore that rf r, r ), twice its Hessian, be positive-definite. r/ (r, r ) thus possesses a minimum eigenvalue and s (r, r ) a maximum, sj. A perturbation theoretic analysis sketched briefly in [3] then establishes that the upper bound of the spectrum of t]x(r, r ) is oo, and therefore the lower bound of the s is zero. 

Now 8G/8n is a continuous linear functional which will be identified with a tangent. This is the basis of the following definition. If for a convex functional G there is... 

The reader may seem surprised with the appearance of +oo in this definition. However, infinite values are well defined in the theory of convex functionals [11] and they are usually introduced to deal in a simple way with domain questions. With this definition the functional J] is a convex lower semicontinuous functional on the whole space L1 fl L3. We are now ready to introduce the following key theorem which we will use to prove differentiability of FL at the set of E-V-densities ... 

According to the definition of convex function Property 2 is very easy to prove. 

One of the most successful QN formulas in practice is associated with the BFGS method (for its developers Broyden, Fletcher, Goldfard, and Shanno). The BFGS update matrix has rank 2 and inherent positive-definiteness (i.e., if B is positive definite then Bj + i is positive definite) as long as yjsk < 0. This condition is satisfied automatically for convex functions but may not hold in general. In practice, the line search must check for the descent property updates that do not satisfy this condition may be skipped. 

The tools for calculating the equilibrium point of a chemical reaction arise from the definition of the chemical potential. If temperature and pressure are fixed, the equilibrium point of a reaction is the point at which the Gibbs free energy function G is at its minimum (Fig. 3.1). As with any convex-upward function, finding the minimum G is a matter of determining the point at which its derivative vanishes. 

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. 

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

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 ... 

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. 

If the function f(xv. .., xn) is convex (concave) on squadratic form (Eq. (2)) in s variables is positive definite (negative definite). The quadratic form (Eq. (2)) in s variables for which is positive definite (negative definite) if [18]... 

The functionals FHK and FEHK have the unfortunate mathematical difficulty that their domains of definition A and B, although they are well defined, are difficult to characterize, i.e., it is difficult to know if a given density n belongs to A or B. It is therefore desirable to extend the domains of definition of FHK and FEHK to an easily characterizable (preferably convex) set of densities. This can be achieved using the constrained search procedure introduced by Levy [19]. We define the Levy-Lieb functional FLL as ... 

These features and definitions permit to express density functions in a compact and extremely general form [92]. Such previous work may be referred to defining a matrix W as bearing a convex coefficient (hyper-)matrix of arbitrary dimensions Dim W)= [S]. In general, this can be obtained by choosing a complex matrix X, with adequate dimension Dim(X)= such that the inward product holds W = X X. Then, symbolically it can be assured that W = eW—>3 eX co = % % j special choice can be... 

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