Convex Sets

We have pointed out the following fact. If if c E is a convex set then the variational inequality... 

Theorem 1.10. A closed convex set of a reflexive Banach space is weakly closed. 

Theorem 1.11. Let V be a reflexive Banach space, and K c V be a closed convex set. Assume that J V R is a coercive and weakly lower semicontinuous functional. Then the problem... 

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

The equilibrium problem for the shell corresponds to minimization of the energy functional over the set of admissible displacements. To this end, introduce the convex sets... 

Proof. To prove the existence of a solution, we implement the idea that was earlier used in a simpler case by (Shi, Shillor, 1992). We introduce two closed convex sets... 

Obviously, for (3.113) to hold, it suffices to choose 5 from the condition We define the bounded closed convex set... 

Mosco U. (1969) Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3 (4), 510-585. 

Notice that those distribution functions that satisfy Eq. (4-179) still constitute a convex set, so that optimization of the E,R curve is still straightforward by numerical methods. It is to be observed that the choice of an F(x) satisfying a constraint such as Eq. (4-179) defines an ensemble of codes the individual codes in the ensemble will not necessarily satisfy the constraint. This is unimportant practically since each digit of each code word is chosen independently over the ensemble thus it is most unlikely that the average power of a code will differ drastically from the average power of the ensemble. It is possible to combine the central limit theorem and the techniques used in the last two paragraphs of Section 4.7 to show that a code exists for which each code word satisfies... 

The mixed and pure states of an A-particle fermion system can be described by positive and normalized operators,, which form a convex set... 

N-particle states,, has values in the convex set,, of positive functions in Ii ( Y) that integrate to the value N... 

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. 

The concept of convexity is useful both in the theory and applications of optimization. We first define a convex set, then a convex function, and lastly look at the role played by convexity in optimization. 

A set of points (or a region) is defined as a convex set in -dimensional space if, for all pairs of points Xj and x2 in the set, the straight-line segment joining them is also entirely in the set. Figure 4.9 illustrates the concept in two dimensions. 

For every pair of points x and x2 in a convex set, the point x given by a linear combination of the two points... 

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... 

The result is illustrated in Figure 4.11 in which a convex quadratic function is cut by the plane /(x) = k. The convex set R projected on to the xt-x2 plane comprises the boundary ellipse plus its interior. 

Analogously, a local maximum is the global maximum of/(x) if the objective function is concave and the constraints form a convex set. 

Illustration of a convex set formed by a plane /(x) = k cutting a convex function. 

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... 

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. 

Problems with concave objective functions to be minimized over a convex set. 

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

After the energy is expressed as a functional of the 2-RDM, a systematic hierarchy of V-representabihty constraints, known as p-positivity conditions, is derived [17]. We develop the details of the 2-positivity, 3-positivity, and partial 3-positivity conditions [21, 27, 34, 33]. In Section II.E the formal solution of V-representability for the 2-RDM is presented through a convex set of two-particle reduced Hamiltonian matrices [7, 21]. It is shown that the positivity conditions correspond to certain classes of reduced Hamiltonian matrices, and consequently, they are exact for certain classes of Hamiltonian operators at any interaction strength. In Section II.F the size of the 2-RDM is reduced through the use of spin and spatial symmetries [32, 34], and in Section II.G the variational 2-RDM method is extended to open-shell molecules [35]. 

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