Positive semidefinite

These necessary conditions for local optimality can be strengthened to sufficient conditions by making the inequality in (3-87) strict (i.e., positive curvature in all directions). Equivalently, the sufficient (necessary) curvature conditions can be stated as follows V /(x ) has all positive (nonnegative) eigenvalues and is therefore defined as a positive (semidefinite) definite matrix. 

If the matrix Q is positive semidefinite (positive definite) when projected into the null space of the active constraints, then (3-98) is (strictly) convex and the QP is a global (and unique) minimum. Otherwise, local solutions exist for (3-98), and more extensive global optimization methods are needed to obtain the global solution. Like LPs, convex QPs can be solved in a finite number of steps. However, as seen in Fig. 3-57, these optimal solutions can lie on a vertex, on a constraint boundary, or in the interior. A number of active set strategies have been created that solve the KKT conditions of the QP and incorporate efficient updates of active constraints. Popular methods include null space algorithms, range space methods, and Schur complement methods. As with LPs, QP problems can also be solved with interior point methods [see Wright (1996)]. 

The status of H can be used to identify the character of extrema. A quadratic form <2(x) = xrHx is said to be positive-definite if Q(x) > 0 for all x = 0, and said to be positive-semidefinite if Q(x) > 0 for all x = 0. Negative-definite and negative-semidefinite are analogous except the inequality sign is reversed. If Q(x) is positive-definite (semidefinite), H(x) is said to be a positive-definite (semidefinite) matrix. These concepts can be summarized as follows ... 

It can be shown from a Taylor series expansion that if/(x) has continuous second partial derivatives, /(x) is concave if and only if its Hessian matrix is negative-semidefinite. For/(x) to be strictly concave, H must be negative-definite. For /(x) to be convex H(x) must be positive-semidefinite and for/(x) to be strictly convex, H(x) must be positive-definite. 

Determine whether the following matrix is positive-definite, positive-semidefinite, negative-definite, negative-semidefinite, or none of the above. Show all calculations. 

If fix) is convex, H(x) is positive-semidefinite at all points x and is usually positive-definite. Hence Newton s method, using a line search, converges. If fix) is not strictly convex (as is often the case in regions far from the optimum), H(x) may not be positive-definite everywhere, so one approach to forcing convergence is to replace H(x) by another positive-definite matrix. The Marquardt-Levenberg method is one way of doing this, as discussed in the next section. 

The second-order necessary conditions require this matrix to be positive-semidefinite on the tangent plane to the active constraints at (0,0), as defined in expression (8.32b). Here, this tangent plane is the set... 

Basis functions between these vector spaces are orthogonal because they are contained in Hilbert spaces with different numbers of particles. Hence the four metric matrices that must be constrained to be positive semidefinite for 3-positivity [17] are given by... 

Because the addition of any two positive semidefinite matrices produces a positive semidefinite matrix, the four 3-positivity conditions [17] imply the following two less stringent constraints ... 

When N = p, the set B simply contains the p-particle reduced Hamiltonians, which are positive semidefinite, but when N = p + 1, because the lifting process raises the lowest eigenvalue of the reduced Hamiltonian, the set also contains p-particle reduced Hamiltonians that are lifted to positive semidefinite matrices. Consequently, the number of Wrepresentability constraints must increase with N, that is, B C B. To constrain the p-RDMs, we do not actually need to consider all pB in B, but only the members of the convex set B, which are extreme A member of a convex set is extreme if and only if it cannot be expressed as a positively weighted ensemble of other members of the set (i.e., the extreme points of a square are the four corners while every point on the boundary of a circle is extreme). These extreme constraints form a necessary and sufficient set of A-representability conditions for the p-RDM [18, 41, 42], which we can formally express as... 

Any arbitrary one-particle reduced Hamiltonian shifted by its A-particle ground-state energy must be expressible by the extreme HamUtonian elements in the convex set As we showed in Eq. (52), keeping the 1-RDM positive semide-finite is equivalent to applying the Al-representability constraints in Eq. (50) for the class of extreme positive semidefinite which may be parameterized by... 

Because is the A -particle energy and not the lowest eigenvalue of K, some of the eigenvalues of C will be negative, and this portion of the reduced Hamiltonian cannot be represented by the positive semidefinite Hamiltonians in Eq. (56). 

A similar argument, however, may also be made from the perspective of the holes. Restricting the one-hole RDM to be positive semidefinite corresponds to applying the A -representability constraints in Eq. (50) to the class of extreme positive semidefinite... 

The three complementary representations of the reduced Hamiltonian offer a framework for understanding the D-, the Q-, and the G-positivity conditions for the 2-RDM. Each positivity condition, like the conditions in the one-particle case, correspond to including a different class of two-particle reduced Hamiltonians in the A-representability constraints of Eq. (50). The positivity of arises from employing all positive semidefinite in Eq. (50) while the Q- and the G-conditions arise from positive semidefinite and B, respectively. To understand these positivity conditions in the particle (or D-matrix) representation, we define the D-form of the reduced Hamiltonian in terms of the Q- and the G-representations ... 

If the G-matrix is positive semidefinite, then the above expectation value of the G-matrix with respect to the vector of expansion coefficients must be nonnegative. Similar analysis applies to G, operators expressible with the D- or Q-matrix or any combination of D, Q, and G. Therefore variationally minimizing the ground-state energy of n (H Egl) operator, consistent with Eq. (70), as a function of the 2-positive 2-RDM cannot produce an energy less than zero. For this class of Hamiltonians, we conclude, the 2-positivity conditions on the 2-RDM are sufficient to compute the exact ground-state A-particle energy on the two-particle space. 

By relaxing the condition that a von Neumann density be positive semidefinite, a graded family of approximations can be constructed. Since an operator can be represented as a polynomial in the annihilation and creation operators, it can be... 

Proposition 3 Let P be the k-matrix for the k-density p. Then P is Hermitian, positive semidefinite, and has unit trace. ... 

We summarize this discussion with the following theorem characterizing the convex set of -matrices. In the statement of this theorem we introduce Pq, the symbol we use to denote the convex set of positive semidefinite matrices with unit trace on the linear space of coefficients of the elements of l/ similarly, we use P to denote the cone of positive semidefinite matrices. 

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