Constraints linearly independent

A sufficient condition for a unique Newton direction is that the matrix of constraint derivatives is of full rank (linear independence... 

Constraint Qualification For a local optimum to satisfy the KKT conditions, an additional regularity condition is required on the constraints. This can be defined in several ways. A typical condition is that the active constraints at x be linearly independent i.e., the matrix [Vh(x ) I VgA(x )] is full column rank, where gA is the vector of inequality constraints with elements that satisfy g x ) = 0. With this constraint qualification, the KKT multipliers (X, v) are guaranteed to be unique at the optimal solution. 

In the rest of this chapter, we assume that the rows of the constraint matrix A are linearly independent, that is, rank (A) = m. If a slack variable is inserted in every row, then A contains a submatrix that is the identity matrix. In the preceding example, if we insert a slack variable x5 into the equality ... 

Problem (8.15) must satisfy certain conditions, called constraint qualifications, in order for Equations (8.17)-(8.18) to be applicable. One constraint qualification (see Luenberger, 1984) is that the gradients of the equality constraints, evaluated at x, should be linearly independent. Now we can state formally the first order necessary conditions. 

Let x be a local minimum or maximum for the problem (8.15), and assume that the constraint gradients Vhj(x ),j — 1,m, are linearly independent. Then there exists a vector of Lagrange multipliers A = (Af,..., A ) such that (x A ) satisfies the first-order necessary conditions (8.17)-(8.18). 

Note that this constraint implies that the (/) columns of Y are orthogonal to the (E) rows of A. Thus, since each column of Y represents one elementary reaction, the maximum number of linearly independent elementary reactions is equal to K — E, i.e., N-f = rank(Y) < K — E. Formost chemical kinetic schemes, Ny = K — E however, this need not be the case. 

Let x be a global minimum of (3.6) at which the gradients of the equality constraints are linearly independent (i.e., x is a regular point). Perturbing the right-hand sides of the equality constraints, we have... 

Remark 2 The linear independence constraint qualification as well as the Slater s imply the Kuhn-Tucker constraint qualification. 

One of the constraints to be imposed (as for AOs) is that the set of MOs for a given system must be linearly independent, i.e. it should not be possible to express any member of the set as a linear combination of the others. The fact that MOs y, y2,..., y satisfy the Schrodinger equation under the one-electron Hamiltonian leads to the conclusion that any linear combination ya ... 

We now multiply Equation (3.10) by e and consider the limit of an infinitely high recycle flow rate (i.e., e —> 0) in the original time scale t. In this limit, we obtain the constraints in Equation (3.12), or equivalently, the linearly independent constraints... 

Note that, owing to the underlying algebraic constraints in the DAE system that describes the slow dynamics, the holdups Mb, Me, and Mr are not independent (there are only two linearly independent constraints among the three holdups, i.e., 0 = wr — and 0 = k2w2 — r, where u, U2, and kr are determined by the proportional control laws in Equation (3.35)). Thus, controlling one of the holdups (e.g., Mb) amounts to regulating the total material holdup in the process. 

Note that the differential equations in Equation (4.36) are not independent, or, equivalently, the corresponding quasi-steady-state constraints are not linearly independent. Specifically, the third constraint can be expressed as a linear combination of the others, i.e., there exist only eight linearly independent constraints that can be written in the form of Equation (4.22), with... 

Assumption 6.4 allows us to isolate a set of linearly independent constraints... 

Equation (7.17) is a description of the fast dynamics of the high-purity distillation column. It involves only the stage temperatures and it can be easily verified that the system of ODEs describing the fast dynamics (as well as the quasi-steady-state conditions that result from setting the left-hand side of (7.17) to zero) are linearly independent. The constraints arising from the fast dynamics can therefore be solved (typically numerically) for the quasi-steady-state values of the stage temperatures, T = [7 (. 7. .. Tp Tg], which can then be substituted into the ODE system (7.8) in order to obtain a description of the dynamics after the fast temperature transient ... 

The description of the dynamics in the intermediate time scale and the corresponding quasi-steady-state constraints involve only the large internal material flows of the column, i.e., V and R. It is easy to verify that these flows do not influence the total material holdup of the column, or the holdup of any of the components in the column and, consequently, the constraints in Equations (7.20) are not linearly independent (more specifically, the last two constraints can be expressed as a linear combination of the remaining constraints). 

It is straightforward to verify that the algebraic constraints in Equations (7.34) are generically linearly independent and hence they can be solved for the quasi-steady-state values Q (M,C) = [T, TR,T ] of the variables [T,TR,TC], Substituting the value for T, we then obtain... 

We would like to emphasize that, due to the closure constraint, there are only (m — 1) linearly independent internal MEC. Thus, the m vectors defined by Eq. (93) in reality span the (m — 1 )-dimensional space of internal MEC. In order to remove this linear dependence one could adopt the relative internal approach of Sect. 2.1.3. Namely, one then selects the electron population of one atom in the system as dependent upon populations of all remaining atoms, and discards the MEC associated with that atom. All remaining MEC can also be constructed directly from the corresponding internal relative softness matrix. Although the sets of independent internal MEC for alternative choices of the dependent atom will differ from one another, they must span the same (m — 1 )-dimensional linear space of independent internal MEC. For example, in the two-AIM system of Fig. 4 there is only one independent internal MEC direction along the P-line. 

Each of the two linearly independent expansions expressed in Eq. (B5), can easily accommodate the conditions at the left and right surface, respectively. However, additional constraints must be imposed on the coefficients so that the solution as a whole satisfies both boundary conditions simultaneously. If the first series is used to match the conditions on the left surface, then the second series in (B5) should vanish there. Similarly the first series should vanish on the right surface where the second series is used. This is explicitly reflected in Eqs. (B9) and (BIO) which are valid for the case of two surfaces held at constant potential. Thus,... 

The choice of appropriate tracers for this constraint. The tracers must be conserved in mixing, and they should be linearly independent (i.e., they should provide unique information). 

For the problem shown in equation 7.83 it is not immediately apparent howto find a particular solution to the constraint equation. However, since the rows of Ai are assumed to be linearly independent, the matrix (A,A ) is invertible. Thus we may take as a particular solution... 

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