Convex conditions

Table 9.1 shows how outer approximation, as implemented in the DICOPT software, performs when applied to the process selection model in Example 9.3. Note that this model does not satisfy the convexity assumptions because its equality constraints are nonlinear. Still DICOPT does find the optimal solution at iteration 3. Note, however, that the optimal MILP objective value at iteration 3 is 1.446, which is not an upper bound on the optimal MINLP value of 1.923 because the convexity conditions are violated. Hence the normal termination condition that the difference between upper and lower bounds be less than some tolerance cannot be used, and DICOPT may fail to find an optimal solution. Computational experience on nonconvex problems has shown that retaining the best feasible solution found thus far, and stopping when the objective value of the NLP subproblem fails to improve, often leads to an optimal solution. DICOPT stopped in this example because the NLP solution at iteration 4 is worse (lower) than that at iteration 3. 

Also note that /i ( ), g1 ( ) are convex functions in xu and hence the required convexity conditions are satisfied. 

Remark 3 Note that the inner problem in (6.46) is v(y) with linearized objective and constraints around xk. The equivalence in solution between (6.45) and (6.46) is true because of the convexity condition and the constraint qualification. 

Since the master problem in OA and its variants has many more constraints than the master problem in GBD and its variants, the lower bound provided by OA is expected to be better than the lower bound provided by the GBD. To be fair, however, in the comparison between GBD and OA we need to consider the variant of GBD that satisfied the conditions of OA, namely, separability and linearity of they variables as well as the convexity condition and the constraint qualification, instead of the general GBD algorithm. The appropriate variant of GBD for comparison is the v2-GBD under the conditions of separability and linearity of they vector. Duran and Grossmann (1986a) showed that... 

It should be noted, however, that we need to check whether Tkh(x) < 0 is quasi-convex at each iteration. If the quasi-convexity condition is not satisfied, then the obtained lower bound by OA/ER may not be a valid one that is, it may be above the global solution of the MINLP model. This may happen due to the potential invalid linearizations which may cut off part of the feasible region. 

If the convexity conditions are satisfied along with the constraint qualification then the GBD variants and the OA attain their global optimum. 

It is important to realize that the convexity conditions only refer to the free energy of mixing. The actual free energy of the mixture equals the free energy of mixing plus the linearly additive term , where is the... 

It should also be noted that another modification to reduce the undesirable effects of nonconvexities in the master problem is to apply global convexity tests, followed by a suitable validation of linearizations. One possibility is to apply the test to all linearizations with respect to the current solution vector (y. v ) (Kravanja and Grossmann, 1994). The convexity conditions that have to be verified for the linearizations are as follows ... 

It is trivial to show that the convexity condition is equivalent to the positiveness of the Flankel-Hadamard determinants for k = 0,1 and I = 0,1,2, or in other words for the first four moments. For higher-order moments the equivalence is lost, since more stringent conditions are required by Theorem 3.4. However, it turns out that the convexity condition is useful for reasons that will be discussed below. 

From the above analysis, it is expected that virtually all present approximations to Exc, including all the GGA s discussed in this paper, violate the convexity condition, because virtually all approximations contain a fd3r n4/3(r) component, and it is, unfortunately, difficult to overcome the concavity of this component as 7 — 0. Along these lines, it has been shown[24] that the LSD approximation and PW91 do not satisfy Eq. (71) for the one-electron densities i(r) = 7r-1 e-2r and n2(r) = (4tt)-1 (1 + 2r) e 2r. All this means that the satisfaction of the low density convexity requirement represents a tough challenge for the future. 

The most important thing to be noted in the context of PD Operator VSS, as well as in the isomorphic VSS companions, is the closed nature of such VSS, when appropriate PD coefficient sets are known. That is PD linear combinations of PD Operators remain PD Operators. Discrete matrix representations of such PD Operators are PD too, and PD linear combinations of PD matrices will remain PD in the same way. Identical properties can be described using convex conditions symbols, if. (pi) Vi and (w) hold, then equation (8) is a convex function fulfilling X(p). 

The set of the vector elements, W — wj, can be used instead in the convex conditions symbol, that is ... 

Equations (A-3)and (A-4) can be considered the discrete counterparts of the continuous convex conditions, defining a convex DF ... 

Convex sets [89] play a leading role in optimisation problems. They have been introduced to deal with some chemical problems [9a], Elementary Jacobi Rotations (EJR) [90] can be applied over a generating vector to obtain optimal coefficients, while preserving convex conditions [39e,41,49,9l]. 

A stationary point for a general Lagrangian function may or may not be a loctil extremum. If, as described in Section 2, suitable convexity conditions hold, then the method of Lagrange multipliers will yield a global minimum. 

Another important concept in molecular quantum similarity is associated with convex conditions. The idea underlying convex conditions, associated with a numerical set, a vector, a matrix, or a function, has been described previously in the initial work on VSS and related issues.Convex conditions correspond to several properties of some mathematical objects. The symbol X(x) means that the conditions (x) = 1 A x V(R ) hold simultaneously for a given mathematical object x, which is present as an argument in the convex conditions symbol. Convex conditions become the same as considering the object as a vector belonging to the unit shell of some VSS. For such kind of elements,... 

Such a property is related to the possibility of constructing approximate atomic and molecular density functions, by means of convex combinations with a basis of structurally simpler functions, which belong to the same VSS a— shell. The ASA described extensively earlier and in Amat et al., " Mestres et al., °° and Carbo-Dorca and Girones is a good illustration of these ideas. Moreover the VSS shell structure can transform density functions into a homothetic construct, a characteristic that is discussed elsewhere. The need to take into account the convex conditions [127] to construct approximate density functions has not been properly performed in the literature, as discussed. 

Thus, the unit shell represents every other a— shell in the VSS. Probability distributions of any kind can be transformed into any other element of the associated VSS in this manner. Any VSS a— shell, S(o), can be considered like a homothetic construct of the unit shell,5(1). Because the elements of the unit shell comply with the form of a probability distribution, they then also fulfill the convenient convex conditions as shown in Eq. [126]. In other words, any probability distribution can be considered an element of the unit shell forming part of a VSS provided with the appropriate dimension. 

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