Generalized cross decomposition

The Generalized Cross Decomposition GCD simultaneously utilizes primal and dual information by exploiting the advantages of Dantzig-Wolfe and Generalized Benders decomposition. 

Remark 1 Formulation (6.52) corresponds to a subclass of the problems which the generalized cross decomposition GCD of Holmberg (1990) can be applied. Holmberg (1990) studied the more general case ofy C and defined the vector y in a similar way that Geoffrion (1972) did for the... 

This section presents the theoretical development of the Generalized Cross Decomposition, GCD. Phase I is discussed first with the analysis of the primal and dual subproblems. Phase II is presented subsequently for the derivation of the problem while the convergence tests are discussed last. 

Figure 6.9 Pictorial representation of generalized cross decomposition...

Lemma 6.9.1 For model (6.52) in which Y is a finite discrete set the convergence test CT will fail after a finite number of iterations, and hence the generalized cross decomposition GBD algorithms will solve (6.52) exactly in a finite number of steps. 

Figure 6.9 presented the generic algorithmic steps of the generalized cross decomposition GCD algorithm, while in the previous section we discussed the primal and dual subproblems, the relaxed primal master problem, the relaxed Lagrange relaxation master problem, and the convergence tests. 

Remark 3 The upper bounding sequence Pk, and the lower bounding sequence pfe will converge to the optimal value of (6.52). This corresponds to the Generalized Benders Decomposition GBD part of the Generalized Cross Decomposition. 

In remark 1 of the formulation section of the GCD we mentioned that problem (6.52) is a subclass of problems for which the Generalized Cross Decomposition can be applied. This is due to having Y = 0, l 9 in (6.52) instead of the general case of the set Y being a continuous, discrete, or discrete-continuous nonempty, compact set. The main objective in this section is to discuss the modifications in the analysis of the GCD for the cases of the Y set being continuous or discrete-continuous. 

MINOPT (Mixed Integer Nonlinear OPTimizer) is written entirely in C and solves MINLP problems by a variety of algorithms that include (i) the Generalized Benders Decomposition GBD, (ii) the Outer Approximation with Equality Relaxation OA/ER, (iii) the Outer Approximation with Equality Relaxation and Augmented Penalty OA/ER/AP, and (iv) the Generalized Cross Decomposition GCD. 

K. Holmberg. Generalized cross decomposition applied to nonlinear integer programming problems duality gaps and convexification in parts. Optimization, 23 341, 1992. 

K. Vlahos. Generalized cross decomposition Application to electricity capacity planning. Technical Report 200, London Business School, 1991. 

Problem Type Linear, mixed-integer, nonlinear, dynamic, and mixed-integer nonlinear programs Method Generalized benders decomposition, outer approximation and variants, genertilized cross decomposition... 

Drier Mechanism. Oxidative cross-linking may also be described as an autoxidation proceeding through four basic steps induction, peroxide formation, peroxide decomposition, and polymerization (5). The metals used as driers are categorized as active or auxiUary. However, these categories are arbitrary and a considerable amount of overlap exists between them. Drier systems generally contain two or three metals but can contain as many as five or more metals to obtain the desired drying performance. 

Although the basic mechanisms are generally agreed on, the difficult part of the model development is to provide the model with the rate constants, physical properties and other model parameters needed for computation. For copolymerizations, there is only meager data available, particularly for cross-termination rate constants and Trommsdorff effects. In the development of our computer model, the considerable data available on relative homopolymerization rates of various monomers, relative propagation rates in copolymerization, and decomposition rates of many initiators were used. They were combined with various assumptions regarding Trommsdorff effects, cross termination constants and initiator efficiencies, to come up with a computer model flexible enough to treat quantitatively the polymerization processes of interest to us. 

An early application of this type of analysis was to decompose Pio/v( ) into its rotational, translational and their cross-correlation subspectra. It was shown through this decomposition that electrostatic solvation spectra for dipole and charge perturbations are dominated by rotational dynamics. More generally, it was shown how the range and symmetry of AP and molecular properties such as masses and moments of inertia are related to the relative contributions of rotational and translational degrees of freedom to SD. INM analysis has also proved useful in comparing the molecular mechaitisms contributing to short-time dynamics observed in different experiments,such as SD, optical Kerr ef-... 

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