Generalized outer approximation

The Generalized Outer Approximation GOA extends the OA to the MINLP problems of types (6.1),(6.2) and introduces exact penalty functions. 

Generalized Outer Approximation with Exact Penalty, GOA/EP 6.7.6.1 The Primal Problem... 

Section 6.7 presents the Generalized Outer Approximation GOA approach. After a brief discussion on the problem formulation, the primal and master subproblem formulations are developed, and the GOA algorithm is stated in section 6.7.4. In Section 6.7.5, the worst case analysis of GOA is discussed, while in section 6.7.6 the Generalized Outer Approximation with exact Penalty GOA/EP and its finite convergence are discussed. 

The new worst-case design algorithm is discussed further below following the general outer-approximation algorithm structure constraint maximization, initialization, and multi-mode design. 

The MINLP-problems were implemented in GAMS [7, 8] and solved by the outer approximation/equality relaxation/augmented penalty-method [9] as implemented in DICOPT. The algorithm generates a series of NLP and MILP subproblems, which were solved by the generalized reduced gradient method [10] as implemented in CONOPT and the integrality relaxation based branch and cut method as... 

Generalized Benders decomposition (GBD), derived in Geoffrion (1972), is an algorithm that operates in a similar way to outer approximation and can be applied to MINLP problems. Like OA, when GBD is applied to models of the form (9.2)-(9.5), each major iteration is composed of the solution of two subproblems. At major iteration K one of these subproblems is NLP(y ), given in Equations (9.6)-(9.7). This is an NLP in the continuous variables x, with y fixed at y The other GBD subproblem is an integer linear program, as in OA, but it only involves the... 

In sections 63-6.1 we discussed the generalized benders decomposition GBD and the outer approximation based algorithms (i.e., OA, OA/ER, OA/ER/AP, GOA), and we identified a number of similarities as well as key differences between the two classes of MINLP algorithms. 

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. 

The general structure of an outer-approximation algorithm for worst-case design is as follows... 

The branch and bound method can be used for MINLP problems, but it requires solving a large number of NLP problems and is, therefore, computationally intensive. Instead, methods such as the Generalized Benders Decomposition and Outer Approximation algorithms are usually preferred. These methods solve a master MILP problem to initialize the discrete variables at each stage and then solve an NLP subproblem to optimize the continuous variables. Details of these methods are given in Biegler et al. (1997) and Diwekar (2003). 

MINLP problems can be solved using several algorithms including branch-and-bound, generalized Benders decomposition (GBD), and the Outer Approximation/Equality-Relaxation (OA/ER). These... 

The most important point to note is that the leading-order solution = 1 does not imply that the heat flux at the body surface is zero, even though this might at first appear to be the case. In general, the condition of matching between the inner and outer approximations within the thermal boundary layer requires not only (11—67c) but also that the spatial derivatives of 9 and should match at each level of approximation for Pr -> 0. Thus, in particular,... 

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

To solve the problem above the branch and bound method (see e.g. [106]), generalized Benders Decomposition [108], Outer Approximation [109, 110], LP/NLP branch and bound [111] and Extended Cutting Plane Method [112] are in use. Grossmann and Kravanja [113] give an extensive compilation of literature on MINLP problems. 

The olefin separation process involves handling a feed stream with a number of hydrocarbon components. The objective of this process is to separate each of these components at minimum cost. We consider a superstructure optimization for the olefin separation system that consists of several technologies for the separation task units and compressors, pumps, valves, heaters, coolers, heat exchangers. We model the major discrete decisions for the separation system as a generalized disjunctive programming (GDP) problem. The objective function is to minimize the annualized investment cost of the separation units and the utility cost. The GDP problem is reformulated as an MINLP problem, which is solved with the Outer Approximation (OA) algorithm that is available in DICOPT++/GAMS. The solution approach for the superstructure optimization is discussed and numerical results of an example are presented. 

Excited states of the hydrogen molecule may be formed from a normal hydrogen atom and a hydrogen atom in various excited states.2 For these the interelectronic interaction will be small, and the Burrau eigenfunction will represent the molecule in part with considerable accuracy. The properties of the molecule, in particular the equilibrium distance, should then approximate those of the molecule-ion for the molecule will be essentially a molecule-ion with an added electron in an outer orbit. This is observed in general the equilibrium distances for all known excited states but one (the second state in table 1) deviate by less than 10 per cent from that for the molecule-ion. It is hence probable that states 3,4, 5, and 6 are formed from a normal and an excited atom with n = 2, and that higher states are similarly formed. 

Figure 13-13. The glycogen molecule. A General structure. B Enlargement of structure at a branch point. The molecule is a sphere approximately 21 nm in diameter that can be visualized in electron micrographs. It has a molecular mass of 10 Da and consists of polysaccharide chains each containing about 13 glucose residues. The chains are either branched or unbranched and are arranged in 12 concentric layers (only four are shown in the figure). The branched chains (each has two branches) are found in the inner layers and the unbranched chains in the outer layer. (G, glycogenin, the primer molecule for glycogen synthesis.)...

At the heart of the AIM theory is the definition of an atom as it exists in a molecule. An atom is defined as the union of a nucleus and the atomic basin that the nucleus dominates as an attractor of gradient paths. An atom in a molecule is thus a portion of space bounded by its interatomic surfaces but extending to infinity on its open side. As we have seen, it is convenient to take the 0.001 au envelope of constant density as a practical representation of the surface of the atom on its open or nonbonded side because this surface corresponds approximately to the surface defined by the van der Waals radius of a gas phase molecule. Figure 6.15 shows the sulfur atom in SC12. This atom is bounded by two interatomic surfaces (IAS) and the p = 0.001 au envelope. It is clear that atoms in molecules are not spherical. The well-known space-filling models are an approximation to the shape of an atom as defined by AIM. Unlike the space-filling models, however, the interatomic surfaces are generally not flat and the outer surface is not necessarily a part of a spherical surface. 

The first point from this development and example is that, although the quasichemical approach is directed towards treating strong attractive - chemical - interactions at short range, it can describe traditional packing problems accurately. The second point is that this molecular-field idea permits us to go beyond the primitive quality noted above of the primitive quasichemical approximation, and specifically to account approximately for the influence of the outer-shell material on the equilibrium ratios Km required by the general theory. This might help with cases of delicate structures noted above with anion hydrates. 

---