Relaxed constraint algorithm

The main idea of the algorithm of Caroe and Schultz [11] is to decompose a 2S-MILP into its scenarios by Lagrangian relaxation of the non-anticipativity constraints. In a Lagrangian relaxation, constraints are removed and included in the objective function with a penalty term. 

In order to apply the aBB algorithm to (17), we must reformulate it as a global optimization problem. This is accomplished by introducing a slack variable s and minimizing its value over an augmented variable set (x,s) subject to a set of relaxed constraints ... 

This basic concept leads to a wide variety of global algorithms, with the following features that can exploit different problem classes. Bounding strategies relate to the calculation of upper and lower bounds. For the former, any feasible point or, preferably, a locally optimal point in the subregion can be used. For the lower bound, convex relaxations of the objective and constraint functions are derived. 

The basis for the determination of solution conformation from NMR data lies in the determination of cross relaxation rates between pairs of protons from cross peak intensities in two-dimensional nuclear Overhauser effect (NOE) experiments. In the event that pairs of protons may be assumed to be rigidly fixed in an isotopically tumbling sphere, a simple inverse sixth power relationship between interproton distances and cross relaxation rates permits the accurate determination of distances. Determination of a sufficient number of interproton distance constraints can lead to the unambiguous determination of solution conformation, as illustrated in the early work of Kuntz, et al. (25). While distance geometry algorithms remain the basis of much structural work done today (1-4), other approaches exist. For instance, those we intend to apply here represent NMR constraints as pseudoenergies for use in molecular dynamics or molecular mechanics programs (5-9). 

The OA/ER algorithm, extends the OA to handle nonlinear equality constraints by relaxing them into inequalities according to the sign of their associated multipliers. 

Appropriate changes need also to be made in the relaxed master problem described in the algorithmic procedure section. Note also that the correct formulation of the master problem via (6.21) increases the number of constraints to be included significantly. 

To handle explicitly nonlinear equality constraints of the form h(x) = 0, Kocis and Grossmann (1987) proposed the outer approximation with equality relaxation OA/ER algorithm for the following class of MINLP problems ... 

The basic idea in OA/ER is to relax the nonlinear equality constraints into inequalities and subsequently apply the OA algorithm. The relaxation of the nonlinear equalities is based upon the sign of the Lagrange multipliers associated with them when the primal (problem (6.21) with fixed y) is solved. If a multiplier A is positive then the corresponding nonlinear equality hi(x) = 0 is relaxed as hi x) <0. If a multiplier A, is negative, then the nonlinear equality is relaxed as -h (jc) < 0. If, however, A = 0, then the associated nonlinear equality constraint is written as 0 ht(x) = 0, which implies that we can eliminate from consideration this constraint. Having transformed the nonlinear equalities into inequalities, in the sequel we formulate the master problem based on the principles of the OA approach discussed in section 6.4. 

The master problem of the OA/ER algorithm is essentially the same as problem (6.20) described in section 6.4.3.2, with the difference being that the vector of inequality constraints will be augmented by the addition of the relaxed equalities ... 

Section 6.5 presents the Outer Approximation with Equality Relaxation OA/ER for handling nonlinear equality constraints. Sections 6.5.1 and 6.5.2 discuss the formulation, assumptions and the basic idea. Section 6.5.3 discusses the equality relaxation, and illustrates it with a simple example, and presents the formulation of the master problem. Section 6.5.4 discusses the OA/ER algorithm and illustrates it with a small planning problem. 

Remark 6 Cerda and Westerberg (1983) developed an MILP model based on the transportation formulation which could also handle all cases of restricted matches. Furthermore, instead of solving the MILP transportation model via an available solver, they proposed several LP relaxations that can avoid the associated combinatorial problems. The drawback, though, of this model is that it requires more variables and constraints. Viswanathan and Evans (1987) proposed a modified transportation model for which they used the out-of-kilter algorithm to deal with constraints. 

A typical run took minutes on a workstation and the predicted conformations agreed with those observed crystallographicallyin all cases. Meadows and Hajduk (102) used experimental constraints with a GA algorithm to dock biotin to stepavidin. Judson et al. (101) also reported docking of flexible molecules into the active sites of thermolysin, car-boxypeptidase, and dihydrofolate reductase. In 9 of the 10 cases examined, the GA found conformations within 1.6 A root-mean-square (rms) of the relaxed crystal conformation. 

Structure optimization of the reactants, products, and some transition states was performed by the Bemy geometry optimization algorithm without symmetry constraints [ 12]. In cases where identification of transition states was rather complicated, the relaxed potential energy surface scan and/or the combined synchronous transit-guided quasi-Newton (STQN) method was employed [13]. 

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