MaxMin

To decide which molecule to add at each iteration requires the dissimilarity values between each molecule remaining in the database and those already placed into the subset to be calculated. Again, this can be achieved in several ways. Snarey et al. investigated two conunon definitions, MaxSum and MaxMin. If there are m molecules in the subset then... 

Dissimilarity-based compound selection (DECS) methods involve selecting a subset of compounds directly based on pairwise dissimilarities [37]. The first compound is selected, either at random or as the one that is most dissimilar to all others in the database, and is placed in the subset. The subset is then built up stepwise by selecting one compound at a time until it is of the required size. In each iteration, the next compound to be selected is the one that is most dissimilar to those already in the subset, with the dissimilarity normally being computed by the MaxMin approach [38]. Here, each database compound is compared with each compound in the subset and its nearest neighbor is identified the database compound that is selected is the one that has the maximum dissimilarity to its nearest neighbor in the subset. 

Finally, applying MaxMin algorithm II, the sets of molecules for which the Euclidean distance is maxima have been determined. 

In the Maximum Dissimilarity (MD) selection method described by Lajiness [40] the first compound is selected at random and subsequent compounds are then chosen iteratively, such that the distance to the nearest of the compounds already chosen is a maximum. This method is known as MaxMin. In this study, the compounds were represented by COUSIN 2-D fragment-based bitstrings. Polinsky et al. [41] use a similar algorithm in the LiBrain system. In this case, the molecules are represented by a feature vector that contains information about the following affinity types—aliphatic hydrophobic, aromatic hydrophobic, basic, acidic, hydrogen bond donor, hydrogen bond acceptor and polarizable heteroatom. 

The basic DBCS algorithm has time complexity 0(n2N), where n compounds are selected from N. Since n is generally a small fraction of N, the time is thus cubic in N. DBCS can also be very computational demanding however, fast implementations have been developed, for example the MaxSum method described by Holliday et al. [42] and a MaxMin method described by Agrafiotis and Lobanov that can be used with low-dimensional descriptors [55],... 

The maxmin approach (23c) uses the shortest nearest-neighbor distance as a measure of diversity in the sample ... 

This approach is particularly efficient when combined with the Cosine coefficient (69) and was used by Pickett et al. in combination with pharmacophore descriptors (70). In lower dimensional spaces the maxsum measure tends to force selection from the comers of diversity space (6b, 71) and hence maxmin is the preferred function in these cases. A similar conclusion was drawn from a comparison of algorithms for dissimilarity-based compound selection (72). 

A possible method of salvaging the validation problem is to solve both a minmax (Eq. 18) and a maxmin optimal control problem ... 

If the maxmin analysis indicates the problem to be infeasible, it is indeed infeasible as there exists some v for which no input sequence can be found to satisfy the performance specification. If the minmax problem indicates feasibility, there is some fixed input sequence which satisfies the constraints for all v. 

For an optimal control problem, the complementary maxmin problem is equivalent to the controller having perfect knowledge of the plant parameters and disturbances, including future disturbances. This formulation could be termed the crystal ball approach to control. For many problems, the crystal ball control will be successful, but this says very little about whether any realizable control system exists that can meet the performance specification. 

The most likely outcome of solving the complementary maxmin/minmax problems is that the maxmin problem is feasible (for each v there exists an input sequence that can satisfy the constraints) and the minmax is infeasible (there is no single input sequence that can satisfy the constraints for all v). This is unfortunate. as this outcome is the least informative as to whether a realizable controller exists. 

For example, optimizing control moves for disturbance rejection in a blending system subject to step disturbances of variable magnitude would give perfect disturbance rejection for a maxmin/perfect-knowledge formulation and a poor... 

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