Optimization unimodal functions

Optimization problems in crystallographic structure refinement are seldom convex, that is very rarely characterized by a unimodal function/(x). Regularization of a two-atom model is an example of such a unimodal function. Fig. 11.2a. in contrast. Fig. 11.2b shows a profile of a function for modelling an amino acid side chain - the peaks correspond to the possible rotamers. In this case, the shape of the function/(x) is called multimodal. Such functions arise naturally in structural macromolecular optimization problems and possess a highly complex multiminima energy landscape that does not lend itself favourably to standard robust optimization techniques. 

Unconstrained optimization deals with situations where the constraints can be eliminated from the problem by substitution directly into the objective function. Many optimization techniques rely on the solution of unconstrained subproblems. The concepts of convexity and concavity will be introduced in this subsection, as well as discussing unimodal versus multimodal functions, singlevariable optimization techniques, and examining multi-variable techniques. 

If the function to be optimized is unimodal, an object from the BzzM In Im 1-zatlonMono class may be used (see Vol. 3, Buzzi-Ferraris and Manenti, 2014). 

When one optimal solution exists in the feasible region, the objective function is uni-modal. When two optimal solutions exist, it is bimodal if more than two, it is multimodal. LP problems are imimodal unless the constraints are inconsistent, such that no feasible region exists. The solutions in Figures 18.1 to 18.3 are unimodal. A two-dimensional, multimodal case is shown in Figure 18.4, taken from Reklaitis et al. (1983) and called the Himmelblau problem. This is an unconstrained problem with the objective function ... 

The golden-section search method determines the optimal solution to a bounded objective function that is one-dimensional and unimodal. However, the function need not be continu-... 

If the function to be optimized is unimodal, an object from the BzzMinimiza-tionMono class may be used. This class can be combined with classes used for the integration of differential systems based on multivalue algorithms (see Vol. 4 -Buzzi-Eerraris and Manenti, in press). The objects from these classes automatically change the integration step and the order of the algorithm as well by adapting them to the problem s features and certain specific requirements. 

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