MVP methods are employed when some optimization variables are discrete (e.g integer), categorical (e.g. color, material), or when the problem dimension is variable. These methods are often successfully applied to problems otherwise considered too difficult because of the unusual variable constraints.

MVP methods are being used successfully in tomography for obtaining a full object description from limited radiographic data and limited computing resources. The object is asumed to belong to a parametrized class described by some geometry (integer and continuous variables) and material composition (categorical variables). Kevin O'Reilly performed the initial study as a master's project, and recent results are extremely encouraging.

MVP must make use of direct search methods because of the lack of derivatives. Categorical variables have no derivatives, and discrete variables only have derivatives if relaxations methods can be employed. Direct search methods are also employed when derivative information is unavailable or unreliable. Techniques of improved efficiency and robustness over the standard simplex-based searches include generalize pattern search (GPS) and mesh-adaptive direct search (MADS). We apply direct search methods to manifolds as well see more

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