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We examine various methods for data clustering and data classification that are based on the minimization of the so-called cluster function and its modifications. These functions are nonsmooth and nonconvex. We use Discrete Gradient methods for their local minimization. We consider also a combination of this method with the cutting angle method for global minimization. We present and discuss results of numerical experiments.

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We examine the validity of the zero duality gap properties for two important dual schemes: a generalized augmented Lagrangian dual scheme and a nonlinear Lagrange-type dual scheme. The necessary and sufficient conditions for the zero duality gap property to hold are established in terms of the lower semicontinuity of the perturbation functions.

The ultimate goal of this paper is to demonstrate that abstract convexity provides a natural language and a suitable framework for the examination of zero duality gap properties and exact multipliers of augmented Lagrangians. We study augmented Lagrangians in a very general setting and formulate the main definitions and facts describing the augmented Lagrangian theory in terms of abstract convexity tools. We illustrate our duality scheme with an application to stochastic semi-infinite optimization.

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We establish some results concerning P-functions from the standpoint of abstract convexity. In particular, we show that the set of all P-functions on a segment is the least set closed under pointwise sum, supremum, and convergence and containing the class of all nonnegative quasi-convex functions on that segment. Further, generalizations are derived of a recent inequality of Hadamard type for P-functions.