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The classical linear prediction problem for a a wide sense stationary process consists of finding an element in the linear span of the past values providing the best possible mean square approximation to the current and future values of the process. In this talk we investigate this and some other similar problems where, in addition to prediction quality, optimization takes into account other features of the objects we search for. One of the most motivating examples of this kind is an approximation of a stationary process by a stationary/differentiable/ process taking into account the kinetic energy that the latter spends in its approximation efforts. We also provide appropriate extensions of the classical Kolmogorov and Krein prediction singularity criteria and Kolmogorov's criterion of error-free interpolation.