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The general case can be handled using variational calculus.
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Nonsmooth convex functions are handled using the concept of subdifferentials.
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This is a frequency approach in which the low-frequency part is obtained through a classical technique while the medium-frequency part is handled through the Variational Theory of Complex Rays (VTCR) initially introduced for vibrations.
For certain categories of cost functions, the problem can be solved analytically, using variational calculus.
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