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Has there been any research on the topic of directional derivatives of functions that are minimums of convex functions?

SecretAgentMan
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Samira Fallah
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    Welcome to OR.SE. You may want to have a look at the Lagrangian relaxation as it works with subgradient direction to optimize the LR dualized function and can be applied on the convex problem. Also, this link might be interested. – A.Omidi Feb 13 '23 at 20:09
  • How can functions be minimums of convex functions? Can you provide an example of what you mean by this? – Henrik Alsing Friberg Feb 14 '23 at 12:02
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    Say z^k:\mathbb{R}^n\rightarrow\mathbb{R}\cup{+\infty} is a given convex function for each k=1,\dots,K. And, let z:\mathbb{R}^n\rightarrow\mathbb{R}\cup{+\infty} be defined as follows: z(x) = \min_{k=1,\dots,K} z^k(x). Intuitively, the directional derivatives of the function $z$ at point $x$ in direction $d$ should match the directional derivatives of the function $z^k$ at point $x$ in direction $d$, where the function $z$ and function $z^k$ coincide at point $x$ (or where the function $z^k$ is active at point $x$). I am seeking a formal proof of this concept in the literature. – Samira Fallah Feb 14 '23 at 15:18
  • @SamiraFallah please [edit] the question to include your updated information. You'll get better answers than having your update sit in the comment section. – SecretAgentMan Feb 15 '23 at 13:09

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