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Let $(E,\mathcal E)$ be a measurable space, $W\subseteq\left\{w:E\to\mathbb R\mid w\text{ is }\mathcal E\text{-measurable}\right\}$ be a Banach space, $k\in\mathbb N$ and $f:W^k\to[0,\infty)$. I'm searching for a multiplier rule (or other kind of useful necessary condition) for the following optimization problem $$\begin{array}{ll}\text{minimize}&f(w)\\\text{subject to}&w_1,\ldots,w_k\ge0\\&w_1+\cdots+w_k\le1.\end{array}\tag1$$ (Of course, assuming that $f$ is sufficiently regular in a suitable sense.)

I'm mainly interested in $W=L^p(\mu)$, for some probability measure $\mu$ on $(E,\mathcal E)$ and $p\in[1,\infty]$. In particular, in the Hilbert space case $p=2$.

I'd be thankful for any suggested approach and/or reference.

We may note that the set $S:=\left\{w\in W:\sum_{i=1}^kw_i\le1\right\}$ is convex and we may reformulate $(1)$ as a minimization problem over $S$ eliminating the second constraint.

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  • $\begingroup$ Is it important for your problem that $W$ is a set of functions? The elements of $L^p(\mu)$ are equivalence classes in general. $\endgroup$
    – Dieter Kadelka
    Commented Nov 10, 2019 at 22:47
  • $\begingroup$ @DieterKadelka Do you see a problem with that? You may take $\mathcal L^p(\mu)$ instead. On the other hand, each functin in the same equivalence class of $L^p(\mu)$ is equal almost surely, so if shouldn't really matter. $\endgroup$
    – 0xbadf00d
    Commented Nov 11, 2019 at 5:19

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