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Given $X$ finite, fix a continuous function $\theta \in \Delta^+ (X) \to [0,1]$, fix a probability measure $\mu^*$, and a $\varepsilon > 0$. Consider: $$ \max_{\mu \in \Delta^+ (X)} \theta (\mu), \quad \text{such that } d(\mu ,\mu^*) \leq \varepsilon $$ where $\Delta^+ (X)$ is the set of all sigma-additive probability measures with finite support, and $d(\mu ,\mu^*)$ is defined as a distance measure.

Does this optimization problem has a solution? How could I tackle such a optimization Problem?

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    $\begingroup$ Depending on $d$, which is not defined, this problem has a solution since $\theta$ is a continuous function defined on a compact set. $\endgroup$
    – Dieter Kadelka
    Commented Jan 16, 2021 at 10:52
  • $\begingroup$ How is $d$ defined? $\endgroup$
    – Aryeh Kontorovich
    Commented Jan 16, 2021 at 18:50
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    $\begingroup$ Since $\Delta^+(X)$ is nothing else as the $|X|$-dimensional simplex $S$, your problem can be reformulated as: Determine $x \in S$ which maximizes $x \to \Theta(x)$ on $\{x \in S \colon d(x,x^*)$, where $x^*$ is some fixed $x^* \in S$. Without knowing $\Theta$, at least its structure (linear, quadratic, ?) and $d$ there is no general answer to your problem! $\endgroup$
    – Dieter Kadelka
    Commented Jan 18, 2021 at 14:14

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