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I have a parametric convex optimization problem:

\begin{array}{cl} \underset{x}{\text{minimize}} & f\left(x,z\right)\\ \text{subject to} & g\left(x\right)\leq0 \end{array}

where $x$ is the variable and $z$ is a parameter. I am looking for known results about the Lipschitz continuity of the solution set mapping $S\left(z\right)$ which is defined as: \begin{equation} S\left(z\right)=\text{argmax}\ f\left(x,z\right)\ \text{s.t. } g{\left(x\right)\leq0} \end{equation}

In my problem, $f$ is also strictly convex (i.e. optimal solution is unique for any z) and I assume that the parameter z takes its value in a convex and compact set ${\cal Z}$. In this case, $S\left(z\right)$ is a function and not just a point-to-set mapping.

Can someone please suggest a classic reference or conditions for the Lipschitz continuity of S to hold? Thank you! Any help is very much appreciated.

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  • $\begingroup$ So you can get rid of the function $g$ and only recall the set $X:=\{g\le0\}$ , taking as domain of $f$ the set $X\times Z$. What are your assumptions on $X$ ? $\endgroup$
    – Pietro Majer
    Commented Nov 26, 2013 at 9:13
  • $\begingroup$ Thanks for the reply. X is assumed to be convex and compact. $\endgroup$
    – Rindra
    Commented Nov 26, 2013 at 10:10

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