Suppose I have a function $f(x,y)$ from $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is convex in both $x$ and $y$. Set


$g(y) = \min\_{x} f(x,y)$


What I would like is for $g(y)$ to be Lipschitz:


$|g(y) - g(y')| \le c \cdot \| y - y' \|$


Unfortunately, $f(x,y)$ may have a very poor Lipschitz constant for general $x$. Are there general conditions on $f$ for which the minima are Lipschitz?


Alternatively, when can we say the minimizer $x^{\ast}(y) = \arg \min\_x f(x,y)$ is Lipschitz in $y$?


I've tried looking in a few convex optimization books for answers, but no luck.



Lipschitz properties of minima/minimizers of convex functions of two variables
I encountered the same problem three years ago and found some relevant literature. Here are a few. See also the refs therein.


Lipschitz Behavior of Solutions to Convex Minimization Problems.
Jean-Pierre Aubin,
Mathematics of Operations Research, Vol. 9, No. 1. (Feb., 1984), pp. 87-111.


Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems. 
O. L. MANGASARIAN and T.-H. SHIAU. 
SIAM J. CONTROL AND OPTIMIZATION, 25(3), 1987.


Lipschitz Continuity of Solutions of Variational Inequalities with a Parametric
Polyhedral Constraint.
N. D. Yen,
Mathematics of Operations Research, Vol. 20, No. 3. (Aug., 1995), pp. 695-708.


On Lipschitzian Stability of Optimal Solutions of Parametrized Semi-Infinite
Programs.
Alexander Shapiro,
Mathematics of Operations Research, Vol. 19, No. 3. (Aug., 1994), pp. 743-752.


SHARP LIPSCHITZ CONSTANTS FOR BASIC OPTIMAL SOLUTIONS AND BASIC FEASIBLE SOLUTIONS OF LINEAR PROGRAMS.
Wu Li, 
SIAM J. CONTROL AND OPTIMIZATION
Vol. 32, No. I, pp. 140-153, January 1994


