Consider $\mathbb{R}^d$, the $d$-dimensional Euclidean space. Let $W_1$ be the $1^{st}$ Wasserstein distance between probability measures $\mu, \nu$ $$W_1(\mu, \nu) = \inf_{\gamma \in \Gamma(\mu, \nu)} \int \|x-y\|_2d\gamma(x,y),$$ where $\Gamma(\mu, \nu)$ is the set of all measures on $\mathbb{R}^d \times \mathbb{R}^d$ with marginals $\mu, \nu$. It is well known that $W_1$ has the following dual representation $$W_1(\mu, \nu) = \sup_{\|f\|_{\text{Lip}} \leq 1} \int f(x)d(\mu-\nu)(x),$$ where $\|f\|_{\text{Lip}} = \sup_{x\neq y} \frac{|f(x)-f(y)|}{\|x-y\|_2}.$ I'm interested in understanding a slight variant of this dual objective $$\sup_{f} \left[\int f(x)d(\mu-\nu)(x) - \frac{M}{2}\|f\|_{\text{Lip}}^2\right],$$ where $M>0$ is a constant. Note that the earlier objective has a constraint on the Lipschitz constant, whereas the latter one has a regularization penalty on the Lipschitz constant.
Is the maximum of the above objective related to the $W_1$ metric? Can we obtain a closed-form expression for the maximum?