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Consider a nonlinear optimization problem of the form \begin{align} \min_{x}&\quad f(x)\\ \nonumber \text{subject to } \quad&h_i(x) = 0,\,i=1,\ldots,I\\ \nonumber \quad&g_j(x) \le 0,\,j=1,\ldots,J\\ \nonumber \quad&x\in X\subseteq\mathbb{R}^n \end{align} Let the Lagrangian be \begin{align} \mathcal{L}(x,\lambda,\mu) = f(x) + \sum_{i=1}^I\lambda_ih_i(x) +\sum_{j=1}^J\mu_jg_j(x) \end{align} with $\mu_j\ge0$ for all $j$. The dual function is defined as \begin{align} \phi(\lambda,\mu) = \min_{x\in X}\mathcal{L}(x,\lambda,\mu) \end{align} What conditions on $f$, $h_i$'s, $g_j$'s, and $X$ will ensure that the dual function is smooth?


EDIT: To simplify this problem, are there any general classes of problems (for example, where $f,g,X$ are convex and $h$ is affine) such that the dual function is smooth?

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The dual function has to be strongly-convex and the overall program has to be convex, i.e. concave constraints (depending how you define them ofc.). You refer to a general concept that $1/L$-smooth functions becomes $L$-strongly convex in dual and vice versa. You can look this up in e.g. Techniques of Variational Analysis by Jonathan M. Borwein, Qiji Jim Zhu or many others.

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