Variational properties of value functions (Q2866203)

The authors consider the family of feasible convex optimization problems NEWLINE\[NEWLINEP(b,\tau):\quad\underset{r,x}{}{\text{minimize}}\,\rho\quad\text{subject to}\quad Ax+r= b,\quad\phi(x)\leq\tau,NEWLINE\]NEWLINE where \(b\in\mathbb{R}^m\), \(A\in\mathbb{R}^{m\times n}\), and the functions \(\phi:\mathbb{R}^n\to\overline{\mathbb{R}}:= (-\infty,\infty)\) and \(\rho: \mathbb{R}^m\to\overline{\mathbb{R}}\) are closed, proper, and convex and continuous relative to their domains and the value function NEWLINE\[NEWLINEv(b,r):= \underset{r,x}{}{\text{inf}}\,\{\rho\mid Ax+ r= b,\, \phi(x)\leq\tau\}NEWLINE\]NEWLINE gives the optimal objective value of problem \(P(b,\tau)\) for fixed parameters \(b\) and \(\tau\).NEWLINENEWLINE The authors characterize the variational properties of the value functions for a broad class of convex formulations, which are not all covered by standard Lagrange multiplier theory. An inverse function theorem is given that links the value functions of different regularization. -- Numerical examples illustrate the theoretical results.