Second-order conditions for non-uniformly convex integrands: quadratic growth in \(L^1\) (Q6566606)

Consider the following optimization problems:\N\[\N\text{Minimize }F(u)+G(u)\quad \text{w.r.t. }u\in L(\lambda ),\N\]\Nwhere \(F:\operatorname*{dom}(G)\rightarrow \mathbb{R}\) is smooth and \(G(u)=\int_{\Omega}g(u)d\lambda\) is defined for some convex but possibly non-smooth \(g: \mathbb{R} \rightarrow \mathbb{R} \cup \left \{ \infty \right \} \). Here, \(\Omega \subset \mathbb{R}^{d}\)is a non-empty, open, and bounded set and \(\lambda\) is the Lebesgue measure.\N\NThe authors, under some assumptions, on the problem data and the potential minimizer \(\overline{u}\in L^{1}(\lambda )\), derived the following second-order condition\N\[\NF^{\prime \prime}(\overline{u})\mu^{2}+G^{\prime \prime}(\overline{u},-F^{\prime}(\overline{u}),\mu )\geq 0\qquad \forall \mu \in \mathcal{M}(\Omega )=(C_{0}(\Omega ))^{\star},\N\]\Nas necessary for local optimality, while the second-order condition\N\[\NF^{\prime \prime}(\overline{u})\mu^{2}+G^{\prime \prime}(\overline{u},-F^{\prime}(\overline{u}),\mu )\geq 0\qquad \forall \mu \in \mathcal{M}(\Omega )\backslash \left \{ 0\right \} ,\N\]\Nis sufficient for local optimality. Here \(F^{\prime \prime}(\overline{u})\) is a weak-\(\star\) continuous and quadratic form which allows for a second-order Taylor-like expansion of \(F\) while \(G^{\prime \prime}(\overline{u},-F^{\prime}(\overline{u}),\cdot )\) is the second subderivative of \(G\). Also, the authors obtain the expression for \(G^{\prime \prime}\). In the main results of the paper, the authors provide no-gap second-order conditions and strictly twice epi-differentiability of \(G^{\prime \prime}\). The paper concludes with a presentation of two applications.