Local Nonconvexity Indices for \(C^{1,1}\) Functions via Generalized Hessians
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911664626991104 |
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| author | Palaisti, Marina |
| author_facet | Palaisti, Marina |
| contents | Davydov, Moldavskaya, and Zitikis introduced local indices for quantifying the lack of convexity of a \(C^2\) function by measuring the nuclear-norm distance of its Hessian from the cone of positive semidefinite matrices. This paper develops a local analogue for functions of class \(C^{1,1}\). At a point \(x\), the classical Hessian is replaced by the Clarke-type generalized Hessian set \(\Hess(h;x)\), defined as the closed convex hull of limiting Hessians at nearby twice differentiability points. Evaluating the same spectral functional over \(\Hess(h;x)\) gives an interval-valued local nonconvexity index whose lower and upper endpoints represent, respectively, the least and greatest visible second-order nonconvexity at \(x\). The construction reduces to the original smooth index when \(h\in C^2\), vanishes for convex \(C^{1,1}\) functions, is invariant under orthogonal changes of variables, satisfies a subadditivity inequality for the upper endpoint under sums, and is upper semicontinuous in its upper endpoint. We also relate the upper endpoint to a pointwise weak-convexity curvature modulus and give explicit \(C^{1,1}\setminus C^2\) examples. The paper is deliberately local in scope: it proposes a scalar diagnostic extracted from generalized Hessian sets, not a replacement for the richer second-order variational theory of nonsmooth convexity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_08491 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Nonconvexity Indices for \(C^{1,1}\) Functions via Generalized Hessians Palaisti, Marina Optimization and Control Classical Analysis and ODEs 49J52, 49J53, 90C30, 26B25 Davydov, Moldavskaya, and Zitikis introduced local indices for quantifying the lack of convexity of a \(C^2\) function by measuring the nuclear-norm distance of its Hessian from the cone of positive semidefinite matrices. This paper develops a local analogue for functions of class \(C^{1,1}\). At a point \(x\), the classical Hessian is replaced by the Clarke-type generalized Hessian set \(\Hess(h;x)\), defined as the closed convex hull of limiting Hessians at nearby twice differentiability points. Evaluating the same spectral functional over \(\Hess(h;x)\) gives an interval-valued local nonconvexity index whose lower and upper endpoints represent, respectively, the least and greatest visible second-order nonconvexity at \(x\). The construction reduces to the original smooth index when \(h\in C^2\), vanishes for convex \(C^{1,1}\) functions, is invariant under orthogonal changes of variables, satisfies a subadditivity inequality for the upper endpoint under sums, and is upper semicontinuous in its upper endpoint. We also relate the upper endpoint to a pointwise weak-convexity curvature modulus and give explicit \(C^{1,1}\setminus C^2\) examples. The paper is deliberately local in scope: it proposes a scalar diagnostic extracted from generalized Hessian sets, not a replacement for the richer second-order variational theory of nonsmooth convexity. |
| title | Local Nonconvexity Indices for \(C^{1,1}\) Functions via Generalized Hessians |
| topic | Optimization and Control Classical Analysis and ODEs 49J52, 49J53, 90C30, 26B25 |
| url | https://arxiv.org/abs/2605.08491 |