AI 中文总结
该研究为赋范空间上扩展实值函数引入范数锥共轭方案，建立超越凸性的Fenchel型对偶理论，证明相关不等式，定义范数锥次微分，发展抽象扰动对偶理论，在无凸性假设下实现弱对偶，由度量下界条件推出强对偶。
AI 中文摘要
我们为赋范空间上的扩展实值函数引入了一种范数锥共轭方案。该构造用形如\(x\mapsto r-\alpha\|x - x_0\|\)（\(\alpha\geq0\)）的平移范数锥取代仿射次小函数，并建立了一个超越凸性的Fenchel型对偶理论的非线性共轭框架。得到的共轭由斜率和中心索引，相关的双共轭是位于函数下方的所有范数锥次小函数的上确界。我们证明了Fenchel-Young型不等式，引入了可允许斜率和可允许高度，并根据范数锥可支撑性刻画了精确双共轭。我们还定义了范数锥次微分，并将其与精确支撑和双共轭联系起来。最后，我们基于扰动变量中的部分范数锥共轭发展了一种抽象扰动对偶理论。在没有凸性假设的情况下弱对偶性成立，而强对偶性则由度量下界条件推出，包括一致的下Lipschitz估计和值函数的下平静性。
英文摘要
We develop a norm-cone conjugation framework, generated by translated norm-cones \[ x\mapsto r-α\|x-x_0\|, \qquad α\geq 0. \] We show that this family generates the same abstract-convex class as the family of Lipschitz continuous concave functions, and exploit its explicit metric structure to obtain a concrete support geometry, a Fenchel--Moreau-type biconjugation theorem, and an associated norm-cone subdifferential. The main optimization consequence is a perturbation-duality theory based on partial norm-cone conjugation. Although the partial conjugate depends on a slope and a centre, the centre can be fixed at the nominal perturbation in the complete biconjugate expression without changing its value. Consequently, the effective dual problem is a scalar program whose only dual variable is the real slope $α\geq0$, even when the primal and perturbation spaces are vector-valued or infinite-dimensional. The resulting dual value is the norm-cone biconjugate of the value function at the nominal perturbation, so that strong duality is characterized by exact pointwise biconjugation. For nonconvex conic inequality problems, exact distance penalization is characterized by norm-cone supportability of the natural value function: the exact penalty parameters form its norm-cone subdifferential, and the least exact parameter is the minimal support slope. Global error bounds, equivalently global metric subregularity, provide a variational-analytic sufficient condition for exactness and hence for strong norm-cone duality. An infinite-dimensional example shows that the results remain applicable with a nonconvex objective and a nonsolid ordering cone.
CommentsSubstantially revised and expanded version. The duality framework and its relation to exact distance penalization have been significantly developed
