Fenchel subdifferential operators: a characterization without cyclic monotonicity

Subdifferential of the supremum function: Moving back and forth between continuous and non-continuous settings

In this paper we develop general formulas for the subdifferential of the pointwise supremum of convex functions, which cover and unify both the compact continuous and the non-compact non-continuous settings. From the non-continuous to the continuous setting, we proceed by a compactification-based approach which leads us to problems having compact index sets and upper semi-continuously indexed mappings, giving rise to new characterizations of the subdifferential of the supremum by means of upper semicontinuous regularized functions and an enlarged compact index set. In the opposite sense, we re…

Weak subdifferentials, $r_L$-density and maximal monotonicity

In this paper, we first investigate an abstract subdifferential for which (using Ekeland's variational principle) we can prove an analog of the Brøndsted-Rockafellar property. We introduce the "$r_L$-density" of a subset of the product of a Banach space with its dual. A closed $r_L$-dense monotone set is maximally monotone, but we will also consider the case of nonmonotone closed $r_L$-dense sets. As a special case of our results, we can prove Rockafellar's result that the subdifferential of a proper convex lower semicontinuous function is maximally monotone.

Note on the Fenchel transform in the Heisenberg group

Given a real-valued function defined on the Heisenberg group, we provide a definition of abstract convexity and Fenchel transform that takes into account the sub-Riemannian structure of the group. In our main result, we prove that, likewise the Euclidean case, a convex function can be characterized via its iterated Fenchel transform; the properties of the H-subdifferential play a crucial role.

非线性 Fenchel 共轭

Nonlinear Fenchel Conjugates

本文将经典 Fenchel 共轭从线性空间推广到任意集合，用非线性测试函数替代线性测试函数，建立双共轭等性质，并在流形与 Lie 群上获得次微分、下确界卷积及凸性结果。

次微分理论中的障碍函数

Barrier functions in the subdifferential theory

本文提出基于障碍函数的新方法证明Correa-Jofré-Thibault定理，即次微分单调性蕴含函数凸性，并借助障碍函数克服处理下半连续函数时的技术困难。

Maximality of the sum of the subdifferential operator and a maximally monotone operator

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar's constraint qualification holds, which is called the "sum problem". In this paper, we establish the maximal monotonicity of $A+B$ provided that $A$ and $B$ are maximally monotone operators such that $\dom A\cap\inte\dom B\neq\varnothing$, and $A+N_{\overline{\dom A}}$ is of type (FPV). This generalizes various current results and also gives an affirmative answer to a problem posed by Borwein and Yao. Moreover, we p…

Intriguing maximally monotone operators derived from nonsunny nonexpansive retractions

Monotone operator theory and fixed point theory for nonexpansive mappings are central areas in modern nonlinear analysis and optimization. Although these areas are fairly well developed, almost all examples published are based on subdifferential operators, linear relations, or combinations thereof. In this paper, we construct an intriguing maximally monotone operator induced by a certain nonexpansive retraction. We analyze this operator, which does not appear to be assembled from subdifferential operators or linear relations, in some detail. Particular emphasis is placed on duality and strong …

An Answer to S. Simons' Question on the Maximal Monotonicity of the Sum of a Maximal Monotone Linear Operator and a Normal Cone Operator

The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar's constraint qualification - that is, whether or not "the sum theorem" is true - is the most famous open problem in Monotone Operator Theory. In his 2008 monograph "From Hahn-Banach to Monotonicity", Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal monotone linear operator and a normal cone operator of a closed convex set provided that the interior of the set makes a nonempty intersection with the domain of the linear operator. In this note, we …

多重共轭凸函数的极大单调性与循环对合性

Maximal monotonicity and cyclic involutivity of multi-conjugate convex functions

本文研究多个凸函数情形下循环对合性与极大单调性的成立条件，证明在一维实数直线上该性质保持成立，而在多维空间中一般需额外正则性假设，并通过实例揭示对偶与多重共轭凸函数及不同维度空间之间的本质差异。

积分预解式与近端混合

Integral Resolvent and Proximal Mixtures

本文基于希尔伯特直积分理论提出积分预解式混合与积分近端混合，将有限族单调算子和凸函数的混合推广到任意族，并应用于复合单调包含系统的松弛。

"Densities" and maximal monotonicity I

We discuss "Banach SN spaces", which include Hilbert spaces, negative Hilbert spaces, and the product of any real Banach space with its dual. We introduce "L-positive" sets, which generalize monotone multifunctions from a Banach space into its dual. We introduce the concepts of "$r_L$-density" and its specialization "quasidensity": the closed quasidense monotone multifunctions from a Banach space into its dual form a (generally) strict subset of the maximally monotone ones, though all surjective maximally monotone and all maximally monotone multifunctions on a reflexive space are quasidense. W…

General Resolvents for Monotone Operators: Characterization and Extension

Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood and analyzed from the alternative viewpoint of firmly nonexpansive mappings, which were found to be precisely the resolvents of monotone operators. For example, the proximal mappings in the sense of Moreau are precisely the resolvents of subdifferential operators. More general notions of "resolvent", "proximal mapping" and "firmly nonexpansive" have been stu…

Rockafellar约束规格下极大单调算子的非极大和

Nonmaximal sums of maximally monotone operators under Rockafellar's constraint qualification

本文构造了Rockafellar和猜想的反例，在$c_0$和$\ell^1$上给出两个极大单调算子满足内部域条件但和非极大的例子，并建立了计算单调极的构造定理。