First Integrals for Problems of Calculus of Variations on Locally Convex Spaces

Linear Monotone Subspaces of Locally Convex Spaces

The main focus of this paper is to study multi-valued linear monotone operators in the contexts of locally convex spaces via the use of their Fitzpatrick and Penot functions. Notions such as maximal monotonicity, uniqueness, negative-infimum, and (dual-) representability are studied and criteria are provided.

A duality theory for non-convex problems in the Calculus of Variations

We present a new duality theory for non-convex variational problems, under possibly mixed Dirichlet and Neumann boundary conditions. The dual problem reads nicely as a linear programming problem, and our main result states that there is no duality gap. Further, we provide necessary and sufficient optimality conditions, and we show that our duality principle can be reformulated as a min-max result which is quite useful for numerical implementations. As an example, we illustrate the application of our method to a celebrated free boundary problem. The results were announced in \cite{BoFr}.

弱凸函数的邻近ε-次微分演算规则与不精确邻近算子

Calculus rules for proximal ε-subdifferentials and inexact proximity operators for weakly convex functions

本文研究弱凸函数的不精确邻近算子，通过将弱凸性模纳入公式推导出邻近ε-次微分的加法规则，从而基于邻近ε-次微分深入分析该算子。

Dual variational formulations for a large class of non-convex models in the calculus of variations

This article develops dual variational formulations for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis and duality theory. The main duality principle is developed as an application to a Ginzburg-Landau type system in superconductivity in the absence of a magnetic field. In the first part final sections, we develop new general dual convex variational formulations, more specifically, dual formulations with a large region of convexity around the critical points which are suitable for the non-convex optim…

变分凸性：新刻画、运算法则及应用

Variational convexity: new characterizations, calculus rules, and applications

本文提出变分凸函数的新刻画（通过近端包络和图），研究保持变分凸性的运算（非线性/线性复合、求和、近端平均），并将其应用于非线性规划问题，得到确保局部极小点而非仅驻点的新条件。

局部凸空间中的拉格朗日乘子

Lagrange Multipliers in locally convex spaces

本文研究了Hausdorff局部凸空间中含无穷多约束的数学规划问题，提出一般拉格朗日乘子规则，用Gateaux可微性替代Fréchet可微性，并引入“容许集”替代闭凸锥以推广现有结果。

Algebraic Lipschitz and Subdifferential Calculus in General Vector Spaces

The main contribution of this paper is that every convex function with non-empty relative algebraic interior of its domain is Lipschitz and subdifferentiable in some algebraic sense without any additional topological constraints. The proposed approach uses slightly modified Clarke's subdifferential for functions defined on a convex symmetric set and Lipschitz with respect to a Minkowski functional. Following this, Clarke's subdifferential calculus is generalized to vector spaces and, where continuity properties are claimed, to topological vector spaces.

局部凸拓扑向量空间中集值优化问题的若干最优性条件

Some optimality conditions of set-valued optimization problems in locally convex topological vector spaces

针对局部凸拓扑向量空间中的集值优化问题，证明不同学者所给广义凸映射定义的等价性，并给出弱有效解、真有效解与最优解的最优性条件。

Unique Minimizers and the Representation of Convex Envelopes in Locally Convex Vector Spaces

It is well known that a strictly convex minimand admits at most one minimizer. We prove a partial converse: Let $X$ be a locally convex Hausdorff space and $f \colon X \mapsto \left( - \infty , \infty \right]$ a function with compact sublevel sets and exhibiting some mildly superlinear growth. Then each tilted minimization problem \begin{equation} \label{eq. minimization problem} \min_{x \in X} f(x) - \langle x' , x \rangle_X \end{equation} admits at most one minimizer as $x'$ ranges over $\text{dom} \left( \partial f^* \right)$ if and only if the biconjugate $f^{**}$ is essentially strictly c…

Sequential Henig proper optimality conditions for multiobjective fractional programming problems via sequential proper subdifferential calculus

In this paper, in the absence of any constraint qualifications, we develop sequential necessary and sufficient optimality conditions for a constrained multiobjective fractional programming problem characterizing a Henig proper efficient solution in terms of the $ε$-subdifferentials and the subdifferentials of the functions. This is achieved by employing a sequential Henig subdifferential calculus rule of the sums of $m \ (m\geq 2)$ proper convex vector valued mappings with a composition of two convex vector valued mappings. In order to present an example illustrating the main results of this p…

Generalized Differentiation and Duality in Infinite Dimensions under Polyhedral Convexity

This paper addresses the study and applications of polyhedral duality of locally convex topological vector (LCTV) spaces. We first revisit the classical Rockafellar's proper separation theorem for two convex sets one which is polyhedral and then present its LCTV extension with replacing the relative interior by its quasi-relative interior counterpart. Then we apply this result to derive enhanced calculus rules for normals to convex sets, coderivatives of convex set-valued mappings, and subgradients of extended-real-valued functions under certain polyhedrality requirements in LCTV spaces by dev…

Piecewise Linear Vector Optimization Problems on Locally Convex Hausdorff Topological Vector Spaces

Piecewise linear vector optimization problems in a locally convex Hausdorff topological vector spaces setting are considered in this paper. The efficient solution set of these problems are shown to be the unions of finitely many semi-closed generalized polyhedral convex sets. If, in addition, the problem is convex, then the efficient solution set and the weakly efficient solution set are the unions of finitely many generalized polyhedral convex sets and they are connected by line segments. Our results develop the preceding ones of Zheng and Yang [Sci. China Ser. A. 51, 1243--1256 (2008)], and …

Explicit bounds for Lipschitz constant of solution to basic problem in calculus of variations

In this paper we present explicit estimate for Lipschitz constant of solution to a problem of calculus of variations. The approach we use is due to Gamkrelidze and is based on the equivalence of the problem of calculus of variations and a time-optimal control problem. The obtained estimate is used to compute complexity bounds for a path-following method applied to a convex problem of calculus of variations with polyhedral end-point constraints.