Agnieszka B. Malinowska, Delfim F. M. Torres
We prove Euler-Lagrange and natural boundary necessary optimality conditions for problems of the calculus of variations which are given by a composition of nabla integrals on an arbitrary time scale. As an application, we get optimality conditions for the product and the quotient of nabla variational functionals.
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We propose a variable metric forward-backward splitting algorithm and prove its convergence in real Hilbert spaces. We then use this framework to derive primal-dual splitting algorithms for solving various classes of monotone inclusions in duality. Some of these algorithms are new even when specialized to the fixed metric case. Various applications are discussed.
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}.
In this note we show how one can obtain results from the nabla calculus from results on the delta calculus and vice versa via a duality argument. We provide applications of the main results to the calculus of variations on time scales.
We introduce a framework for quasi-Newton forward--backward splitting algorithms (proximal quasi-Newton methods) with a metric induced by diagonal $\pm$ rank-$r$ symmetric positive definite matrices. This special type of metric allows for a highly efficient evaluation of the proximal mapping. The key to this efficiency is a general proximal calculus in the new metric. By using duality, formulas are derived that relate the proximal mapping in a rank-$r$ modified metric to the original metric. We also describe efficient implementations of the proximity calculation for a large class of functions;…
The Duality Theory of Fractional Calculus and a New Fractional Calculus of Variations Involving Left Operators Only
本文利用对偶性建立仅含左或右算子的分数阶分部积分公式，并据此提出只涉及单侧分数阶导数的变分法，进而从仅含左导数的拉格朗日量导出耗散系统运动方程。
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Duality Theory on Vector Spaces
本文在无拓扑向量空间中利用几何方法建立共轭演算规则，并给出 Fenchel-Rockafellar 与 Lagrange 强对偶及凸优化最优解的条件。
Numerical Methods and Analysis via Random Field Based Malliavin Calculus for Backward Stochastic PDEs
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We show how a linear control systems theory for the backward nabla differential operator on an arbitrary time scale can be obtained via Caputo's duality. More precisely, we consider linear control systems with outputs defined with respect to the backward jump operator. Kalman criteria of controllability and observability, as well as realizability conditions, are proved.
This article develops duality principles applicable to non-convex models in the calculus of variations. The results here developed are applied to Ginzburg-Landau type equations. For the first and second duality principles, through an optimality criterion developed for the dual formulations, we qualitatively classify the critical points of the primal and dual functionals in question. We formally prove there is no duality gap between the primal and dual formulations in a local extremal context. Finally, in the last sections, we present a global existence result, a duality principle and respectiv…
Bilateral facial reduction: qualification-free subdifferential calculus and exact duality
本文提出了一种无资格条件的双侧面部缩减方法，用于解决一般凸优化问题中的对偶性、最优性条件和子微分计算。
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