Source Conditions for non-quadratic Tikhonov Regularisation

A Joint Tikhonov Regularization and Augmented Lagrange Approach for Ill-posed State Constrained Control Problems with Sparse Controls

We provide a modified augmented Lagrange method coupled with a Tikhonov regularization for solving ill-posed state-constrained elliptic optimal control problems with sparse controls. We consider a linear quadratic optimal control problem without any additional $L^2$ regularization terms. The sparsity is guaranteed by an additional $L^1$ term. Here, the modification of the classical augmented Lagrange method guarantees us uniform boundedness of the multiplier that corresponds to the state constraints. We present a coupling between the regularization parameter introduced by the Tikhonov regulari…

A globally convergent and locally quadratically convergent modified B-semismooth Newton method for $\ell_1$-penalized minimization

We consider the efficient minimization of a nonlinear, strictly convex functional with $\ell_1$-penalty term. Such minimization problems appear in a wide range of applications like Tikhonov regularization of (non)linear inverse problems with sparsity constraints. In (2015 Inverse Problems (31) 025005), a globalized Bouligand-semismooth Newton method was presented for $\ell_1$-Tikhonov regularization of linear inverse problems. Nevertheless, a technical assumption on the accumulation point of the sequence of iterates was necessary to prove global convergence. Here, we generalize this method to …

Modified Tikhonov regularization for identifying several sources

We study whether a modified version of Tikhonov regularization can be used to identify several local sources from Dirichlet boundary data for a prototypical elliptic PDE. This paper extends the results presented in [5]. It turns out that the possibility of distinguishing between two, or more, sources depends on the smoothing properties of a second or fourth order PDE. Consequently, the geometry of the involved domain, as well as the position of the sources relative to the boundary of this domain, determines the identifiability. We also present a uniqueness result for the identification of a si…

连续时间线性二次强化学习中熵正则化器的最优调度

Optimal scheduling of entropy regulariser for continuous-time linear-quadratic reinforcement learning

本研究基于熵正则化松弛随机控制框架，针对具有未知漂移系数的连续时间线性二次强化学习问题，提出通过调整松弛策略采样频率与熵正则化强度，证明两种算法的遗憾达 $\mathcal{O}(\sqrt{N})$ 阶，匹配已知最佳结果。

Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference

Operator inference learns low-dimensional dynamical-system models with polynomial nonlinear terms from trajectories of high-dimensional physical systems (non-intrusive model reduction). This work focuses on the large class of physical systems that can be well described by models with quadratic nonlinear terms and proposes a regularizer for operator inference that induces a stability bias onto quadratic models. The proposed regularizer is physics informed in the sense that it penalizes quadratic terms with large norms and so explicitly leverages the quadratic model form that is given by the und…

Multiscale Linear-Quadratic Stochastic Optimal Control With Multiplicative Noise

We investigate the asymptotic properties of a finite-time horizon linear-quadratic optimal control problem driven by a multiscale stochastic process with multiplicative Brownian noise. We approach the problem by considering the associated differential Riccati equation and reformulating it as a classical and deterministic singular perturbation problem. Asymptotic properties of this deterministic problem can be gathered from the well-known Tikhonov Theorem. Consequently, we are able to propose two approximation methods to the value function of the stochastic optimal control problem. The first is…

具有能量正则化的狄利克雷控制问题，由非 coercive 偏微分方程所 governed

Dirichlet control problems with energy regularization governed by non-coercive elliptic equations

本文研究了非凸多边形域上非 coercive 偏微分方程的线性二次狄利克雷控制问题，通过能量正则化和分级网格分析，得出了最优误差估计。

On the p-regularized trust region subproblem

The $p$-regularized subproblem (p-RS) is a regularisation technique in computing a Newton-like step for unconstrained optimization, which globally minimizes a local quadratic approximation of the objective function while incorporating with a weighted regularisation term $\fracσ{p} \|x\|^p$. The global solution of the $p$-regularized subproblem for $p=3$, also known as the cubic regularization, has been characterized in literature. In this paper, we resolve both the global and the local non-global minimizers of (p-RS) for $p>2$ with necessary and sufficient optimality conditions. Moreover, we p…

一种用于非线性规划问题的共轭梯度方法：使用Caputo分数梯度

A Conjugate Gradient Method for Nonlinear Programming Problems using Caputo Fractional Gradients

本文提出了一种基于Caputo分数梯度的共轭梯度方法，用于解决非线性规划问题，通过改进的线搜索技术和正则化方法，提升了优化效率和稳定性。

非凸最小化下无界方差中的下界与近似锚定SGD

Lower Bounds and Proximally Anchored SGD for Non-Convex Minimization Under Unbounded Variance

本文研究了非凸优化中无界方差条件下SGD变种的下界，并提出PASTA算法在多种非凸情形下达到最优复杂度。