Integer optimal control problems with total variation regularization: Optimality conditions and fast solution of subproblems

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS

Esaim-Control Optimisation and Calculus of Variations Pub Date : 2023-01-01 DOI:10.1051/cocv/2023065

Jonas Marko, Gerd Wachsmuth

引用次数: 3

Abstract

We investigate local optimality conditions of first and second order for integer optimal control problems with total variation regularization via a finite-dimensional switching-point problem. We show the equivalence of local optimality for both problems, which will be used to derive conditions concerning the switching points of the control function. A non-local optimality condition treating back-and-forth switches will be formulated. For the numerical solution, we propose a proximal-gradient method. The emerging discretized subproblems will be solved by employing Bellman’s optimality principle, leading to an algorithm which is polynomial in the mesh size and in the admissible control levels. An adaption of this algorithm can be used to handle subproblems of the trust-region method proposed in Leyffer and Manns, ESAIM: Control Optim. Calc. Var. 28 (2022) 66. Finally, we demonstrate computational results.

查看原文本刊更多论文

全变分正则化整数最优控制问题:子问题的最优性条件和快速解

通过有限维切换点问题，研究了具有全变分正则化的整数最优控制问题的一阶和二阶局部最优性条件。我们证明了这两个问题的局部最优性的等价性，这将用于导出关于控制函数的开关点的条件。一个处理来回切换的非局部最优性条件将被提出。对于数值解，我们提出了一种近似梯度法。采用Bellman最优性原理对出现的离散子问题进行求解，得到一个在网格大小和允许控制水平上都是多项式的算法。该算法的改进可用于处理Leyffer和Manns在ESAIM: Control Optim中提出的信任域方法的子问题。中国生物医学工程学报，28 (2022)最后，给出了计算结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics

自引率

7.10%

发文量

期刊介绍： ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.