We investigate an extension of Mixed-Integer Optimal Control Problems by adding switching costs, which enables the penalization of chattering and extends current modeling capabilities. The decomposition approach, consisting of solving a partial outer convexification to obtain a relaxed solution and using rounding schemes to obtain a discrete-valued control can still be applied, but the rounding turns out to be difficult in the presence of switching costs or switching constraints as the underlying problem is an Integer Program. We therefore reformulate the rounding problem into a shortest path problem on a parameterized family of directed acyclic graphs (DAGs). Solving the shortest path problem then allows to minimize switching costs and still maintain approximability with respect to the tunable DAG parameter $$\theta $$ . We provide a proof of a runtime bound on equidistant rounding grids, where the bound is linear in time discretization granularity and polynomial in $$\theta $$ . The efficacy of our approach is demonstrated by a comparison with an integer programming approach on a benchmark problem.

中文翻译：

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具有切换成本的混合整数最优控制问题：一种最短路径方法

我们通过增加转换成本来研究混合整数最优控制问题的扩展，这可以惩罚抖动并扩展当前的建模能力。分解方法，包括求解部分外凸化以获得松弛解并使用舍入方案获得离散值控制仍然可以应用，但在存在切换成本或切换约束的情况下，舍入变得困难，如根本问题是整数程序。因此，我们将舍入问题重新表述为参数化有向无环图 (DAG) 族上的最短路径问题。解决最短路径问题然后允许最小化切换成本并且仍然保持关于可调 DAG 参数 $$\theta $$ 的近似性。我们提供了等距舍入网格上的运行时边界的证明，其中边界在时间离散化粒度和 $$\theta $$ 中的多项式上是线性的。通过在基准问题上与整数规划方法进行比较，证明了我们方法的有效性。