Efficient Solution of Discrete Subproblems Arising in Integer Optimal Control with Total Variation Regularization

Abstract

We consider a class of integer linear programs (IPs) that arise as discretizations of trust-region subproblems of a trust-region algorithm for the solution of control problems, where the control input is an integer-valued function on a one-dimensional domain and is regularized with a total variation term in the objective, which may be interpreted as a penalization of switching costs between different control modes. We prove that solving an instance of the considered problem class is equivalent to solving a resource-constrained shortest-path problem (RCSPP) on a layered directed acyclic graph. This structural finding yields an algorithmic solution approach based on topological sorting and corresponding run-time complexities that are quadratic in the number of discretization intervals of the underlying control problem, the main quantifier for the size of a problem instance. We also consider the solution of the RCSPP with an [Formula: see text] algorithm. Specifically, the analysis of a Lagrangian relaxation yields a consistent heuristic function for the [Formula: see text] algorithm and a preprocessing procedure, which can be employed to accelerate the [Formula: see text] algorithm for the RCSPP without losing optimality of the computed solution. We generate IP instances by executing the trust-region algorithm on several integer optimal control problems. The numerical results show that the accelerated [Formula: see text] algorithm and topological sorting outperform a general-purpose IP solver significantly. Moreover, the accelerated [Formula: see text] algorithm is able to outperform topological sorting for larger problem instances. We also give computational evidence that the performance of the superordinate trust-region algorithm may be improved if it is initialized with a solution obtained with the combinatorial integral approximation. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms–Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2023.1294 .

Input Regularization for Integer Optimal Control in BV with Applications to Control of Poroelastic and Poroviscoelastic Systems

“…In each inner iteration, a trust-region subproblem TR(𝑤 𝑛−1 , Δ 𝑛,𝑘 ) is solved for the current trust-region radius, Δ 𝑛,𝑘 , and the previously accepted iterate 𝑤 𝑛−1 or the input 𝑤 0 (if 𝑛 − 1 = 0). We highlight that the trust-region subproblems become integer linear programs after discretization, see [40], which can be solved to optimality with a pseudo-polynomial algorithm as detailed in [45,53]. This is a deviation from the standard literature, where the convergence theory is developed using a Cauchy point which only guarantees a sufficient decrease and not optimality for the trust-region subproblem, see for example chapter 12 in [19].…”

“…Algorithm 1 is implemented in MATLAB. C++ is used for the subproblem solver implementation, which follows [53]. All computations were executed on a workstation with an AMD Epic 7742 CPU and 96 GB RAM.…”

“…Therein, the non-smoothness of the subproblems that arises from the distributed integer variables is handled explicitly so that the trust-region subproblems are integer linear programs after discretization. They can be solved efficiently with graph-based [53] and dynamic programming-based approaches [45]. For one-dimensional time domains (0,𝑇 ), like the one in (P), the TV-term of a 𝑊 -valued function 𝑤 is the sum of the jump heights of the function 𝑤, implying that only finitely many jumps occur because the height of a single jump is bounded below (by 1 in the case of 𝑊 ⊂ ℤ; always by some constant if |𝑊 | < ∞).…”