On Discrete Subproblems in Integer Optimal Control with Total Variation Regularization in Two Dimensions

TL;DR

The paper analyzes two-dimensional discretized trust-region subproblems arising in integer optimal control with total variation regularization. It establishes structural properties of the underlying polyhedron, connects the discretized problem to graph problems (notably MBFH and s-t cuts), and proves a strong NP-hardness connection under a grid-graph reduction. Leveraging these insights, the authors derive cutting planes, a primal heuristic, and branching rules, and validate them with solver-based experiments that show significant speedups for medium-sized instances. The results point toward scalable approaches and future work on sharper cuts, decomposition, and higher-dimensional extensions. Overall, the work advances both the theoretical understanding and practical solution of two-dimensional IOCP subproblems with TV penalties.

Abstract

We analyze integer linear programs which we obtain after discretizing two-dimensional subproblems arising from a trust-region algorithm for mixed integer optimal control problems with total variation regularization. We discuss NP-hardness of the discretized problems and the connection to graph-based problems. We show that the underlying polyhedron exhibits structural restrictions in its vertices with regards to which variables can attain fractional values at the same time. Based on this property, we derive cutting planes by employing a relation to shortest-path and minimum bisection problems. We propose a branching rule and a primal heuristic which improves previously found feasible points. We validate the proposed tools with a numerical benchmark in a standard integer programming solver. We observe a significant speedup for medium-sized problems. Our results give hints for scaling towards larger instances in the future.

Figures (9)

• Figure 1: Example of an underlying grid with $N=3$ and $M=4$. For each node there exists a corresponding entry in the control $x+d$. For each edge there exists a corresponding absolute value term (horizontal: $\beta_{i,j}$, vertical: $\gamma_{i,j}$ in Subsection \ref{['subsec:Ip form']}) modeling the contribution of the control jump between neighbouring cells to the total variation.
• Figure 2: The original graph consisted of $6$ nodes. Each node becomes one square of $36 \times 36$ nodes in $V_1$. The set $V_3$ contains the black nodes. The set $V_1$ is illustrated in dark grey. Two squares are connected by a node in $V_2$ if the corresponding nodes in the original graph were connected. The set $V_2$ contains the light grey nodes on the interfaces between these squares of nodes of $V_1$.
• Figure 3: We use a shortest-path approach to determine the optimal control in each node in the order $1$ to $12$. To guarantee optimality we need to take the control value in node $1$ into account when choosing the control value in node $4$ in order to accurately model the red edge between the nodes. Thus we need to encode the last $\min\{N,M\}=3$ control values in a graph construction which means the size grows exponentially in $\min\{N,M\}$. If all row edges are ignored, the resulting problem can be solved by a pseudo-polynomial algorithm in the same manner as the one-dimensional \ref{['eq:trip']}.
• Figure 4: Visualization of the connection between the capacity used on the fractional component and the amount of jumps (cut edges) in the fully connected graph. For the example visualized above the values are given by $\Delta=8$, $\Delta_r = 3$ and $|G|=10$. We can see that $\tilde{\rho}$ and $M$ are chosen such that the amount of jumps is always affinely underestimated.
• Figure 5: Performance plots for $N=64$ (left) and $N=96$ (right). The plots visualize what fraction of the instances are solved after a given time. For $N=64$ nearly all instances are solved after $600$ seconds while for $N=96$ a significant number of instances are not solved within the time limit of $1$ hour.

Theorems & Definitions (46)

• Remark 1
• Lemma 1
• Theorem 1
• proof
• Corollary 1
• proof
• Theorem 2: Conditional $p$-approximation
• proof
• Remark 2
• Corollary 2