arXiv subjects

Marvin Severitt

Publications and source records attributed to Marvin Severitt.

4 recordsLinked to original sources

arXiv2026

Randomization and Performance Improvement for Integer Optimal Control with Total Variation Regularization

Mixed-integer PDE-constrained optimization problems are computationally challenging due to both the combinatorial nature of integer programming as well as evaluation of the model. Many algorithms and subsequent performance improvements have been developed to solve these problems, but they are often limited by problem size. We numerically analyze two such algorithms: SLIP and Patch-SLIP, which solve trust-region subproblems over either the full or partial domain, respectively. We additionally propose and prove convergence of a randomized third algorithm, Randomized-Patch-SLIP, which solves trust-region subproblems over randomly selected patches of the domain. We compare performance of all three algorithms with various improvement techniques found throughout the literature; the purpose of this work is to document the best combinations of these improvements in conjunction with various solvers. We additionally establish benchmark problems in image denoising and cloaking. Computational results are reported on combinations of algorithms and improvements, and code used in the experiments is provided as a package.

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arXiv2025

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

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.

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arXiv2024

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

We revisit a class of integer optimal control problems for which a trust-region method has been proposed and analyzed in arXiv:2106.13453v3 [math.OC]. While the algorithm proposed in arXiv:2106.13453v3 [math.OC] successfully solves the class of optimization problems under consideration, its convergence analysis requires restrictive regularity assumptions. There are many examples of integer optimal control problems involving partial differential equations where these regularity assumptions are not satisfied. In this article we provide a way to bypass the restrictive regularity assumptions by introducing an additional partial regularization of the control inputs by means of mollification and proving a $Γ$-convergence-type result when the support parameter of the mollification is driven to zero. We highlight the applicability of this theory in the case of fluid flows through deformable porous media equations that arise in biomechanics. We show that the regularity assumptions are violated in the case of poro-visco-elastic systems, and thus one needs to use the regularization of the control input introduced in this article. Associated numerical results show that while the homotopy can help to find better objective values and points of lower instationarity, the practical performance of the algorithm without the input regularization may be on par with the homotopy.

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arXiv2022

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

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 $A^*$ algorithm. Specifically, the analysis of a Lagrangian relaxation yields a consistent heuristic function for the $A^*$ algorithm and a preprocessing procedure, which can be employed to accelerate the $A^*$ 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 $A^*$ algorithm and topological sorting outperform a general purpose IP solver significantly. Moreover, the accelerated $A^*$ algorithm is able to outperform topological sorting for larger problem instances.

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