arXiv subjects

Nils-Christian Kempke

Publications and source records attributed to Nils-Christian Kempke.

7 recordsLinked to original sources

arXiv2026

CHAP: A Hybrid GPU-CPU Heuristic for MIP

We present CHAP (Coordinating Heuristics Across Platforms) a GPU-CPU-hybrid primal heuristic framework for mixed-integer programming. CHAP adopts a portfolio approach where it coordinates a set of primal heuristics, including Local Search, Fix-and-Propagate, and Feasibility Pump, via a shared solution pool. The solution pool is used to exchange feasible incumbent solutions, LP solutions, along with promising infeasible solution candidates, enabling a more comprehensive exploration of the solution space. On the GPU side, we implement a native tabu search featuring a novel best-shift algorithm built on sort, scan, and reduce primitives, along with specialized kernel designs. We additionally leverage cuPDLPx as an approximate LP solver. On the CPU side, we employ various Fix-and-Propagate strategies, guided by information from the solution pool, complemented by a CPU-based tabu search and a Feasibility Pump. All components operate collaboratively, iteratively repairing and improving candidate solutions maintained in the pool. We evaluate our framework on the 50-instances benchmark from the 2026 Land-Doig MIP Competition under competition constraints, including a five-minute time limit. In these settings, CHAP finds solutions to 47 instances outperforming both Gurobi (44) in default mode and NVIDIA cuOpt (43) in heuristics-only mode. The results demonstrate that coordinated cross-platform portfolios offer a promising direction for the integration of GPU heuristics into modern high-performance MIP solvers. The code will be made available on GitHub.

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arXiv2026

A Massively Parallel Interior-Point Method for Arrowhead Linear Programs with Local Linking Structure

In practice, non-specialized interior point algorithms often cannot utilize the massively parallel compute resources offered by modern many- and multi-core compute platforms. However, efficient distributed solution techniques are required, especially for large-scale linear programs. This article describes a new decomposition technique for systems of linear equations implemented in the parallel interior-point solver PIPS-IPM++. The algorithm exploits a matrix structure commonly found in optimization problems: a doubly-bordered block-diagonal or arrowhead structure. This structure is preserved in the linear KKT systems solved during each iteration of the interior-point method. We present a hierarchical Schur complement decomposition that distributes and solves the linear optimization problem; it is designed for high-performance architectures and scales well with the availability of additional computing resources. The decomposition approach uses the border constraints' locality to decouple the factorization process. Our approach is motivated by large-scale unit commitment problems. We demonstrate the performance of our method on a set of mid-to large-scale instances, some of which have more than 10^9 nonzeros in their constraint matrix.

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arXiv2026

GPU accelerated variant of Schroeppel-Shamir's algorithm for solving the market split problem

The market split problem (MSP), introduced by Cornuejols and Dawande (1998), is a challenging binary optimization problem that performs poorly on state-of-the-art linear programming-based branch-and-cut solvers. We present a novel algorithm for solving the feasibility version of this problem, derived from Schroeppel-Shamir's algorithm for the one-dimensional subset sum problem. Our approach is based on exhaustively enumerating one-dimensional solutions of MSP and utilizing GPUs to evaluate candidate solutions across the entire problem. The resulting hybrid CPU-GPU implementation efficiently solves instances with up to 10 constraints and 90 variables. We demonstrate the algorithm's performance on benchmark problems, solving instances of size (9, 80) in less than fifteen minutes and (10, 90) in up to one day.

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arXiv2026

Fix-and-Propagate Heuristics Using Low-Precision First-Order LP Solutions for Large-Scale Mixed-Integer Linear Optimization

We investigate the use of low-precision first-order methods (FOMs) within a fix-and-propagate (FP) framework for solving mixed-integer programming problems (MIPs). We employ GPU-accelerated PDLP, a variant of the Primal-Dual Hybrid Gradient (PDHG) method specialized to LP problems, to solve the \LP-relaxation of our MIPs to low accuracy. This solution is used to motivate fixings within our FP framework. We evaluate the performance of our heuristic on MIPLIB 2017, demonstrating that low-accuracy LP solutions do not lead to a loss in the quality of the FP heuristic solutions. Further, we use our FP framework to produce high-accuracy solutions for large-scale (up to 243 million nonzeros and 8 million decision variables) unit commitment-based dispatch and expansion planning problems created with the modeling framework REMix. For the largest problems, we can generate solutions with a primal-dual gap of under 2% in less than 4 hours, whereas state-of-the-art commercial solvers cannot produce feasible solutions within 2 days of runtime.

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arXiv2026

Distributed Parallel Structure-Aware Presolving for Arrowhead Linear Programs

We present a structure-aware parallel presolve framework specialized to arrowhead linear programs (AHLPs) and designed for high-performance computing (HPC) environments, integrated into the parallel interior point solver PIPS-IPM++. Large-scale LPs arising from automated model generation frequently contain redundancies and numerical pathologies that necessitate effective presolve, yet existing presolve techniques are primarily serial or structure-agnostic and can become time-consuming in parallel solution workflows. Within PIPS-IPM++, AHLPs are stored in distributed memory, and our presolve builds on this to apply a highly parallel, distributed presolve across compute nodes while keeping communication overhead low and preserving the underlying arrowhead structure. We demonstrate the scalability and effectiveness of our approach on a diverse set of AHLPs and compare it against state-of-the-art presolve implementations, including PaPILO and the presolve implemented within Gurobi. Even on a single machine, our presolve significantly outperforms PaPILO by a factor of 18 and Gurobi's presolve by a factor of 6 in terms of shifted geometric mean runtime, while reducing the problems by a similar amount to PaPILO. Using a distributed compute environment, we outperform Gurobi's presolve by a factor of 13.

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arXiv2025

Developing heuristic solution techniques for large-scale unit commitment models

Shifting towards renewable energy sources and reducing carbon emissions necessitate sophisticated energy system planning, optimization, and extension. Energy systems optimization models (ESOMs) often form the basis for political and operational decision-making. ESOMs are frequently formulated as linear (LPs) and mixed-integer linear (MIP) problems. MIPs allow continuous and discrete decision variables. Consequently, they are substantially more expressive than LPs but also more challenging to solve. The ever-growing size and complexity of ESOMs take a toll on the computational time of state-of-the-art commercial solvers. Indeed, for large-scale ESOMs, solving the LP relaxation -- the basis of modern MIP solution algorithms -- can be very costly. These time requirements can render ESOM MIPs impractical for real-world applications. This article considers a set of large-scale decarbonization-focused unit commitment models with expansion decisions based on the REMix framework (up to 83 million variables and 900,000 discrete decision variables). For these particular instances, the solution to the LP relaxation and the MIP optimum lie close. Based on this observation, we investigate the application of relaxation-enforced neighborhood search (RENS), machine learning guided rounding, and a fix-and-propagate (FP) heuristic as a standalone solution method. Our approach generated feasible solutions 20 to 100 times faster than GUROBI, achieving comparable solution quality with primal-dual gaps as low as 1% and up to 35%. This enabled us to solve numerous scenarios without lowering the quality of our models. For some instances that GUROBI could not solve within two days, our \FP method provided feasible solutions in under one hour.

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arXiv2019

First Experiments with Structure-Aware Presolving for a Parallel Interior-Point Method

In linear optimization, matrix structure can often be exploited algorithmically. However, beneficial presolving reductions sometimes destroy the special structure of a given problem. In this article, we discuss structure-aware implementations of presolving as part of a parallel interior-point method to solve linear programs with block-diagonal structure, including both linking variables and linking constraints. While presolving reductions are often mathematically simple, their implementation in a high-performance computing environment is a complex endeavor. We report results on impact, performance, and scalability of the resulting presolving routines on real-world energy system models with up to 700 million nonzero entries in the constraint matrix.

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