Inverse Optimization Techniques in Network Systems

Inverse optimisation in network systems addresses the problem of adjusting model parameters so that a prespecified solution attains optimality under the modified conditions. This paradigm inverts the traditional optimisation workflow by treating a feasible flow, cut or path as the target outcome and seeking minimal changes—measured by suitable distance metrics—to link capacities, costs or weights. Methods range from exact polynomial-time algorithms for special cases to heuristic and approximation schemes for NP-hard formulations. Recent advances have explored variants with gain and loss factors on arcs, multi-weight objectives across distinct network problems and the handling of both sum-type and max-type norms for parameter deviation. Applications span transportation planning, telecommunications traffic engineering, energy distribution and logistics, where practitioners can infer cost or capacity settings consistent with observed network behaviour. Emerging research is emphasising robust and data-driven extensions, integrating statistical learning to estimate model coefficients from historical operations. Overall, inverse optimisation offers a unifying framework for network calibration, sensitivity analysis and prescriptive decision support under practical budget or resource constraints.

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Research from all publishers

Work on the inverse generalised maximum flow problem has introduced efficient feasibility checks and strongly polynomial-time routines for cases governed by bottleneck-type distances, while proposing heuristic approaches for sum-type metrics in NP-hard scenarios. These techniques enable minimal capacity adjustments so that a chosen flow becomes the network’s maximum under gain-loss factors on arcs.

Investigations into inverse maximum capacity path problems have developed unified algorithmic treatments for both sum-type and max-type distance measures, delivering polynomial-time solutions in preserved-capacity variants and tailored secant-Newton hybrids for exact optimal adjustment. Practical deployment to real-world transportation networks has demonstrated the approach’s capacity to retrofit routing metrics with minimal disruption.

A recent extension to inverse problems with multiple weight functions formulates a broad class encompassing shortest paths, bipartite matchings and arborescences. By employing linear programming duality, this work provides min–max characterisations of the minimal deviation vector across k weight sets, highlights challenges in integrality of optimal adjustments and offers a unified conceptual framework for multi-objective network calibration.

Inverse Optimization Techniques in Network Systems publication trend

The graph below shows the total number of articles in inverse optimization techniques in network systems across all publications each year (not limited to Nature Index journals).

Technical terms

Inverse optimisation: A modelling approach that seeks to modify input parameters of an optimisation problem so that a given feasible solution becomes optimal under minimal total adjustment.

Maximum capacity path: A path between two nodes whose minimum-capacity arc is as large as possible, often used to ensure high-throughput routes in network design.

Sum-type distance: A metric quantifying the aggregate magnitude of parameter changes, typically expressed via ℓ1-norm or weighted Hamming measures.

Max-type distance: A metric capturing the largest single modification among all parameters, often formalised through ℓ∞-norms or bottleneck distance formulations.

References

1. Inverse Generalized Maximum Flow Problems. Mathematics (2019).
2. Inverse Maximum Capacity Path Problems Under Sum-Type and Max-Type Distances and Their Practical Application to Transportation Networks. IEEE Access (2020).
3. Inverse optimization problems with multiple weight functions. Discrete Applied Mathematics (2023).
4. Machine Learning and Inverse Optimization for Estimation of Weighting Factors in Multi-Objective Production Scheduling Problems. Applied Sciences (2022).