This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| Cutting-plane method | |
|---|---|
| Name | Cutting-plane method |
| Field | Mathematical optimization |
Cutting-plane method The cutting-plane method is an iterative optimization technique for solving constrained optimization problems by generating linear constraints that progressively restrict a feasible region. It finds use in integer programming, convex optimization, and combinatorial optimization, connecting to branch-and-bound, subgradient, and interior-point strategies. The method underpins advances in operations research, algorithmic game theory, and computational geometry.
The cutting-plane method iteratively refines a feasible polyhedron by adding hyperplane constraints derived from violated optimality or feasibility conditions to exclude infeasible or suboptimal points while preserving all feasible optima. Early theoretical framing links to separation theorems in convex analysis and computational frameworks used in tackling mixed-integer programs in industrial scheduling, network design, and resource allocation. Practitioners often combine cutting planes with branching heuristics and preprocessing routines from major solver ecosystems.
Cutting-plane derivations rely on convexity results like the separating hyperplane theorem and duality theorems such as those in linear programming and convex optimization. For integer programming, Gomory's fractional cuts emerge from tableau-based algebraic manipulations connected to lattice theory and Diophantine approximation. In convex nonsmooth optimization, Kelley’s method and Kelley–Rockafellar style outer-approximation build on subgradient calculus and Fenchel conjugation. Theoretical guarantees invoke polyhedral theory, extreme point characterization from Minkowski–Weyl representations, and facet-defining inequalities used in polytope studies.
Numerous algorithmic variants address different model classes: Gomory cuts for pure integer linear programs; Chvátal–Gomory and split cuts for mixed-integer programming; lift-and-project procedures associated with Sherali–Adams hierarchies; and Benders decomposition applying L-shaped cuts for block-structured stochastic programs. For convex continuous problems, Kelley’s cutting-plane and analytic center cutting-plane methods produce central cuts using log-barrier concepts from interior-point literature. Combinatorial optimization benefits from problem-specific cuts like clique, odd-cycle, and subtour elimination inequalities for set-partitioning and routing problems.
Convergence properties differ by variant: for finite-dimensional polyhedral programs, finitely many facet-defining cuts suffice to identify an optimal vertex, while for general convex problems, asymptotic convergence may require diminishing cut selection or stabilization via bundle methods. Complexity analyses interlink with NP-hardness results for integer programs and polynomial-time solvability for rational linear programs via the ellipsoid method and interior-point algorithms. Worst-case iteration counts can be exponential for pure cutting-plane schemes on combinatorial instances, prompting hybrid frameworks combining cuts with branching to achieve practical tractability.
Effective implementation integrates cut generation, cut management, and numerical stabilization within solver infrastructures. Practical solvers use presolve modules, cut pools, and activation strategies to avoid cut proliferation and to maintain numerical conditioning. Separation routines often call combinatorial subroutines and call external oracles informed by polyhedral studies, while globalization tactics borrow from trust-region and bundle frameworks to control step selection. Parallelization, warm-starts leveraging basis information, and compatibility with sparse matrix formats are critical for scalability in large-scale industrial instances.
Cutting-plane methods underpin advances across domains: integer and mixed-integer programming for supply-chain Walmart, airline crew scheduling for Delta Air Lines, vehicle routing in logistics, and telecommunication network design for carriers like AT&T. In computational biology, cuts aid genome assembly and haplotype phasing used in projects at institutions such as Broad Institute. Finance applications include portfolio optimization and risk management in firms like Goldman Sachs. Energy grid operations and unit commitment models employ cuts in regional systems overseen by organizations such as PJM Interconnection.
Key contributors include researchers associated with foundational developments: Ralph Gomory introduced Gomory cuts at IBM, Kelley and Shor developed early convex cutting methods influenced by work at Bell Labs and academic groups at Massachusetts Institute of Technology and University of California, Berkeley. Geoffrion advanced decomposition and L-shaped methods while scholars at Carnegie Mellon University and Stanford University contributed to lift-and-project hierarchies and polyhedral combinatorics. Commercial solver evolution involved teams at FICO and Gurobi Optimization integrating cutting-plane frameworks into industrial products.
Category:Optimization methods