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Linear Programming

by N/A – mathematical method, not a product • Founded 1947

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Linear programming (LP) is a mathematical optimization technique for maximizing or minimizing a linear objective function subject to linear equality and inequality constraints. It provides a systematic way to allocate limited resources—such as time, money, or materials—across competing activities. LP is foundational in operations research, powering decision-making in logistics, finance, manufacturing, and many AI/ML pipelines for constrained optimization.

Key Features

  • •Linear objective and constraints: models problems with a linear objective function and linear equality/inequality constraints.
  • •Convexity and global optimality: feasible region is a convex polytope, guaranteeing any local optimum is a global optimum.
  • •Standardized formulations: canonical forms (standard, slack, dual) enable systematic modeling and analysis.
  • •Efficient solvers: mature algorithms like simplex, interior-point, and barrier methods with highly optimized commercial and open-source solvers.
  • •Duality theory: every LP has a dual problem, providing sensitivity analysis and economic interpretation of shadow prices.
  • •Scalability: modern solvers can handle very large-scale problems with millions of variables and constraints in practice.
  • •Rich ecosystem: widely supported in modeling languages (AMPL, GAMS), libraries (PuLP, Pyomo, CVXOPT), and commercial tools (CPLEX, Gurobi).

Use Cases

  • •Supply chain and logistics optimization (routing, production planning, distribution).
  • •Workforce and shift scheduling under labor and regulatory constraints.
  • •Portfolio optimization and asset allocation with linear risk/return approximations.
  • •Manufacturing planning: capacity planning, blending, cutting stock, and resource allocation.
  • •Energy systems optimization (unit commitment, power dispatch approximations).
  • •Telecommunications and network flow optimization (max flow, min cost flow).
  • •Ad allocation, pricing, and revenue management with linearized constraints.
  • •As a subroutine in ML pipelines (e.g., constrained regression, feature selection with linear constraints).

Adoption

Market Stage
Late Majority

Used By

Manufacturing and logistics enterprises (e.g., automotive, retail, airlines)Financial institutions and hedge fundsEnergy utilities and grid operatorsTech companies using LP-based optimization in ad serving and logistics

Alternatives

Mixed-Integer Linear Programming (MILP)

Optimization

Extends linear programming by allowing some variables to be integer-valued, enabling modeling of discrete decisions at the cost of NP-hard complexity.

Models discrete choices and combinatorial structuresVery expressive for real-world planning and scheduling

Quadratic Programming (QP)

Optimization

Generalizes LP by allowing a quadratic objective with linear constraints, useful when costs or risks are better modeled quadratically.

Captures curvature in cost/risk functionsWidely used in finance and control

Nonlinear Programming (NLP)

Optimization

Handles nonlinear objectives and constraints, enabling more realistic but potentially non-convex models.

Very flexible modeling of real-world phenomenaCan capture complex physics and engineering constraints

Constraint Programming (CP)

Optimization

Focuses on combinatorial problems with logical and discrete constraints, using search and propagation rather than purely algebraic methods.

Excellent for scheduling and combinatorial searchRich constraint types (logical, global constraints)

Heuristics and Metaheuristics (e.g., Genetic Algorithms, Simulated Annealing)

Optimization

Approximate optimization methods that do not rely on linearity or convexity, often used when exact methods are infeasible.

Applicable to very complex, black-box problemsEasy to adapt to new problem structures

Industries

ManufacturingLogistics and TransportationFinance and InsuranceEnergy and UtilitiesTelecommunicationsRetail and E-commerceAerospace and AirlinesPublic Sector and Defense