Linear Optimization Overview

Principles

Definition

Linear optimization is a mathematical method within the field of mathematical programming focused on finding the optimal value (maximum or minimum) of a linear objective function subject to a set of linear equality and inequality constraints. This fundamental technique provides a rigorous framework for modeling and solving resource allocation, planning, and decision-making problems across various scientific, engineering, and economic disciplines.

Ontological type

Core Methods

Applications

Modeling Formulations

Key developments

Key Figures

Decomposition and Duality era

R. T. Rockafellar [1] emerges as a leading figure in this era, with affiliations including the University of Washington [2] and Princeton University [3]. His key contributions, notably monotone operator theory [4] and the proximal point algorithm [4], provided a rigorous convergence framework for iterative methods central to decomposition and duality in linear optimization. Rockafellar [1]'s work also helped unify multiplier-based updates with convergence theory, informing augmented Lagrangian and proximal strategies that bridged dual formulations with practical solvers in the late 1970s. This body of work by Rockafellar [1] underpinned scalable decomposition approaches for production, planning, and location problems characteristic of the era.

Relaxation and Lifting era

Robert Tibshirani[1] is a statistician closely associated with Stanford University[2] and the University of Toronto[3] during the Relaxation and Lifting era. In this era, his key contributions are embodied by the 1996 paper 'Regression Shrinkage and Selection Via the Lasso'[4], which introduced the Lasso[4]—an L1-penalized regression that performs variable selection by shrinking coefficients toward zero, enabling sparse models and convex optimization that improved tractability for high-dimensional data. Tibshirani[1]'s subsequent work further fortified the role of sparsity-promoting regularization in high-dimensional statistics, reinforcing the practical appeal of convex relaxations and scalable optimization in this era. Moreover, through the Lasso framework, Tibshirani[1] helped bridge statistical theory and algorithmic practice, influencing how researchers approached model selection, error control, and interpretable solutions during the era.

Scalable Solver Ecosystem era

David L. Donoho[1] is associated with Stanford University[3] and the University of California, San Diego[4] in this era. His key contributions include compressed sensing[7], which opened new avenues for recovering sparse signals from undersampled data and underpinned scalable, robust inference in high-dimensional optimization. Robert Tibshirani[2] is affiliated with Harvard University[5] and the Massachusetts Institute of Technology[6] during this era. Regularization Paths for Generalized Linear Models via Coordinate Descent[8] provided a practical way to compute model fits across a continuum of regularization strengths, a capability central to data-driven, scalable optimization.