Optimization

Continuous and discrete optimization theory and algorithms.

foundation tier

Optimization. Continuous and discrete optimization theory and algorithms. The literature on optimization divides naturally along several axes: the foundational structures that organise the subject, the techniques that drive proofs and computations, the questions about classification or representation that animate current research, and the bridges to neighbouring areas of mathematics and science. The references below trace those axes through the canonical textbook treatments and recent technical contributions.

Foundations and canonical references

The standard treatments of optimization approach the subject from complementary angles. Boyd, Convex Optimization (2004) is the anchor reference for the subject and lays out the core definitions, theorems, and worked examples that practitioners return to. Nocedal, Numerical Optimization (2006) gives a parallel, more proof-oriented exposition of the same material and is widely used as a graduate text. Bertsekas, Nonlinear Programming (1999) offers an alternative presentation that complements the primary references and is useful for triangulating definitions and proof techniques.

Open methodological questions for optimization include sharpening the bridges between foundational theory and computational practice, extending classical results to broader or more structured settings, and integrating the techniques surveyed above with adjacent mathematical disciplines. The references listed in this page are the entry points that current work builds on.

Prerequisites

Sources

  • textbook · primary · 2004
    Convex Optimization
    boyd-2004, vandenberghe-2004
  • textbook · primary · 2006
    Numerical Optimization
    nocedal-2006, wright-2006
  • textbook · supporting · 1999
    Nonlinear Programming
    bertsekas-1999

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  1. 01

    Convex Optimization

    Duality, KKT conditions, and interior-point methods.
  2. 02

    Linear and Quadratic Programming

    Simplex, interior-point, and active-set methods.
  3. 03

    Semidefinite Programming

    SDP relaxations and applications to combinatorial optimization.
  4. 04

    Nonconvex Optimization

    Global optimization, branch-and-bound, and landscape analysis.
  5. 05

    Integer and Combinatorial Programming

    Cutting planes, branch-and-cut, and Lagrangian relaxation.
  6. 06

    First-Order Methods

    Subgradient, proximal, accelerated, and stochastic gradient algorithms.
  7. 07

    Second-Order Methods

    Newton, quasi-Newton, and trust-region methods.
  8. 08

    Conic Programming

    Second-order cone programming and copositive cones.
  9. 09

    Submodular Optimization

    Submodular functions, greedy guarantees, and continuous relaxations.
  10. 10

    Derivative-Free Optimization

    Bayesian optimization, evolutionary strategies, and trust-region DFO.
  11. 11

    Optimization on Manifolds

    Riemannian gradient methods and retractions.
  12. 12

    Min-Max and Variational Inequalities

    Saddle-point problems, GAN training, and equilibrium computation.
  13. 13

    Online Learning and Bandits

    Multi-armed bandits, contextual bandits, and best-arm identification.

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