Numerical Optimization

Nocedal, J., & Wright, S. J. (2006). Numerical Optimization. Springer.

Type

Book

Intellectual base

Review or monograph

Year

2006

DOI

10.1007/978-0-387-40065-5

ISBN

9780387303031

DOI registrant

Springer Nature

Link

https://doi.org/10.1007/978-0-387-40065-5

Cited from

computer_science, mathematics

Cited by

5 citations across 5 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Optimization Landscape
  • In mathematical optimisation, convexity, multimodality, and conditioning are landscape properties that determine which algorithms — gradient descent, Newton, BFGS, simulated annealing, basin-hopping — succeed, and the landscape framing is the working vocabulary of modern numerical optimisation.

    This sourceStandard reference treating convexity, conditioning, and modality as the landscape properties governing gradient, Newton, and quasi-Newton methods.

• Termination Condition
  • The unfolding process is the iteration \(x_{n+1} = x_n - f(x_n)/f'(x_n)\), which under good conditions doubles the number of correct digits each step.

    This sourceCovers Newton's method, its local quadratic convergence, and the composite stopping criteria (gradient/step-size tolerance plus iteration cap) of practical solvers.

Domain-specific

• Gauss–Newton Algorithm
  • Gauss–Newton drops the residual-weighted second-derivative term and uses \(J^{\mathsf T}J\) as the Hessian model.
• Limited-Memory BFGS
  • Storage and per-iteration linear algebra scale as \(O(mn)\), making quasi-Newton curvature practical when \(n\) is large and \(m\ll n\).

Mechanisms

• Gradient Descent or Ascent Search
  • Its failure mode is structural and famous: a purely local rule finds a local optimum, not necessarily the best one — it walks confidently into the nearest valley and calls it home, blind to a deeper valley one ridge over.

    This sourceExplains that optimization algorithms generally locate only local minima and can miss a lower value in an unsampled region, becoming trapped among multiple local minima.

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