Line Search Algorithm Assignment Help

Line search algorithms

  1. Uniform search
  2. Dichotomous search
  3. Golden section search
  4. Quadratic fit search

1. Uniform Search

Uniform Search

Pick the search points α12,.....,αn so that they are uniformly spaced over some preset range α

  • α* = argminα∈α12,.....,αn ƒ(α)
  • + No assumptions about convexity or shape of ƒ(α)
  • + Finds (nearly) a global minimum
  • - Relatively inefficient

2. Dichotomous search

Suppose you know that α is in the range of αmin to αmax

  • calculate the following evaluation points
    Dichotomous search
  • Otherwise, set αmin = b
  • Repeat until convergence
  • If the derivative dƒ(α)⁄dα can be calculated, the computation can be reduced to one evaluation of the derivative per an iteration.
  • If the derivative is used, this is called bisection method.

3. Golden section line search

Suppose you know that α is in range of αmin to αmax

Golden section line search

Loop to 2 until convergence

4. Quadratic fit line search

  • Find α1, α2 and α3 such that ƒ(α1) ≥ ƒ(α2) and ƒ(α2) ≤ ƒ(α3)
  • Find α* as the minimum to the quadratic fit of ƒ(α1), ƒ(α2), and ƒ(α3)
  • If α α2,
    if ƒ(α*) ƒ (α2), then αnew = α1, α2, α*
    if ƒ(α*) ≤ ƒ (α2), then αnew = α2, α*, α3


    if ƒ(α*) ƒ (α2), then αnew = α*, α2, α3
    if ƒ(α*) ≤ ƒ(α2), then αnew = α1, α*, α2
  • Loop to 2 until convergence
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