International Scholarly Research Notices / 2011 / Article / Alg 1

Research Article

Gradient-Type Methods: A Unified Perspective in Computer Science and Numerical Analysis

Algorithm 1

(a) Given { 𝛾 π‘˜ } , { 𝜎 π‘˜ } , π‘˜ = 0 , 1 … s.t.
i n f { 𝛾 π‘˜ } > 0 , i n f { 𝜎 π‘˜ } > 0
Let 𝐱 ( 0 ) ∈ 𝑅 𝑛 be a starting point
For a given vector 𝐬 ( π‘˜ ) ∈ 𝐷 ( 𝛾 π‘˜ , 𝐱 ( π‘˜ ) ) , set
𝐱 ( π‘˜ + 1 ) = 𝐱 ( π‘˜ ) βˆ’ πœ† π‘˜ 𝐬 ( π‘˜ ) , with πœ† π‘˜ ∈ ( 0 , 𝜎 π‘˜ β€– βˆ‡ 𝑓 ( 𝐱 ( π‘˜ ) ) β€– ) :
𝑓 ( 𝐱 ( π‘˜ + 1 ) ) β‰ˆ m i n πœ‡ { 𝑓 ( 𝐱 ( π‘˜ ) βˆ’ πœ‡ 𝐬 ( π‘˜ ) ) , 0 < πœ‡ ≀ 𝜎 π‘˜ β€– βˆ‡ 𝑓 ( 𝐱 ( π‘˜ ) ) β€– }

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