An Efficient Approach to Solve the Large-Scale Semidefinite Programming Problems

Algorithm 2

The EG algorithm: solving the special structure problem shown in (2.17).

Initialize:

 (i) 𝜽⁰ ∈ the interior of Δ_𝐽={𝜽∈𝑅^𝐽∶𝟏^⊤𝜽=𝑘,𝜽≻0}.

 (ii) The maximum number of integrations 𝑖_max

While  iteration 𝑖=1,2,…,𝑖_max  do

  (1) Generate the sequence {𝜽^𝑖} with:

            𝜽^𝑖_𝑗=𝜽_𝑗^𝑖−1exp[−𝜏_𝑖𝑓^′_𝑗(𝜽^𝑖−1)]∑^𝐽_𝑗=1𝜽_𝑗^𝑖−1exp[−𝜏_𝑖𝑓^′_𝑗(𝜽^𝑖−1)]

  where 𝜏_𝑖 is step-size, which can be determined by 𝜏_𝑖=(2log𝐽/𝐿_𝑓)(1/√𝑖) following [15],

  𝑓′(𝜽)=[𝑓^′₁(𝜽),…,𝑓^′_𝐽(𝜽)]^𝑇 is the gradient of 𝑓(⋅).

  (2) Stop if some stopping criteria are met.

Output: 𝜽^𝑖 and 𝑓(𝜽).