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

Algorithm 1

The matrix generation.

Initialize:

 (i) The maximum number of integrations 𝐽_max.

 (ii) The pre-set tolerance value 𝜀 (e.g., 10⁻⁵).

 (iii) 𝑡^∗₀=0.

Iteration 1:

(1) randomly select 𝑍₁∈𝒬_𝑛; 𝜽=𝑘.

(2) then compute: 𝑋=𝜽𝑍₁, 𝑈^∗=𝑓′(𝑋), 𝑡^∗₁=⟨𝑈,𝑍₁⟩.

While  iteration 𝐽=2,3,…,𝐽_max  do

  (1) If abs(𝑡^∗_𝐽−𝑡^∗_𝐽−1)<𝜀 then break (problem solved);

  (2) Find a new 𝑍_𝐽 by finding the eigenvector 𝑣 corresponding to the smallest eigenvalue of

  𝑈^∗ (𝑣 is normalized by 𝑣⟵(𝑣/‖𝑣‖₂) and 𝑍_𝐽=𝑣∗𝑣^⊤).

  (3) Find a new 𝜽=[𝜃₁,…,𝜃_𝐽]^⊤∈Δ_𝐽={𝜽∈𝑅^𝐽∶𝟏^⊤𝜽=𝑘,𝜽≻0}, by solving the following

  spectral structure problem by EG algorithm as described in Section 2.3.3:

          min_𝜽𝑓(𝜽),s.t.𝜽∈𝑅^𝐽,𝟏^⊤𝜽=𝑘,𝜽≻0

  where 𝑓(𝜽) is obtained from 𝑓(𝑋) by substituting 𝑋 with ∑^𝐽_𝑗=1𝜃_𝑗𝑍_𝑗

  (4) Compute 𝑋=∑^𝐽_𝑗=1𝜃_𝑗𝑍_𝑗, 𝑈^∗=𝑓′(𝑋), 𝑡^∗_𝐽=min{⟨𝑈^∗,𝑍_𝐽⟩}

Output: The p.s.d. matrix 𝑋=∑^𝐽_𝑗=1𝜃_𝑗𝑍_𝑗 and minimal value 𝑓(𝑋).