diff --git a/docs/src/references.bib b/docs/src/references.bib index c03f5fff345..fbc566a5769 100644 --- a/docs/src/references.bib +++ b/docs/src/references.bib @@ -173,6 +173,18 @@ @article{Peng2007 doi = {10.1137/050641983}, } +@article{RechtFazelParrilo2010, + author = {Recht, Benjamin and Fazel, Maryam and Parrilo, Pablo A.}, + title = {Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization}, + journal = {SIAM Review}, + volume = {52}, + number = {3}, + pages = {471--501}, + year = {2010}, + issn = {0036-1445}, + url = {https://www.jstor.org/stable/20780168}, +} + @article{Zimmerman2011, author = {Zimmerman, Ray Daniel and Murillo-Sánchez, Carlos Edmundo and Thomas, Robert John}, journal = {IEEE Transactions on Power Systems}, diff --git a/docs/src/tutorials/conic/simple_examples.jl b/docs/src/tutorials/conic/simple_examples.jl index 9e2b383bcb3..ea06c1da0b3 100644 --- a/docs/src/tutorials/conic/simple_examples.jl +++ b/docs/src/tutorials/conic/simple_examples.jl @@ -115,6 +115,33 @@ S, T = solve_max_cut_sdp([0 5 7 6; 5 0 0 1; 7 0 0 1; 6 1 1 0]) S, T = solve_max_cut_sdp([0 1 5 0; 1 0 0 9; 5 0 0 2; 0 9 2 0]) +# ## Low-rank matrix completion + +# The matrix completion problem seeks to find the missing entries of a matrix +# with a given (possibly random) subset of fixed entries, such that the completed +# matrix has the lowest attainable rank. +# +# For more details, see [RechtFazelParrilo2010](@cite). + +function example_matrix_completion(; svdtol = 1e-6) + rng = Random.MersenneTwister(1234) + n = 20 + mask = rand(rng, 1:25, n, n) .== 1 + B = randn(rng, n, n) + model = Model(SCS.Optimizer) + @variable(model, X[1:n, 1:n]) + @constraint(model, X[mask] .== B[mask]) + @variable(model, t) + @constraint(model, [t; vec(X)] in MOI.NormNuclearCone(n, n)) + @objective(model, Min, t) + optimize!(model) + @assert is_solved_and_feasible(model) + ## Return the approximate rank of the completed matrix to a given tolerance: + return sum(LinearAlgebra.svdvals(value.(X)) .> svdtol) +end + +example_matrix_completion() + # ## K-means clustering via SDP # Given a set of points ``a_1, \ldots, a_m`` in ``\mathbb{R}^n``, allocate them to ``k`` clusters.