Convexification of Permutation-Invariant Sets and an Application to Sparse Principal Component Analysis

Jinhak Kim, Mohit Tawarmalani, Jean Philippe P. Richard

Research output: Contribution to journalArticlepeer-review


We develop techniques to convexify a set that is invariant under permutation and/or change of sign of variables and discuss applications of these results. First, we convexify the intersection of the unit ball of a permutation and sign-invariant norm with a cardinality constraint. This gives a nonlinear formulation for the feasible set of sparse principal component analysis (PCA) and an alternative proof of the K-support norm. Second, we characterize the convex hull of sets of matrices defined by constraining their singular values. As a consequence, we generalize an earlier result that characterizes the convex hull of rank-constrained matrices whose spectral norm is below a given threshold. Third, we derive convex and concave envelopes of various permutation-invariant nonlinear functions and their level sets over hypercubes, with congruent bounds on all variables. Finally, we develop new relaxations for the exterior product of sparse vectors. Using these relaxations for sparse PCA, we show that our relaxation closes 98% of the gap left by a classical semidefinite programming relaxation for instances where the covariance matrices are of dimension up to 50 × 50.

Original languageEnglish (US)
Pages (from-to)2547-2584
Number of pages38
JournalMathematics of Operations Research
Issue number4
StatePublished - Nov 2022

Bibliographical note

Funding Information:
Funding: The second and third authors wish to acknowledge the support from the National Science Foundation Division of Civil, Mechanical and Manufacturing Innovation [Grants 1727989 and 1917323].

Publisher Copyright:
Copyright: © 2021 INFORMS.


  • convexification
  • majorization
  • permutation-invariant sets
  • sparse PCA


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