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We develop techniques to convexify a set that is invariant under permutationand/or change of sign of variables and discuss applications of these results.First, we convexify the intersection of the unit ball of a permutation andsign-invariant norm with a cardinality constraint. This gives a nonlinearformulation for the feasible set of sparse principal component analysis (sparsePCA) and an alternative proof of the $K$-support norm. Second, we characterizethe convex hull of sets of matrices defined by constraining their singularvalues. As a consequence, we generalize an earlier result that characterizesthe convex hull of rank-constrained matrices whose spectral norm is below agiven threshold. Third, we derive convex and concave envelopes of variouspermutation-invariant nonlinear functions and their level-sets over hypercubes,with congruent bounds on all variables. Finally, we develop new relaxations forthe 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 SDPrelaxation for instances where the covariance matrices are of dimension up to$50\times 50$.

Convexification of Permutation-Invariant Sets and an Application to Sparse PCA