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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.
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Basic Convex Analysis in Metric Spaces with Bounded Curvature
Differentiable structure ensures that many of the basics of classical convex analysis extend naturally from Euclidean space to Riemannian manifolds. Without such structure, however, extensions are more challenging. Nonetheless, in Alexandrov spaces with curvature bounded above (but possibly positive), we develop several basic building blocks. We define subgradients via pro- jection and the normal cone, prove their existence, and relate them to the classical affine minorant property. Then, in what amounts to a simple calculus or duality result, we develop a necessary optimality condition for minimizing the sum of two convex functions.
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- Award ID(s):
- 2006990
- PAR ID:
- 10495483
- Publisher / Repository:
- SIAM
- Date Published:
- Journal Name:
- SIAM Journal on Optimization
- Volume:
- 34
- Issue:
- 1
- ISSN:
- 1052-6234
- Page Range / eLocation ID:
- 366 to 388
- Subject(s) / Keyword(s):
- subdifferential, normal cone, Alexandrov spaces
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1137/23M1551389
