Distributionally robust optimization with principal component analysis

Jianqiang Cheng, Richard Li Yang Chen, Habib N. Najm, Ali Pinar, Cosmin Safta, Jean Paul Watson

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


Distributionally robust optimization (DRO) is widely used because it offers a way to overcome the conservativeness of robust optimization without requiring the specificity of stochastic programming. On the computational side, many practical DRO instances can be equivalently (or approximately) formulated as semidefinite programming (SDP) problems via conic duality of the moment problem. However, despite being theoretically solvable in polynomial time, SDP problems in practice are computationally challenging and quickly become intractable with increasing problem sizes. We propose a new approximation method to solve DRO problems with moment-based ambiguity sets. Our approximation method relies on principal component analysis (PCA) for optimal lower dimensional representation of variability in random samples. We show that the PCA approximation yields a relaxation of the original problem and derive theoretical bounds on the gap between the original problem and its PCA approximation. Furthermore, an extensive numerical study shows the strength of the proposed approximation method in terms of solution quality and runtime. As examples, for distributionally robust conditional value-at-risk and risk-averse production-transportation problems the proposed PCA approximation using only 50% of the principal components yields near-optimal solutions (within 1%) with a one to two order of magnitude reduction in computation time.

Original languageEnglish (US)
Pages (from-to)1817-1841
Number of pages25
JournalSIAM Journal on Optimization
Issue number2
StatePublished - 2018
Externally publishedYes


  • Distributionally robust optimization
  • Principal component analysis
  • Semidefinite programming
  • Stochastic programming

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science


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