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# Optimal Uncertainty Quantification via Convex Optimization and Relaxation

## Citation

Han, Shuo (2014) Optimal Uncertainty Quantification via Convex Optimization and Relaxation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/X00K-T615. https://resolver.caltech.edu/CaltechTHESIS:10162013-111333269

## Abstract

Many engineering applications face the problem of bounding the expected value of a quantity of interest (performance, risk, cost, etc.) that depends on stochastic uncertainties whose probability distribution is not known exactly. Optimal uncertainty quantification (OUQ) is a framework that aims at obtaining the best bound in these situations by explicitly incorporating available information about the distribution. Unfortunately, this often leads to non-convex optimization problems that are numerically expensive to solve.

This thesis emphasizes on efficient numerical algorithms for OUQ problems. It begins by investigating several classes of OUQ problems that can be reformulated as convex optimization problems. Conditions on the objective function and information constraints under which a convex formulation exists are presented. Since the size of the optimization problem can become quite large, solutions for scaling up are also discussed. Finally, the capability of analyzing a practical system through such convex formulations is demonstrated by a numerical example of energy storage placement in power grids.

When an equivalent convex formulation is unavailable, it is possible to find a convex problem that provides a meaningful bound for the original problem, also known as a convex relaxation. As an example, the thesis investigates the setting used in Hoeffding's inequality. The naive formulation requires solving a collection of non-convex polynomial optimization problems whose number grows doubly exponentially. After structures such as symmetry are exploited, it is shown that both the number and the size of the polynomial optimization problems can be reduced significantly. Each polynomial optimization problem is then bounded by its convex relaxation using sums-of-squares. These bounds are found to be tight in all the numerical examples tested in the thesis and are significantly better than Hoeffding's bounds.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:concentration inequalities; convexity; optimization; uncertainty quantification
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Electrical Engineering
Thesis Availability:Public (worldwide access)
• Murray, Richard M.
Thesis Committee:
• Murray, Richard M. (chair)
• Chandrasekaran, Venkat
• Hassibi, Babak
• Low, Steven H.
Defense Date:26 September 2013
Funders:
Funding AgencyGrant Number
U.S. Army Research OfficeW911NF09-0001
BoeingUNSPECIFIED
National Science FoundationCNS-0931746
U.S. Air Force Office of Scientific ResearchFA9550-12-1-0389
U.S. Dept. of Energy. Predictive Science Academic Alliance ProgramUNSPECIFIED
Record Number:CaltechTHESIS:10162013-111333269
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:10162013-111333269
DOI:10.7907/X00K-T615
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7991
Collection:CaltechTHESIS
Deposited By: Shuo Han
Deposited On:16 Dec 2013 23:51