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Nonconvex Robust Optimization for Problems with Constraints

Operations Research Center and Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Operations Research Center and Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Operations Research Center and Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
dbertsim{at}mit.edu
nohadani{at}mit.edu
kwongmeng{at}alum.mit.edu

We propose a new robust optimization method for problems with objective functions that may be computed via numerical simulations and incorporate constraints that need to be feasible under perturbations. The proposed method iteratively moves along descent directions for the robust problem with nonconvex constraints and terminates at a robust local minimum. We generalize the algorithm further to model parameter uncertainties. We demonstrate the practicability of the method in a test application on a nonconvex problem with a polynomial cost function as well as in a real-world application to the optimization problem of intensity-modulated radiation therapy for cancer treatment. The method significantly improves the robustness for both designs.

Key words: optimization; robust optimization; nonconvex optimization; constraints
History: received August 2007; revised September 2008; accepted December 2008.