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Robust convex optimization. (English) Zbl 0977.90052

Summary: We study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set \({\mathcal U}\) yet the constraints must hold for all possible values of the data from \({\mathcal U}\). The ensuing optimization problem is called robust optimization. In this paper we lay the foundation of robust convex optimization. In the main part of the paper we show that if \({\mathcal U}\) is an ellipsoidal uncertainty set, then for some of the most important generic convex optimization problems (linear programming, quadratically constrained programming, semidefinite programming and others) the corresponding robust convex program is either exactly, or approximately, a tractable problem which lends itself to efficient algorithms such as polynomial time interior point methods.

MSC:

90C31 Sensitivity, stability, parametric optimization
65K05 Numerical mathematical programming methods
90C25 Convex programming
90C60 Abstract computational complexity for mathematical programming problems
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