The Legendre-Fenchel Conjugate of the Product of Two Positive Definite Quadratic Forms
SIAM Journal on Matrix Analysis and Applications
Convexity conditions of Kantorovich function and related semi-infinite linear matrix inequalities
Journal of Computational and Applied Mathematics
Robust univariate spline models for interpolating interval data
Operations Research Letters
Minmax regret bottleneck problems with solution-induced interval uncertainty structure
Discrete Optimization
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We consider a rather general class of mathematical programming problems with data uncertainty, where the uncertainty set is represented by a system of convex inequalities. We prove that the robust counterparts of this class of problems can be reformulated equivalently as finite and explicit optimization problems. Moreover, we develop simplified reformulations for problems with uncertainty sets defined by convex homogeneous functions. Our results provide a unified treatment of many situations that have been investigated in the literature and are applicable to a wider range of problems and more complicated uncertainty sets than those considered before. The analysis in this paper makes it possible to use existing continuous optimization algorithms to solve more complicated robust optimization problems. The analysis also shows how the structure of the resulting reformulation of the robust counterpart depends both on the structure of the original nominal optimization problem and on the structure of the uncertainty set.