Abstract
In this paper we discuss the robust counterpart (RC) methodology to handle the optimization under uncertainty problem as proposed by Ben-Tal and Nemirovskii. This optimization methodology incorporates the uncertain data in U a so-called uncertainty set and replaces the uncertain problem by its so-called robust counterpart. We apply the RC approach to uncertain Conic Optimization (CO) problems, with special attention to robust linear optimization (RLO) problem and include a discussion on parametric uncertainty for that case. Some new supported examples are presented to give a clear description of the used of RC methodology theorem.
Highlights
The Robust Counterpart (RC) Methodology of BenTal and Nemirovskii, is one of the existing methodologies for handling uncertainty in the data of an optimization problem
Citing from Ben-Tal [9], the main challenge in this RC methodology is how and when we can reformulate the robust counterpart of uncertain problems as a computationally tractable optimization problem or at least approximate the robust counterpart by a tractable problem
By the same authors, many good results were obtained for robust linear optimization problems Ben-Tal and Nemirovskii [6, 7], robust quadratic and conic quadratic optimization problems Ben-Tal and Nemirovskii [10] and robust semidefinite optimization problems and together with El Ghaoui in [20]
Summary
The Robust Counterpart (RC) Methodology of BenTal and Nemirovskii, is one of the existing methodologies for handling uncertainty in the data of an optimization problem. Citing from Ben-Tal [9], the main challenge in this RC methodology is how and when we can reformulate the robust counterpart of uncertain problems as a computationally tractable optimization problem or at least approximate the robust counterpart by a tractable problem. Due to its definition the robust counterpart highly depends on how we choose the uncertainty set. A recent comprehensive survey on the works of Robust Optimization (RO) is discussed by Gabrel et al [21]. Ben-Tal and Nemirovskii [5, 6, 7, 9], and in Ben-Tal et al [11] applied their RC methodology to the truss topology design (TTD) problem (Ben-Tal and Nemirovskii [4]). By the same authors, many good results were obtained for robust linear optimization problems Ben-Tal and Nemirovskii [6, 7], robust quadratic and conic quadratic optimization problems Ben-Tal and Nemirovskii [10] and robust semidefinite optimization problems (see in Boyd and Vandenberghe [16]) and together with El Ghaoui in [20]
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