In the research papers on uncertain linear complementarity problems, stochastic versions have been applied and attracted much attention, and the stochastic linear complementarity problem was established. In the study of the stochastic linear complementarity problems, three types of appropriate deterministic formulations have been proposed. According to these formulations, several methods and techniques have been proposed and studied.
However, if probabilistic methods are adopted to deal with uncertain linear complementarity problems, there arise the following problems. First, the probability distributions of random matrix and random vector are known in advance, which may not be appropriate in many real situations. Next, the solutions to the three formulations may not satisfy some conditions of the problem, and thus, there is no guarantee that the solutions to satisfy some “hard” conditions, i.e., those which must be satisfied in some practical problems. Moreover, the difficulty with quick computation due to the growing size of the problem is another challenge.
Ri Won Ju, a researcher at the Faculty of Management of Industrial Economy, has investigated uncertain linear complementarity problems by adopting the robust optimization technique. He focused on the solutions to Uncertain Linear Complementarity Problems (ULCP) different from the best well-known technique based on stochastic linear complementarity problems.
He proposed the notion of the ρ-robust counterpart and the ρ-robust solutions of ULCP. For three important examples of uncertainty set, namely, the unknown-but-bounded uncertainty set, the simple ellipsoidal uncertainty set and the intersection-of-ellipsoids uncertainty set, he obtained some necessary and sufficient conditions, and sufficient conditions which the ρ-robust solutions of ULCP satisfy, respectively, and discussed some special cases.
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