We propose a robust quadratic regression model to handle the statistics inaccuracy. Unlike the traditional robust statistic approaches that mainly focus on eliminating the effect of outliers, the proposed model employs the recently developed robust optimization methodology and tries to minimize the worst-case residual errors. First, we give a solvable equivalent semidefinite programming for the robust least square model with ball uncertainty set. Then the result is generalized to robust models underl1- andl∞-norm critera with general ellipsoid uncertainty sets. In addition, we establish a robust regression model for per capital GDP and energy consumption in the energy-growth problem under the conservation hypothesis. Finally, numerical experiments are carried out to verify the effectiveness of the proposed models and demonstrate the effect of the uncertainty perturbation on the robust models.

We propose a robust probability classifier model to address classification problems with data uncertainty. A class-conditional probability distributional set is constructed based on the modifiedχ2-distance. Based on a “linear combination assumption” for the posterior class-conditional probabilities, we consider a classification criterion using the weighted sum of the posterior probabilities. An optimal robust minimax classifier is defined as the one with the minimal worst-case absolute error loss function value over all possible distributions belonging to the constructed distributional set. Based on the conic duality theorem, we show that the resulted optimization problem can be reformulated into a second order cone programming problem which can be efficiently solved by interior algorithms. The robustness of the proposed model can avoid the “overlearning” phenomenon on training sets and thus keep a comparable accuracy on test sets. Numerical experiments validate the effectiveness of the proposed model and further show that it also provides promising results on multiple classification problems.

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