<p style='text-indent:20px;'>In this paper, we survey the primary research on the theory and applications of distributionally robust optimization (DRO). We start with reviewing the modeling power and computational attractiveness of DRO approaches, induced by the ambiguity sets structure and tractable robust counterpart reformulations. Next, we summarize the efficient solution methods, out-of-sample performance guarantee, and convergence analysis. Then, we illustrate some applications of DRO in machine learning and operations research, and finally, we discuss the future research directions.</p>

We propose 1 norm regularized quadratic surface support vector machine models for binary classification in supervised learning. We establish some desired theoretical properties, including the existence and uniqueness of the optimal solution, reduction to the standard SVMs over (almost) linearly separable data sets, and detection of true sparsity pattern over (almost) quadratically separable data sets if the penalty parameter on the 1 norm is large enough. We also demonstrate their promising practical efficiency by conducting various numerical experiments on both synthetic and publicly available benchmark data sets.

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We suggest a new greedy strategy for convex optimization in Banach spaces and prove its convergent rates under a suitable behavior of the modulus of uniform smoothness of the objective function.