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High-dimensional data sets, with fewer observations than variables, pose a challenge for statistical methods, particularly if outlying observations are present. Several proposals for robust and sparse estimation in the context of multivariate statistical methods are available, together with algorithms for the computation. We present a unified computational approach based on reformulating the problem as a constrained optimization problem, also incorporating sparsity constraints. Recent developments with adaptive gradient descent algorithms can efficiently solve such problems, and they are also scalable with data dimensionality.The procedures are illustrated in the example of canonical correlation analysis, where also higher-order directions can be directly computed, and the sparsity can be controlled easily. Extensions to other multivariate methods are possible.