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Consider estimation of average treatment effects with multi-valued treatments using augmented inverse probability weighted (IPW) estimators, depending on outcome regression and propensity score models in high-dimensional settings. These regression models are often fitted by regularized likelihood-based estimation, while ignoring how the fitted functions are used in the subsequent inference about the treatment parameters. Such separate estimation can be associated with known difficulties in existing methods. We develop regularized calibrated estimation for fitting propensity score and outcome regression models, where sparsity-including penalties are employed to facilitate variable selection but the loss functions are carefully chosen such that valid confidence intervals can be obtained under possible model misspecification. Unlike in the case of binary treatments, the usual augmented IPW estimator is generalized by allowing different copies of coefficient estimators in outcome regression to ensure just-identification. For propensity score estimation, the new loss function and estimating functions are directly tied to achieving covariate balance between weighted treatment groups. We develop practical numerical algorithms for computing the regularized calibrated estimators with group Lasso by innovatively exploiting Fisher scoring, and provide rigorous high-dimensional analysis for the resulting augmented IPW estimators under suitable sparsity conditions, while tackling technical issues absent or overlooked in previous analyses. We present simulation studies and an empirical application to estimate the effects of maternal smoking on birth weights. The proposed methods are implemented in the R package mRCAL.