We here provide a comprehensive study of the utility-deviation-risk portfolio selection problem. By considering the first-order condition for the corresponding objective function, we first derive the necessary condition that the optimal terminal wealth satisfying two mild regularity conditions solves for a primitive static problem, called the nonlinear moment problem. We then illustrate the application of this general necessity result by revisiting the nonexistence of the optimal solution for the mean-semivariance problem. Second, we establish an alternative version of the verification theorem serving as the sufficient condition that the solution, which satisfies another mild condition different from that for necessity, of the nonlinear moment problem is the optimal terminal wealth of the original utility-deviation-risk portfolio selection problem. We then apply this general sufficiency result to revisit the various well-posed mean-risk problems already known in the literature and to also establish the existence of the optimal solutions for both utility-downside-risk and utility-strictly-convex-risk problems under the assumption that the underlying utility satisfies the Inada condition. To the best of our knowledge, positive answers to the latter two problems have long been absent in the literature. In particular, the existence result in the utility-downside-risk problem is in contrast to the well-known nonexistence of an optimal solution for the mean-downside-risk problem. As a corollary, the existence result in utility-semivariance problem allows us to utilize the semivariance as a proper risk measure in the classical portfolio management paradigm.