In this paper, we consider the weighted sum-power minimization under quality-of-service (QoS) constraints in the multi-user multi-input-single-output (MISO) uplink wireless network assisted by intelligent reflecting surface (IRS). We perform a comprehensive investigation on various aspects of this problem. First, when users have sufficient transmit powers, we present a new sufficient condition guaranteeing arbitrary information rate constraints. This result strengthens the feasibility condition in existing literature. Then, we design novel penalty dual decomposition (PDD) based and nonlinear equality constrained alternative direction method of multipliers (neADMM) based solutions to tackle the IRS-dependent-QoS-constraints, which effectively solve the feasibility check and power minimization problems. Besides, we further extend our proposals to the cases where channel status information (CSI) is imperfect and develop an online stochastic algorithm that satisfy QoS constraints stochastically without requiring prior knowledge of CSI errors. Extensive numerical results are presented to verify the effectiveness of our proposed algorithms.