  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.
