The paper derives minimum theorems that characterise the steady state cyclic state of a body subjected to cyclic load and temperature. The inelastic material behaviour is described by a convex flow potential. The model is chosen to provide an intermediary description between perfect plasticity, for which general minimum theorems are already known, and more complex and realistic creep constitutive relationships involving internal state variable. The results presented here provide generalisations of the upper and lower bound shakedown theorems and the general result of Ponter and Chen (2001). The Linear Matching method is also discussed and its role as a general programming method is clarified. This allows a discussion of the method as both a kinematic and an equilibrium method. Sufficient conditions for convergence are derived and are shown to correspond to realistic material creep properties only in the case of the kinematic method. This emphasises the view that the method exists as a useful computational tool only as an upper bound method. In an accompanying paper, the minimum theorems are applied to the evaluation of design related properties of the cyclic state of a creeping body.