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In this work, we study discrete-time Markov decision processes (MDPs) underconstraints with Borel state and action spaces and where all the performancefunctions have the same form of the expected total reward (ETR) criterion overthe infinite time horizon. One of our objective is to propose a convexprogramming formulation for this type of MDPs. It will be shown that the valuesof the constrained control problem and the associated convex program coincideand that if there exists an optimal solution to the convex program then thereexists a stationary randomized policy which is optimal for the MDP. It will bealso shown that in the framework of constrained control problems, the supremumof the expected total rewards over the set of randomized policies is equal tothe supremum of the expected total rewards over the set of stationaryrandomized policies. We consider standard hypotheses such as the so-calledcontinuity-compactness conditions and a Slater-type condition. Our assumptionsare quite weak to deal with cases that have not yet been addressed in theliterature. An example is presented to illustrate our results with respect tothose of the literature.

A convex programming approach for discrete-time Markov decision processes under the expected total reward criterion