Duality plays an important role in scheduling optimization, primarily as a source of bounds and algorithmic insights. In most scheduling problems, however, duality is used only indirectly through relaxations such as Lagrangian or LP relaxations, while explicit dual formulations for the original discrete models are rarely considered. A structural approach to this question was proposed in "On the dual and inverse problems of scheduling jobs to minimize maximum penalty" (Lazarev, Pravdivets, Werner, 2018), where dual problems were introduced for a single-machine scheduling model with release dates and the maximum penalty objective. In that framework the dual problem takes the form of a max-min problem corresponding to the original min-max objective. In this work we extend this approach to several classical single-machine scheduling models with min-sum objectives and without precedence constraints. For each problem we formulate the corresponding dual problem of the sum-min structure and study its structural properties. We prove that the dual formulations provide valid lower bounds for the original objectives (weak duality) and can be computed in polynomial time.