What are the problems of interest for several reasons in cost minimization?

Cost Minimization:

A significant implication of the firm choosing a profit-maximizing production plan is which there is no method to produce similar amounts of outputs at a lower total input cost. Therefore, cost minimization is an essential condition for profit maximization. Such observation inspires us to an independent study of the firm’s cost minimization. The problem is of interest for several purposes. First, this leads us to a number of results and constructions which are technically very helpful. Second, as extensive as the firm is a price taker into its input market, the results flowing through the cost minimization carry on to be valid whether or not the output market is competitive and therefore whether or not the firm takes the output price as specified as. Third, while the production set demonstrates nondecreasing returns to scale, then the cost function and optimizing vectors of the cost minimization problem that keep the levels of outputs fixed, those are better behaved than the profit function.