On semidefinite linear fractional optimization problems

Recently, semidefinite optimization problems have been intensively studied since many optimization problem can be transformed into the problems and the problems are computationally tractable. In this paper, we consider a semidefinite linear fractional optimization problem (SLF), and obtain optimality conditions for (SLF) which hold without any constraint qualification and which are expressed by sequences. By using the optimality conditions, we formulate the nonfractional dual problem (D) for (SLF), which is expressed by sequences, and prove the weak duality theorem and the strong duality theorem which hold between (SLF) and (D). The strong duality theorem holds without any constraint qualification. Furthermore, we characterize the solution set of (SLF) by using the optimality conditions.