Due to the semi-infinite constraints at the relay node, the objective problem (59) is non-convex.
Problem (27) has semi-infinite constraints (27b)–(27e) and non-convex rank-one constraint (27h), which are intractable.
Specifically, the semi-infinite constraints can be reformulated into a linear matrix inequality (LMI), which can be efficiently solved by using an alternating optimization algorithm.
In this scenario, we consider a two-stage solution where in the first step, the semi-infinite constraints are converted to linear matrix inequalities (LMI) and in the second step, we use our proposed iterative algorithm to solve it.
In order to maximize the weighted SR, the objective problem is first converted into a decoupled one by employing the Cauchy-Schwarz inequality and S-lemma, and then, the optimal relay beamforming design is also investigated with the LMI versions of the semi-infinite constraints.
Then, we consider robust secure transmit scheme design where the CSI uncertainties are molded by the worst-case model, and S-procedure and its extension are employed to transform the semi-infinite constraint problem into finite constraint problem.
The purpose of this paper is to study the Levitin-Polyak well-posedness for generalized semi-infinite multiobjective programming problems, where the objective function is vector-valued and the generalized semi-infinite constraint functions are real-valued.
It would be interesting to consider the Levitin-Polyak well-posedness for semi-infinite vector optimization problems, where the objective function and the semi-infinite constraint functions are also vector-valued.
Problem (16) is challenging to be solved due to the non-convex objective function (16a) (the objective function is the difference of two logarithm functions) and the semi-infinite constraint (16b) (due to the norm-bounded CSI errors).
