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A Non-convex Optimization Approach of Searching Algebraic Degree Phase-type Representations for General Phase-type Distributions

For a continuous-time phase-type distribution, starting with its Laplace-Stieltjes transform, we obtain a necessary and sufficient condition for its minimal phase-type representation to have the same order as the algebraic degree of the Laplace-Stieltjes transform. To facilitate finding this minimal representation, we transform this condition equivalently into a quadratic nonconvex optimization problem, which can be effectively addressed using an alternating minimization algorithm. The algorithm convergence is also proved. Moreover, the method we develop for the continuous-time phase-type distributions can be directly used to the discrete-time phase-type distributions after establishing an equivalence between the minimal representation problems for continuous-time and discrete-times phase-type distributions.