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

Yujie Liu, Dacheng Yao, Hanqin Zhang

TL;DR

This work addresses the problem of when a PH distribution has a minimal representation whose order matches the algebraic degree of its Laplace-Stieltjes transform ${\cal L}(s)=\frac{p(s)}{q(s)}$, by deriving an iff condition encoded as a nonconvex quadratic optimization. The authors develop an alternating minimization algorithm that alternates between solving convex subproblems for matrices $P$ and $A$, with a proof of convergence to a critical point and a clear translation from the transfer-function form to a PH representation via the Jordan form ${\cal J}$ and residue vector $\beta$. They show that the same approach extends to discrete-time PH representations through a rigorous CT↔DT equivalence, yielding a unified method for minimal PH representations across time scales. Numerical experiments demonstrate the method’s effectiveness when a minimal algebraic-degree representation exists (zero objective) and its limitations when such a representation does not exist or when poles are complex, while highlighting robustness and sensitivity to initialization. Overall, the paper provides a practical, convergent framework for identifying minimal PH representations and connects CT and DT cases in a cohesive theory.

Abstract

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.

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

TL;DR

This work addresses the problem of when a PH distribution has a minimal representation whose order matches the algebraic degree of its Laplace-Stieltjes transform , by deriving an iff condition encoded as a nonconvex quadratic optimization. The authors develop an alternating minimization algorithm that alternates between solving convex subproblems for matrices and , with a proof of convergence to a critical point and a clear translation from the transfer-function form to a PH representation via the Jordan form and residue vector . They show that the same approach extends to discrete-time PH representations through a rigorous CT↔DT equivalence, yielding a unified method for minimal PH representations across time scales. Numerical experiments demonstrate the method’s effectiveness when a minimal algebraic-degree representation exists (zero objective) and its limitations when such a representation does not exist or when poles are complex, while highlighting robustness and sensitivity to initialization. Overall, the paper provides a practical, convergent framework for identifying minimal PH representations and connects CT and DT cases in a cohesive theory.

Abstract

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.
Paper Structure (12 sections, 10 theorems, 147 equations, 2 tables)

This paper contains 12 sections, 10 theorems, 147 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Minimal PH Representation Problem: Continuous-time Case
  3. Alternating Minimization Algorithm
  4. Some Properties and An Alternative Expression
  5. Non-convex Optimization Problem
  6. Minimal Discrete-time PH Representation
  7. Numerical Experiments
  8. Concluding Remarks
  9. Proof of Proposition \ref{['prop_Jordan']}
  10. Proof of Proposition \ref{['prop_P']}
  11. Proof of Lemma \ref{['obj-convexity']}
  12. Proof of Lemma \ref{['Alter-optimal-solution']}

Key Result

Proposition 1

For rational function ${\cal L}(s)$ given by prob1, suppose that there exists an $n$-order PH representation $(\alpha, A)$ such that equ_Lapl holds. Then each distinct eigenvalue of $A$ has exactly one Jordan block, that is, for $1\leq i\leq \ell$, $J^{(i)}(\lambda_i)={\cal J}_{n_i}(\lambda_i)$, and which is denoted by ${\cal J}$ in the following analysis.

Theorems & Definitions (24)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Theorem 1
  • proof
  • Lemma 1
  • Lemma 2
  • Proposition 4
  • proof
  • ...and 14 more