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A polynomial time infeasible interior-point arc-search algorithm for convex optimization

Yaguang Yang

TL;DR

An infeasible interior-point algorithm for the convex optimization problem using arc-search techniques, aiming at optimizing the performance in every iteration, is proposed and a polynomial bound is established.

Abstract

This paper proposes an infeasible interior-point algorithm for the convex optimization problem using arc-search techniques. The proposed algorithm simultaneously selects the centering parameter and the step size, aiming at optimizing the performance in every iteration. Analytic formulas for the arc-search are provided to make the arc-search method very efficient. The convergence of the algorithm is proved and a polynomial bound of the algorithm is established. The preliminary numerical test results indicate that the algorithm is efficient and effective.

A polynomial time infeasible interior-point arc-search algorithm for convex optimization

TL;DR

An infeasible interior-point algorithm for the convex optimization problem using arc-search techniques, aiming at optimizing the performance in every iteration, is proposed and a polynomial bound is established.

Abstract

This paper proposes an infeasible interior-point algorithm for the convex optimization problem using arc-search techniques. The proposed algorithm simultaneously selects the centering parameter and the step size, aiming at optimizing the performance in every iteration. Analytic formulas for the arc-search are provided to make the arc-search method very efficient. The convergence of the algorithm is proved and a polynomial bound of the algorithm is established. The preliminary numerical test results indicate that the algorithm is efficient and effective.
Paper Structure (10 sections, 23 theorems, 91 equations, 3 algorithms)

This paper contains 10 sections, 23 theorems, 91 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem description
  3. The interior-point algorithm with arc-search
  4. Convergence analysis
  5. Implementation and numerical test
  6. Matlab implementation
  7. Preliminary numerical test
  8. Conclusions
  9. acknowledgements
  10. Selection of the centering parameter $\sigma_k$ and step size $\alpha_k$

Key Result

Proposition 2.1

Assume that (a) ${\bf A}_E$ is full rank, (b) the constraints set of system (CP) is not empty, (c) $f({\bf x})$ is differentiable up to the third order and is locally Lipschitz continuous at optimal solution $\bar{{\bf x}}$, then system (CP) has the following properties.

Theorems & Definitions (28)

  • Remark 2.1
  • Proposition 2.1
  • Theorem 2.1
  • Remark 2.2
  • Theorem 3.1: yang11
  • Lemma 3.1: yiy21
  • Proposition 3.1
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.2
  • ...and 18 more