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Theorem of Alternative for Extended Homogeneous Linear System and its Application in Conic Optimization

Yurii Nesterov

Abstract

In this paper, we develop a new framework for constructing infeasible-start primal-dual methods for Conic Optimization. Our approach can be seen as a straightforward consequence of Gordan Theorem of Alternative. Given by the target upper bound $ε> 0$ for the duality gap as the only input parameter, we form an auxiliary convex problem of minimizing barrier function with linear equality constraints. Its solution can be easily transformed to the requested output. This function can be minimized by different schemes of Unconstrained Optimization, with possible quadratic convergence in the end of the process. In our paper, we analyze the Damped Newton Method and a short-step path-following scheme. For both of them, we prove polynomial-time complexity results. Our methods are able to benefit from the hot-start opportunities. We can ensure the residual of the linear equality constraints in the primal and dual problems to be at the level of machine accuracy, independently on the accuracy parameter $ε$.

Theorem of Alternative for Extended Homogeneous Linear System and its Application in Conic Optimization

Abstract

In this paper, we develop a new framework for constructing infeasible-start primal-dual methods for Conic Optimization. Our approach can be seen as a straightforward consequence of Gordan Theorem of Alternative. Given by the target upper bound for the duality gap as the only input parameter, we form an auxiliary convex problem of minimizing barrier function with linear equality constraints. Its solution can be easily transformed to the requested output. This function can be minimized by different schemes of Unconstrained Optimization, with possible quadratic convergence in the end of the process. In our paper, we analyze the Damped Newton Method and a short-step path-following scheme. For both of them, we prove polynomial-time complexity results. Our methods are able to benefit from the hot-start opportunities. We can ensure the residual of the linear equality constraints in the primal and dual problems to be at the level of machine accuracy, independently on the accuracy parameter .
Paper Structure (9 sections, 11 theorems, 109 equations)

This paper contains 9 sections, 11 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Notations and Generalities
  3. Theorem of Alternative for Convex Cones
  4. Infeasible-Start Formulation for Conic Optimization
  5. Infeasible Start Newton Method
  6. Infeasible-Start Short-Step Path-Following Scheme
  7. Hot-Start Possibilities
  8. Conclusion
  9. Appendix: Complementary Facts on Conic Optimization

Key Result

Theorem 1

Cone ${\cal F}_p$ is not empty if and only if ${\cal F}_d\equiv \{0\}$.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 3 more