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On a robust approach to "split" feasibility problems: solvability and global error bound conditions

Amos Uderzo

TL;DR

The paper tackles robust split feasibility problems (RSFP) with data uncertainty by reformulating the robust counterpart as a set-valued inclusion using the map $ ext{A}_{ ext U}$ and its realizations. It derives a sufficient solvability condition and a global error bound expressed via the residual $oldsymbol u_{ ext{A}_{ ext U}}$, using convex-analytic tools and subdifferential calculus. A detailed treatment of the polyhedral case yields SCQ and a $Q$-increasing property that guarantee explicit error bounds under reasonable regularity (including a positive exact covering bound). The work provides a robust, non-probabilistic framework for ensuring feasibility and stable solutions under uncertainty, with concrete conditions and illustrative examples, while leaving room for extensions to broader uncertainty and non-linearities.

Abstract

In the present paper, a robust approach to a special class of convex feasibility problems is considered. By techniques of convex and variational analysis, conditions for the existence of robust feasible solutions and related error bounds are investigated. This is done by reformulating the robust counterpart of a split feasibility problem as a set-valued inclusion, a problem for which one can take profit from the solvability and stability theory that has been recently developed. As a result, a sufficient condition for solution existence and error bounds is established in terms of problem data and discussed through several examples. A specific focus is devoted to error bound conditions in the case of the robust counterpart of polyhedral split feasibility problems.

On a robust approach to "split" feasibility problems: solvability and global error bound conditions

TL;DR

The paper tackles robust split feasibility problems (RSFP) with data uncertainty by reformulating the robust counterpart as a set-valued inclusion using the map and its realizations. It derives a sufficient solvability condition and a global error bound expressed via the residual , using convex-analytic tools and subdifferential calculus. A detailed treatment of the polyhedral case yields SCQ and a -increasing property that guarantee explicit error bounds under reasonable regularity (including a positive exact covering bound). The work provides a robust, non-probabilistic framework for ensuring feasibility and stable solutions under uncertainty, with concrete conditions and illustrative examples, while leaving room for extensions to broader uncertainty and non-linearities.

Abstract

In the present paper, a robust approach to a special class of convex feasibility problems is considered. By techniques of convex and variational analysis, conditions for the existence of robust feasible solutions and related error bounds are investigated. This is done by reformulating the robust counterpart of a split feasibility problem as a set-valued inclusion, a problem for which one can take profit from the solvability and stability theory that has been recently developed. As a result, a sufficient condition for solution existence and error bounds is established in terms of problem data and discussed through several examples. A specific focus is devoted to error bound conditions in the case of the robust counterpart of polyhedral split feasibility problems.
Paper Structure (10 sections, 11 theorems, 165 equations)

This paper contains 10 sections, 11 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standing assumptions on the problem data
  4. Convex analysis tools
  5. A subdifferential condition for solvability and error bounds for convex inequalities
  6. Functional characterization of ${\rm S}_{C,\mathcal{U},Q}$
  7. Subtransversality of sets
  8. Conditions for solvability and error bounds
  9. Error bounds in the polyhedral case
  10. Conclusions

Key Result

proposition 1

Let $\varphi:\mathbb R^n\longrightarrow\mathbb R$ be a convex function. Assume that $[\varphi>0]\ne\varnothing$ and that Then, it is $[\varphi\le 0]\ne\varnothing$ and

Theorems & Definitions (29)

  • remark 1
  • proposition 1
  • proof
  • proposition 2
  • proof
  • proposition 3
  • proof
  • lemma 1: Properties of $\mathcal{A}_\mathcal{U}$
  • proof
  • remark 2
  • ...and 19 more