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arxiv: 2309.00515 · v2 · pith:PQB37CR3new · submitted 2023-09-01 · 🧮 math.OC

A New Notion of Tykhonov Well-Posedness for Optimization Problems

J. S. Chen , V. K. Hoang , V. S. T. Long This is my paper

Pith reviewed 2026-05-24 07:01 UTC · model grok-4.3

classification 🧮 math.OC
keywords Tykhonov well-posednessdirectional well-posednessminimal time functionoptimization problemslevel setsadmissible functionsstability and convergence
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The pith

Optimization problems not well-posed classically can become Tykhonov well-posed along suitable directions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a new notion of Tykhonov well-posedness for optimization problems defined with respect to a set of directions. This generalizes the classical version by requiring existence of minimizers, stability, and convergence of minimizing sequences only along those directions instead of the whole space. It uses the minimal time function to define directional convergence and provides characterizations in terms of the diameter of level sets and admissible functions. Several examples demonstrate that choosing appropriate directions can make problems well-posed in this sense even if they fail the classical criterion, which has implications for making numerical methods reliable.

Core claim

By building on the minimal time function, the authors define Tykhonov well-posedness with respect to a set of directions, establish its characterizations via diameter of level sets and admissible functions, and show through examples that this notion can hold for direction sets even when the problem is not classically Tykhonov well-posed over the entire space.

What carries the argument

Tykhonov well-posedness with respect to a set of directions, defined using the minimal time function to measure convergence along directions.

If this is right

  • Optimization problems that fail classical well-posedness can succeed with a suitable direction set.
  • Numerical methods can be adapted to ensure sequences converge to minimizers along the chosen directions.
  • Characterizations in terms of level set diameters and admissible functions remain valid for directional versions.
  • The theoretical framework for well-posedness is broadened beyond the full space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach might allow directional analysis in problems from infinite-dimensional spaces where full-space well-posedness is hard to achieve.
  • Connections could be explored to other directional concepts in variational analysis.
  • Practical testing on specific non-well-posed problems like certain ill-posed inverse problems could validate the adaptation of solvers.

Load-bearing premise

The minimal time function supplies a workable metric for directional convergence and the diameter and admissible function characterizations hold when the direction set is only a subset of the space.

What would settle it

Finding an optimization problem and a direction set where the diameter of level sets goes to zero but minimizing sequences do not converge along those directions would disprove the characterizations.

read the original abstract

Building upon the minimal time function, we propose and study a novel notion of Tykhonov well-posedness with respect to a set of directions for optimization problems. This concept generalizes the classical Tykhonov well-posedness by focusing on existence, stability and convergence along specific directions, rather than over the entire space. We first establish several characterizations of Tykhonov well-posedness with respect to a set of directions, formulated in terms of the diameter of level sets and admissible functions. We then investigate relationships between these level sets and admissible functions. To highlight the advantages of the proposed framework, we present several illustrative examples. In particular, we show that by selecting a suitable set of directions, optimization problems that are not well-posed in the classical sense may still be Tykhonov well-posed with respect to those directions. This viewpoint not only broadens the theoretical landscape of well-posedness but also has practical implications, as it allows numerical methods to be effectively adapted so that the generated sequences converge reliably to minimizers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes a directional generalization of Tykhonov well-posedness for optimization problems, defined via the minimal-time function relative to an arbitrary closed direction set D. It establishes equivalences between this notion and the condition that the diameter of the level sets tends to zero, as well as characterizations via admissible functions; it further examines relationships between these objects and supplies examples in which problems that fail classical Tykhonov well-posedness remain well-posed when restricted to a suitable proper subset D.

Significance. The directional framework enlarges the scope of well-posedness theory by permitting analysis along chosen directions when full-space well-posedness fails. The direct-estimate proofs of the equivalences for arbitrary closed D, together with the concrete examples that separate the directional and classical notions, constitute the main technical contribution and could inform the design of directionally convergent numerical schemes.

minor comments (3)
  1. [§3] The precise definition of the minimal-time function and the admissible-function class should be stated explicitly at the beginning of §3 (or in a preliminary subsection) rather than being introduced only through the characterizations.
  2. Notation for the direction set D and the associated level sets should be made uniform across the statements of the main theorems and the examples; currently the same symbol appears with slightly different qualifiers in different places.
  3. [Theorem 3.2] The paper would benefit from a short remark clarifying whether the closedness assumption on D is essential for the diameter characterization or whether the results extend to non-closed sets by a simple closure argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation of minor revision. We are pleased that the directional generalization of Tykhonov well-posedness, its characterizations, and the separating examples are viewed as a valuable contribution to the theory.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces directional Tykhonov well-posedness as a definition built on the existing minimal-time function to a direction set D, then derives equivalences to diameter-of-level-sets and admissible-function conditions via direct estimates that hold for arbitrary closed D. These steps are self-contained mathematical proofs without reduction to fitted parameters, self-citations as load-bearing premises, or renaming of known results. The illustrative examples demonstrate the distinction from classical well-posedness without internal inconsistency or circular dependence on D equaling the whole space.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The construction rests on the minimal time function (presumably from prior literature) and standard properties of level sets in optimization; no free parameters or new entities beyond the definition itself are visible in the abstract.

axioms (1)
  • domain assumption The minimal time function is well-defined and lower semicontinuous for the optimization problems under consideration.
    Invoked to build the directional well-posedness notion (abstract, sentence 1).
invented entities (1)
  • Tykhonov well-posedness with respect to a set of directions no independent evidence
    purpose: To relax classical well-posedness to directional convergence only.
    This is the central new object introduced in the paper.

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Reference graph

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