arxiv: 2604.12045 · v1 · submitted 2026-04-13 · 🧮 math.OC

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On the Connectedness of Sublevel Sets in Invex Optimization

Vinzenz Thoma , Zebang Shen , Niao He

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Pith reviewed 2026-05-10 15:08 UTC · model grok-4.3

classification 🧮 math.OC

keywords invex optimizationsublevel setsconnectednessmountain pass theoremminimax problemsnonconvex functionsoptimization landscapeglobal minimizers

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The pith

Sublevel sets of invex functions are connected under mild assumptions, aiding global optimization.

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

The paper seeks to establish that sublevel sets of invex functions are connected when mild assumptions hold. This would matter because it suggests local search methods have a better chance of reaching global minimizers without being confined to separate regions. The authors create a toolkit using the topological mountain pass theorem for this purpose and apply it to invex functions, which cover several non-convex cases. They also use it to prove connectedness of solution sets in invex-incave minimax problems and incave games.

Core claim

The sublevel sets of invex functions are connected under mild assumptions. This is shown with a mathematical toolkit based on the topological mountain pass theorem. The result is further used to establish connectedness of solution sets for invex-incave minimax problems and incave games.

What carries the argument

The topological mountain pass theorem toolkit, which proves connectedness of sublevel sets for invex functions.

If this is right

Local search algorithms become less likely to be trapped in isolated valleys.
Convergence to global minimizers is facilitated for invex functions.
The solution sets of invex-incave minimax problems are connected.
The solution sets of incave games are connected.

Where Pith is reading between the lines

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

The result could lead to improved global convergence guarantees for optimization methods on invex functions.
Similar toolkits might be developed for other classes of functions beyond invex ones.
Practical verification could involve checking the connectedness on specific invex functions arising in applications.

Load-bearing premise

Invex functions must satisfy the mild assumptions that enable the application of the mountain pass theorem toolkit.

What would settle it

An invex function that meets the mild assumptions but has a disconnected sublevel set for some value would show the claim is false.

Figures

Figures reproduced from arXiv: 2604.12045 by Niao He, Vinzenz Thoma, Zebang Shen.

**Figure 2.** Figure 2: Construction of the set S (in yellow) in the proof of Theorem 2. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: For x, y ∈ R, define a = max (|x| − 1, 0) and b = max (|y| − 1, 0). The plot shows the function f(x, y) = a 2 + 3 sin2 (b) sin2 (a) − 4b 2 − 10 sin2 (b). This function is nonconvex-nonconcave, but satisfies the assumptions of Proposition 3 (verified in Section A.7). Therefore E = M = M and this common set is connected. A direct computation shows E = [−1, 1] × [−1, 1]. 3.2 Incave Potential Games In this sec… view at source ↗

**Figure 4.** Figure 4: Plot of the utility functions u1(a1, a2) = − 1 2 a 2 1+a1a2 and u2(a1, a2) = − 1 2 a 2 2 + (a 3 1 − 2a1) of a two-player game. The utilities are incave in ai and have three disconnected Nash equilibria (0, 0), (− √ 3, − √ 3) and (√ 3, √ 3). arise naturally from assumptions on players’ rationality and common knowledge thereof, (2) S, k-rationalizable actions are good predictors of human behavior, and (3) S-… view at source ↗

**Figure 5.** Figure 5: Mixing coordinates creates new equilibria. [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗

**Figure 6.** Figure 6: Plot of the function f(x, y) = a 2 exp −b 2 − b 2 with E = [−1, 1] × [−1, 1], from Example 2. for κ = Å S αx−1η 1−αx αx x µ 1−αx αx x ã . Example 2. For x, y ∈ R, define a = max (|x| − 1, 0) and b = max (|y| − 1, 0) and consider the function f(x, y) = a 2 exp −b 2 − b 2 , plotted in [PITH_FULL_IMAGE:figures/full_fig_p035_6.png] view at source ↗

read the original abstract

Understanding the topology of sublevel sets yields crucial insights into the optimization landscape of non-convex functions. If sublevel sets are connected, local search algorithms are less likely to be trapped in isolated valleys, facilitating convergence to global minimizers. However, few results exist to establish connectedness in the nonconvex setting. In this work, we present a mathematical toolkit based on the topological mountain pass theorem and use it to study invex functions, a class of functions that includes those satisfying the Polyak-{\L}ojasiewicz inequality and generalizations thereof. We show that their sublevel sets are connected under mild assumptions. We further leverage our result to establish the connectedness of different solution sets for invex-incave minimax problems and incave games.

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.

Desk Editor's Note private letter to a colleague

The paper applies mountain pass to get connectedness for invex sublevel sets, but the argument needs a compactness condition that invexity does not automatically supply.

read the letter

The main point is that this paper uses the mountain pass theorem to prove that sublevel sets of invex functions are connected, yet the proof step that produces the critical point rests on a compactness assumption the authors do not fully verify for the class. The new contribution is the direct application of this topological argument to invex functions and the extension to connectedness of solution sets in invex-incave minimax problems and incave games. That extension is the part most likely to be picked up in game theory or robust optimization work. The paper does a clean job of spelling out the contradiction: if a sublevel set above the infimum were disconnected, mountain pass would force a critical point at a higher level, but invexity turns every critical point into a global minimizer. The writing stays focused and does not overclaim the scope. The soft spot is the Palais-Smale or equivalent compactness requirement. Standard mountain pass statements need some form of it at the mountain-pass level to guarantee the critical point exists. The manuscript refers to mild assumptions that make the toolkit apply, but it does not show that invexity implies this condition or supply checks that rule out sequences where it fails. If such a sequence exists for some invex function, the contradiction does not hold and disconnected sublevels remain possible. This is a central rather than peripheral issue. The work is aimed at researchers who already know invexity and topological methods in optimization. A reader looking for concrete guarantees on landscape connectivity in nonconvex settings will get value from the minimax part once the assumptions are tightened. It deserves a serious referee because the core logic is coherent and the extension adds scope, even though revisions on the compactness step are needed. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a toolkit based on the topological mountain pass theorem to prove that sublevel sets of invex functions (including those satisfying the Polyak-Łojasiewicz inequality) are connected under mild assumptions. It further applies the result to establish connectedness of solution sets for invex-incave minimax problems and incave games.

Significance. If rigorously established, the result would offer a useful topological characterization of optimization landscapes for a wide class of non-convex functions, potentially explaining why local search methods can reach global minimizers without being trapped in disconnected valleys.

major comments (2)

[Main theorem and its proof (Section 3)] The proof of the main connectedness result proceeds by contradiction: a disconnected sublevel set {f ≤ c} (c > inf f) would produce, via the mountain-pass theorem, a critical point at some level d > inf f. Invexity then forces every critical point to be a global minimizer, yielding the desired contradiction. However, standard statements of the mountain-pass theorem (including the version invoked in the manuscript) require a compactness condition such as Palais-Smale at the mountain-pass level. The paper states only that the functions are invex and satisfy “mild assumptions” for the toolkit; it neither proves that invexity implies Palais-Smale nor supplies verifiable conditions under which the compactness holds uniformly for the class. This assumption is load-bearing for the contradiction step.
[Introduction and statement of main results (Section 1)] The “mild assumptions” required for the mountain-pass toolkit are never listed explicitly or shown to be satisfied by the invex functions under consideration. Without a precise statement of these assumptions (e.g., a concrete Palais-Smale-type condition or a coercivity requirement), it is impossible to determine the precise scope of the claimed connectedness result.

minor comments (2)

[Abstract] The abstract refers to “mild assumptions” without any indication of what they are; a brief parenthetical or forward reference to the precise hypotheses would improve readability.
[Section 2] Notation for the mountain-pass level and the critical-value set is introduced without a dedicated preliminary subsection; a short “Notation and preliminaries” paragraph would help readers track the topological arguments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points regarding the explicit statement of assumptions in our use of the mountain-pass theorem. We will revise the manuscript to address these concerns by providing a clear enumeration of the mild assumptions and discussing conditions under which they hold for invex functions.

read point-by-point responses

Referee: [Main theorem and its proof (Section 3)] The proof of the main connectedness result proceeds by contradiction: a disconnected sublevel set {f ≤ c} (c > inf f) would produce, via the mountain-pass theorem, a critical point at some level d > inf f. Invexity then forces every critical point to be a global minimizer, yielding the desired contradiction. However, standard statements of the mountain-pass theorem (including the version invoked in the manuscript) require a compactness condition such as Palais-Smale at the mountain-pass level. The paper states only that the functions are invex and satisfy “mild assumptions” for the toolkit; it neither proves that invexity implies Palais-Smale nor supplies verifiable conditions under which the compactness holds uniformly for the class. This assumption is load-bearing for the contradiction step.

Authors: We agree that the Palais-Smale condition is essential for the mountain-pass theorem and is implicitly included among the mild assumptions. In the revised manuscript, we will explicitly list all assumptions in a new paragraph in Section 1 and restate them at the beginning of Section 3. We will clarify that the functions are assumed to satisfy a Palais-Smale-type condition at levels strictly above the infimum. We will also add a remark providing verifiable sufficient conditions (e.g., coercivity on reflexive Banach spaces or the Polyak-Łojasiewicz inequality together with C^1 regularity) under which invex functions satisfy this compactness requirement. This makes the scope of the result precise while preserving the proof structure. revision: yes
Referee: [Introduction and statement of main results (Section 1)] The “mild assumptions” required for the mountain-pass toolkit are never listed explicitly or shown to be satisfied by the invex functions under consideration. Without a precise statement of these assumptions (e.g., a concrete Palais-Smale-type condition or a coercivity requirement), it is impossible to determine the precise scope of the claimed connectedness result.

Authors: We accept this criticism. The revised introduction will contain an explicit bullet-point list of the mild assumptions, including the Palais-Smale condition at mountain-pass levels and any auxiliary topological requirements. We will also include a short discussion of concrete classes (such as invex functions satisfying the Polyak-Łojasiewicz inequality) where these assumptions are known to hold, thereby delineating the applicability of the connectedness theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external theorem to invex definition

full rationale

The derivation proceeds by applying the topological mountain-pass theorem (an external result) to the standard property of invex functions that every critical point is a global minimizer. A contradiction argument then shows that sublevel sets above the infimum cannot be disconnected. This chain depends on the definition of invexity plus the stated mild assumptions needed for the mountain-pass toolkit; it does not reduce the claimed connectedness result to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation whose content is itself unverified. The assumptions are treated as prerequisites rather than derived from the conclusion, and no ansatz or renaming of known results occurs. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract relies on the standard definition of invex functions (including PL inequality generalizations) and the topological mountain pass theorem; no free parameters, invented entities, or ad-hoc axioms are introduced.

axioms (2)

domain assumption Invex functions include those satisfying the Polyak-Lojasiewicz inequality and generalizations thereof
Stated directly in the abstract as the class under study.
standard math The topological mountain pass theorem can be applied to establish connectedness of sublevel sets
Described as the basis for the mathematical toolkit.

pith-pipeline@v0.9.0 · 5421 in / 1184 out tokens · 79737 ms · 2026-05-10T15:08:34.816521+00:00 · methodology

discussion (0)

Reference graph

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