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Extreme flows: where physics meets mathematically rigorous bounds

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This essay argues that the S1–S3 framework has closed two model problems—maximum enstrophy growth in 1D viscous Burgers flows and enstrophy dissipation in unforced 2D Navier-Stokes flows—by finding flows that saturate rigorous bounds.

desk verdict A readable synthesis of a real research program; the Burgers sharpness is genuine, but the 2D 'closed' claim outruns the evidence. read the letter →

arxiv 2608.04859 v1 pith:Z762ESNV submitted 2026-08-05 physics.flu-dyn math.AP

Bartosz Protas This is my paper · ORCID
classification physics.flu-dynmath.AP MSC 76D0576B0335Q3549J20
keywords extremeflowssharpaprioriboundsenstrophygrowthviscousBurgersequationtwo-dimensionalNavier-Stokesdissipationanomalyvariationaloptimizationsingularityformation
topics Navier-Stokes Regularity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The essay argues that extreme behavior in fluid models can be studied systematically by combining rigorous a priori bounds with numerical optimization: derive the sharpest bound, look for flows that attain it, and read off the physical mechanism. It reports two problems where this program has succeeded. For 1D viscous Burgers flows, the maximum finite-time enstrophy growth scales as $E_0^{3/2}$, and this exponent is now proven sharp. For unforced 2D Navier-Stokes flows, the enstrophy dissipation in the inviscid limit is governed by a sharp combined estimate that leaves no room for improvement beyond a possible logarithmic correction. The same route is then used to search for finite-time singularities in 3D Euler flows, producing a candidate flow whose $\dot{H}^3$ norm grows consistently with singularity formation as long as the computation remains resolved.

What carries the argument

The engine is the S1-S3 loop: energy-method inequalities (e.g., $dE/dt \le C\nu^{-1/3}E^{5/3}$ for Burgers and the vorticity-difference bound (34)-(35) for 2D Navier-Stokes) provide upper bounds; variational problems such as Problem 3.2 and Problem 3.4 maximize the quantity of interest over constraint manifolds with fixed enstrophy, palinstrophy, or $L^q$ norm; and adjoint-based Riemannian gradient methods with continuation solve these nonconvex problems to produce maximizer branches. The central identities are the enstrophy growth rate $r(u)=dE/dt=-\nu\|\partial_{xx}u\|^2_{L^2}+\tfrac12\int(\partial_xu)^3dx$ and the enstrophy dissipation rate $\varepsilon_\nu(\varphi)\le \frac{2}{T}\|\varphi\|_{L^2}\|\omega(T)-\omega_\nu(T)\|_{L^2}$, which connects dissipation to inviscid-limit vorticity convergence. Sharpness means that a family of maximizers saturates the bound's exponent (or prefactor), and the physical mechanism is read off from the maximizing flows.

What would settle it

For the same $P_0$, $\nu$, and $T$ as in figure 3a, find an initial condition in the constraint set $S$ whose enstrophy dissipation exceeds the reported upper envelope $\hat{\varepsilon}_\nu^T$; if such a state exists, the combined estimate (34)-(35) is not saturated and the sharpness claim fails.

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Extended reading notes

Core claim

The central claim is that the S1-S3 loop (deduce a priori bounds, verify sharpness by variational maximization, extract mechanisms) yields closed solutions to two model problems. In the Burgers problem, the instantaneous bound $dE/dt \le C\nu^{-1/3}E^{5/3}$ is sharp in its exponent, and the finite-time numerical maximizers found by Ayala & Protas (2011) grow like $E_0^{3/2}$; Albritton & Nitti (2023) then proved the matching upper bound, so the problem is mathematically closed. In the 2D Navier-Stokes problem, Matharu et al. (2022) showed that the combined estimate (34)-(35) is saturated by six branches of extreme initial conditions that maximize enstrophy dissipation, so the estimate is declared sharp and offers no room for improvement other than, perhaps, a logarithmic correction. For 3D Euler flows, maximizing the $\dot{H}^3$ seminorm over Gevrey-class initial data yields a flow whose norm growth is consistent with finite-time singularity formation, with the near-singular structure being two colliding jets forming a flattened vortex-ring gap.

Load-bearing premise

The load-bearing premise is that the numerical search found every relevant branch of maximizers for the nonconvex Problem 3.4, so that the upper envelope over branches equals the true supremum of enstrophy dissipation; the paper itself concedes that all its maximizers are generally only local.

Editorial extensions

If this is right

  • The maximum finite-time enstrophy growth in 1D viscous Burgers flows is now known to scale as $E_0^{3/2}$; no improvement in the exponent is possible.
  • Enstrophy dissipation in unforced 2D Navier-Stokes flows vanishes in the inviscid limit at a rate consistent with the sharp bound (34)-(35), ruling out an enstrophy dissipation anomaly in this setting.
  • The instantaneous 3D Navier-Stokes bounds on enstrophy growth and on $L^q$ norm growth are sharp in their exponents, but no single flow saturates both, suggesting a finite-time singularity would occur along a trajectory that does not saturate either bound.
  • The variational search for extreme 3D Euler flows identifies a candidate finite-time singularity whose mechanism is nearly axisymmetric and emerges unprescribed from the optimization.
  • Convex sum-of-squares upper bounds independently match the numerically observed Burgers extremes, giving a route to certify global sharpness for problems where local search alone is not exhaustive.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the 2D sharpness claim is right, then one expects analogous optimization-based saturation to reveal sharp bounds in forced 2D Navier-Stokes and in models such as the generalized Constantin-Lax-Majda and surface quasi-geostrophic equations, where anomalous dissipation or blow-up questions remain open.
  • The recurring $3/2$ exponent for enstrophy amplification across Burgers and 3D Navier-Stokes optimized flows may be a general scaling law for extreme enstrophy growth; this is testable by computing higher-precision exponents at larger $E_0$ and $B$ values.
  • A testable extension is to use the time-reversibility of Euler flows to formulate the singularity search from a near-blowup terminal state backward in time, which could sharpen the candidate geometry and give a concrete falsifiable prediction of the singular structure.
  • The nonexhaustive branch search means the reported sharpness is conditional; a certified global-optimization upper bound matching the same envelope would remove that condition and elevate the 2D sharpness claim from numerical evidence to a verified statement.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript is an essay presenting a three-step research program (S1–S3): derive rigorous a priori bounds on extreme growth/dissipation quantities in fluid models, probe sharpness of those bounds by solving variational optimization problems, and extract the physical mechanisms from the saturating flows. It surveys two model problems claimed to be 'closed': maximum enstrophy growth in 1D viscous Burgers flows, where the sharp exponent 3/2 is supported by the independent rigorous result of Albritton & Nitti (2023), and enstrophy dissipation in unforced 2D Navier–Stokes flows, where sharpness of the bound (34)–(35) is inferred from numerical maximizers of Problem 3.4 and fits to a power law in ν. The essay also reviews local-maximizer searches for potential singularities in 3D Navier–Stokes and Euler flows, and closes with open problems and methodological outlook.

Significance. If the two 'closed' claims were fully established, the paper would demonstrate a valuable template for connecting rigorous a priori estimates with numerical variational optimization. The Burgers half of that claim is genuinely strong: the upper bound, the independently proven exponent 3/2, and the numerically identified saturating family are mutually consistent. The paper also deserves credit for stating its nonconvexity limitation explicitly in Section 7 and for reporting computational details in Appendix B that make the optimization results reproducible in principle. However, the 2D sharpness claim is not supported to the same standard: it rests on local maximizers of a nonconvex problem and on a fit to only part of the rigorous bound. The essay is therefore best read as a programmatic survey whose flagship 2D conclusion needs substantial qualification or additional evidence.

major comments (3)
  1. [§3.2, specifically the paragraph after Eq. (37) and the concluding paragraph] The claim that 'the combined estimate (34)–(35) is sharp and does not offer any room for improvement' is not supported by the evidence presented. The comparison is made to the ansatz f2(ν)=Cν^α, which the paper itself identifies with only the second argument of the max in (35); the first argument θ_{φ,p,M}(Cν e^{-CT}/2) is left untested because it is 'not given explicitly enough'. Moreover, the bound (35) carries the prefactor M^{1-1/p} with M=∥φ∥_{L∞}, and the constraint set S in Problem 3.4 fixes only P(φ)=P0 in H^1. Since H^1(T^2) does not embed in L∞, M is uncontrolled and could carry an additional ν-dependence. Agreement of the numerical envelope with a fitted power law therefore does not demonstrate saturation of the full rigorous bound; it only establishes consistency with one term of one side. A complete sharpness argument would require control of M and a comparison with the full expression, including the θ term.
  2. [§3.2 and §7, with Appendix B.4] The sharpness conclusion also rests on an unverified completeness of the branches of maximizers. Problem 3.4 is nonconvex, and the methods described in Appendix B compute local maximizers; Section 7 explicitly concedes that 'the maximizers found with the approach described in Appendix B are generally only local'. The upper envelope q̂ν^T in Fig. 3a is the maximum over the branches that were found, not a certified global supremum over S. Continuation from a limited set of seeds, described in Appendix B.4, cannot rule out the existence of another branch with larger enstrophy dissipation. If such a branch existed, the saturation claim would fail. Thus, unlike the Burgers problem in §3.1 — where the exponent 3/2 is independently proved by Albritton & Nitti (2023) — the 2D problem is not mathematically closed on the basis of this manuscript.
  3. [§3.2, fitting procedure around Eqs. (37)–(38) and Figures 3–4] Even taking the computed upper envelope at face value, the fit has limited evidentiary weight. Only five viscosity values are used, the fitting error is reported only as a mean absolute deviation over those five points, and the exponent α̃(T) is itself a fitted parameter determined by a bracketing procedure. No confidence intervals or sensitivity analysis are given for C(T) or α̃(T). Furthermore, no estimate is provided for how the omitted first argument of the max in (35) or the uncontrolled factor M^{1-1/p} would affect the prefactor. Consequently, the quantitative agreement in Fig. 3b cannot distinguish between saturation of the full bound and agreement with an effective power law over a narrow range of ν. A sharpness claim of this strength requires either a rigorous lower bound matching the full upper-bound expression or a certified global solution of Problem 3.4.
minor comments (5)
  1. [§2, Eq. (15)] The displayed definition of the Sobolev norm appears to contain a typographical error: it reads [1+(2πk)^s]^2, whereas the standard H^s norm on the torus uses (1+(2π|k|)^2)^s |û_k|^2, with |k| rather than the vector k in the scalar factor.
  2. [§2, Eqs. (14) and (17)] The definitions of kinetic energy and enstrophy appear to omit the square on the L^2 norms: K(u) should be (1/2)∥u∥_{L^2}^2 and E(u) should be (1/2)∥ω∥_{L^2}^2. The text later uses these quantities consistently with the squared norms, so this is a formatting issue rather than a substantive error.
  3. [§1.1.1] The sentence describing the Clay Millennium problem says the challenge was posed 'at the beginning of the 20th century'; the Navier–Stokes prize problem was posed in 2000, which is the beginning of the 21st century.
  4. [§3.1, Fig. 2(d) caption versus Eq. (28)] The caption to Fig. 2(d) states that the observed power laws have exponents 1 and 3/2, whereas Eq. (28) in the text reports a fitted exponent 1.531. These should be reconciled; the value 1.531 is presumably a finite-range fit estimate, but the present wording invites confusion about whether the claimed asymptotic exponent is 3/2 or 1.53.
  5. [§4.2.2 and §4.3] The Euler singularity search is presented with appropriate caution in most places, but the phrases in §4.3 'our search did produce a solution with a behavior consistent with singularity formation' and the abstract's mention of singularity search could be read as giving the Euler numerical evidence the same status as the rigorously supported Burgers result. A sentence explicitly stating that the Euler result is a resolution-dependent numerical indication, not a proof, would help calibrate expectations.

Circularity Check

1 steps flagged · score 6.0 of 10

The 2D sharpness claim rests on a fit of ansatz f2(nu) to the very data it is then said to confirm.

  1. fitted input called prediction [Section 3.2, Eqs. (37)-(38), Figs. 3b and 4; sharpness conclusion at end of Section 3.2]
    "To find out which of the functions (37a)-(37c) best describes the actual dependence of the data shown in figure 3a on nu, ... ansatz (37b) also involves an a priori undefined exponent alpha in (0,1). ... The most accurate fits were obtained with ansatz (37b) and the ratio qbar_nu^T/f2(nu) is plotted ... close to unity over the entire range of nu indicating that ansatz function f2(nu) accurately captures the dependence of qbar_nu^T on nu. ... these caveats notwithstanding, we conclude that the combined estimate (34)-(35) is sharp and does not offer any room for improvement."

    The function f2(nu)=C nu^alpha is not fixed by bound (35): the first argument of the max contains an unspecified function theta_{phi,p,M}, and M=||phi||_{L^infty} is not controlled on the H^1 constraint manifold. The constants C(T) and alpha(T) are chosen by minimizing fitting error (38) against qbar_nu^T, which is exactly the quantity whose viscosity scaling is being assessed. Thus plotting qbar_nu^T/f2(nu) displays the data divided by its own best-fit curve, so near-unity values are ensured by the least-squares construction rather than by saturation of the rigorous bound. The later claim that alpha(T) is approximately exponential in T is likewise a fit to the already-fitted exponents (Fig. 4).

full rationale

Most of the essay is not circular. The Burgers enstrophy-growth problem (Section 3.1) is closed by the independent rigorous result of Albritton & Nitti (2023); the earlier numerics of Ayala & Protas (2011) are explicitly corroborated by that theorem and by the independent sum-of-squares bounds of Fantuzzi & Goluskin (2020). The 3D searches are presented as exploratory, with explicit caveats that Problems 4.1-4.7 are nonconvex and only local maximizers are found, so no closed-form claim is made there. The circularity is concentrated in Section 3.2. There, the ansatz f2(nu) is chosen to mimic one argument of the external bound (35), but its exponent alpha is a free parameter fitted to the optimization data, and the ratio qbar/f2 close to unity is then reported as evidence that the bound is sharp. Because f2 is the best-fit curve to qbar, the agreement is tautological; the unspecified function theta_{phi,p,M} and uncontrolled M=||phi||_{L^infty} prevent (35) from fixing alpha. Thus the sharpness conclusion "does not offer any room for improvement" is not independently supported. The nonconvexity and branch-completeness gap is a separate correctness risk and is not itself circularity. Overall, there is one load-bearing fitted-input-as-confirmation step, so the paper is partially circular.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claims of sharpness rest on numerical fits and on nonconvex optimization solved only locally; no new entities are introduced. The main 'axiom' is that local maximizers saturate the true supremum, which the paper itself flags as a limitation.

free parameters (5)
  • Exponent alpha_tilde(T) in 2D dissipation ansatz f2 = decreasing from about 0.8 to about 0.3 over T range (Figure 4)
    Fitted via (38) and bracketing to numerical maxima of enstrophy dissipation; used to claim consistency with the upper bound (34)-(35).
  • Prefactor C1 and exponent alpha1 of enstrophy growth fit, Eq. (69) = C1 = 3.72e-3, alpha1 = 2.97 +/- 0.02
    Least-squares fit to numerical maximizers of Problem 4.1; used to infer sharpness of bound (46).
  • Fitted exponents alpha2 and prefactors C2 for Lq growth, Table 2 = e.g., q=4: alpha2 = 11.88 +/- 0.03, C2 = 2.9e-15
    Fits to numerical data from Problem 4.2 (Bleitner & Protas 2026, under review).
  • Exponents in finite-time fits (71a)-(71c) = 1.490, 1.18, 1.19
    Least-squares fits to max-over-T values; used to characterize transient growth in Navier-Stokes flows.
  • Prefactor C(t) = 0.0568 (ln ||u||)^{0.5742} for Euler singularity ansatz = 0.0568 and 0.5742
    Fitted to growth rate of ||u||_{dot H^3} in the resolution-1024 Euler computation; used to argue blow-up consistency.
assumptions (4)
  • standard math Standard inequalities: Young, Cauchy-Schwarz, Poincare, Gagliardo-Nirenberg, Gronwall (Appendix A).
    Used in all bound derivations.
  • domain assumption Regularity and periodic boundary conditions: solutions are classical on [0,T] on T^d, d=1,2,3.
    Assumed throughout (e.g., in Section 1.1, equations (1)-(5)).
  • ad hoc to paper The numerical maximizers of nonconvex PDE optimization problems represent the global extreme behavior.
    Needed to conclude sharpness of bounds in Sections 3.2 and 4.1; admitted as unproven in Section 7.
  • ad hoc to paper Resolution refinement of the pseudospectral discretization yields converging approximations to the variational problems and Euler trajectories.
    Underlies Figures 13-14 and the singularity inference.

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Cite this review

Pith. "Pith review of Extreme flows: where physics meets mathematically rigorous bounds." pith.science (2026). https://pith.science/paper/Z762ESNV

@misc{pith2026260804859,
  author       = {Pith},
  title        = {Pith review of: Extreme flows: where physics meets mathematically rigorous bounds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z762ESNV}},
  note         = {Machine review of arXiv:2608.04859}
}
read the original abstract

Extreme flows realize the largest possible growth, either instantaneously or in finite time, of certain quantities of interest which is achieved by a suitable choice of the initial condition or the applied forcing. The quantities of interest usually measure some small-scale properties and therefore provide information about the regularity of the flow. Extreme behavior is at the heart of several open problems in fluid mechanics including the dissipation anomaly in turbulence and formation of singularities in various models of fluid flow. In this essay we describe a framework making it possible to study such extreme behavior systematically by combining mathematical analysis, scientific computation and physics. As a first step, one aims to deduce rigorous upper bounds on the growth of the quantities of interest in the solutions of a given model. These inequalities express fundamental limitations on the most extreme behavior possible among {\em all} admissible solutions. However, given how they are obtained, these bounds may be conservative and overestimate the growth actually realizable in the system. In order to probe this possibility, as the next step, we set up variational optimization problems where the growth of the quantity of interest is maximized under suitable constraints. Solution of such problems is enabled by modern methods of numerical optimization. When properties of the thus obtained maximizers match the bounds, the bounds are declared sharp and therefore cannot be fundamentally improved. Finally, properties of the solutions saturating the bounds reveal insights about the physical mechanisms realizing the extreme behavior. We survey problems where this research program has produced sharp bounds together with extreme flows saturating these bounds. A collection of open problems is then presented and we close the essay with a discussion of possible methodological improvements.

Figures

Figures reproduced from arXiv: 2608.04859 by the authors.

Figure 1
Figure 1. (a) Solution y(t) of model problem (8) exhibiting blow-up as t → t0 = 1/y0; (b) Solutions u(ti , x) of system (5) with the initial condition u0(x) = sin(x) at different times ti ∈ [0, 1]; a singularity occurs at xs = 1/2 where limt→1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. (a) Optimal initial conditions ue0;E0,T obtained by solving Problem 3.2 with fixed enstrophy E0 = 103 and different time intervals: (thick solid line) T = 10−3 , (thin solid line) T = 10−2 , (thin dashed line) T = 10−1.5 , (thin dotted line) T = 10−1 and (thick dotted line) T = 100 ; the arrow indicates the trend with increasing T. (b) (red dashed lines) optimal initial conditions ue0;E0,T E0 max and (black solid li… view at source ↗
Figure 3
Figure 3. (a) Dependence of the maximum enstrophy dissipation χν(ϕq T ν ) on T for dif￾ferent indicated values of ν with distinct branches of local maximizers, cf [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Dependence of the optimal exponents αe = αe(T) in ansatz f2(ν) on the length T of the time window. The dashed line represents the exponential fit, in the form indicated, to the values of αe = αe(T). dependence of the exponent αe on T reveals an approximately exponentia…
Figure 5
Figure 5. Figure 5: Dependence of (a) p in (41) and (b) s in (56) on the index q with solid symbols representing the values considered in this essay. The dashed horizontal lines correspond to the limiting values of p and s obtained when q → ∞. Condition (43) implies that, if blow-up occur…
Figure 6
Figure 6. Figure 6: Local maximizers of Problem (4.1) in the limit E0 → 0 given by the eigen￾functions u1 of (67) with (a,d) |k| 2 = 1, (b,e) |k| 2 = 2 and (c,f) |k| 2 = 3. Figures (a–c) represent the isosurfaces defined by the the relation |∇ × u1|(x) = 0.95||∇ × u1||L∞, whereas figures …
Figure 7
Figure 7. Figure 7: (a,b) The dependence of the maximum rate of growth of enstrophy RE0 (ueE0 ) on E0 for (a) small E0 and (b) large E0 in solutions of Problem 4.1; the dashed lines in panels (a) and (b) represent, respectively, the asymptotic relation (68) and the RHS in estimate (46). (…
Figure 8
Figure 8. Figure 8: Maximizers (a) ueE0 of Problem 4.1 obtained for E0 = 20 and (b) ueB of Problem 4.2 obtained for q = 5 and B = 177.8. Both figures show the volume rendering of the vorticity magnitude |ω(x)| (in red) and selected vortex lines (in blue, with the color intensity proportio…
Figure 9
Figure 9. Figure 9: (a) Time evolution of the enstrophy E(u(t)) in the Navier-Stokes flows with the initial condition given by (black dashed line) the maximizer ueE0 of Problem 4.1 and (red solid lines) the maximizers ue0;E0,T of Problem 4.3 for E0 = 200 and T = 0.15, 0.23, 0.3 (the curve…
Figure 10
Figure 10. Figure 10: (a) Dependence of the exponent γ in expressions (72) on q for Navier-Stokes flows with the optimal initial conditions found by solving (blue) Problems 4.4 and 4.6 and (red) Problems 4.5 and 4.7. (b) Dependence of ∆E, cf. (73), on the minimum enstrophy Emin in Navier-S…
Figure 11
Figure 11. Figure 11: (a) Flow trajectories corresponding to the optimal initial data ue0;E0,TeE0 ob￾tained by solving Problem 4.3 with different E0 ∈ [100, 1000] shown using the coordinates {E, dE/dt} (blue solid lines with the arrow indicating the trend with the increase of E0); the thic…
Figure 12
Figure 12. Figure 12: (a–c) Vorticity components of the optimal initial condition ue0;E0,TeE0 obtained by solving Problem 4.3 for the initial enstrophy E0 = 500 and the corresponding optimal length TeE0 = 0.17 of the time interval; the time evolution of the flow corresponding to this initi…
Figure 13
Figure 13. Figure 13: Dependence of the maximum value of the objective functional ΦT (ue N 0;T )) obtained in the solution of Problem 4.10 with (a) T = 25 and (b) T = 75 on the numerical resolution N (Zhao & Protas, 2023). inferred from the slope of the tangent to the curves at ∥u(t)∥H˙ 3 …
Figure 14
Figure 14. Figure 14: (a) Dependence of (d/dt)∥u(t)∥H˙ 3 on ∥u(t)∥H˙ 3 in the Euler flows with the optimal initial conditions ue N 0;75 for t ∈ [0, 75] and different resolutions N3 and (b) the corresponding exponent α(t) in ansatz (89) obtained for N = 1024 with a local fit as a function o…
Figure 15
Figure 15. Figure 15: (a) Isosurfaces of |ω| and log10(||D| 3u|) in the terminal state u 1024 75; ue 1024 0;75  together with selected streamlines. (b) The vorticity component ω ⊥ normal to the symmetry plane x1 = x2 in the terminal state u 1024 75; ue 1024 0;75  . Animated versions of t…
Figure 16
Figure 16. Figure 16: Schematic representation of the operations performed at each iteration in (104) with the projection PTnMB and retraction RMB defined, respectively, in (119) and (120) (Kang & Protas, 2022). It can be regarded as a generalization of the standard line-search problem wit…

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