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arxiv: 1906.11580 · v1 · pith:EQ4EYDFEnew · submitted 2019-06-27 · 🧮 math.OC

Gradient projection and conditional gradient methods for constrained nonconvex minimization

Maxim Balashov , Boris Polyak , Andrey Tremba This is my paper

Pith reviewed 2026-05-25 14:57 UTC · model grok-4.3

classification 🧮 math.OC
keywords gradient projectionconditional gradientmanifold optimizationnonconvex minimizationlinear convergenceLojasiewicz conditionFrank-Wolfe method
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The pith

Gradient projection on manifolds achieves global linear convergence when the objective meets the Lezanski-Polyak-Lojasiewicz condition.

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

The paper develops versions of the gradient projection algorithm suited to minimizing smooth functions on smooth manifolds such as spheres. It shows that this algorithm converges globally to a stationary point at a linear rate once the objective satisfies the Lezanski-Polyak-Lojasiewicz condition on the manifold. The authors also study the conditional gradient (Frank-Wolfe) algorithm and give conditions under which its full-step version converges globally with a linear rate. These guarantees apply under minimal natural assumptions on the problem data and supply convergence rates for two classical first-order methods on nonconvex constrained problems.

Core claim

Under the Lezanski-Polyak-Lojasiewicz condition on the manifold the gradient projection algorithm converges globally to a stationary point with linear rate; certain additional conditions likewise guarantee global linear convergence for the full-step conditional gradient method.

What carries the argument

The Lezanski-Polyak-Lojasiewicz condition on a manifold, which supplies a lower bound relating the gradient norm to the suboptimality gap.

If this is right

  • Gradient projection becomes a globally convergent method with an explicit linear rate for manifold-constrained problems obeying the condition.
  • The conditional gradient method can achieve global linear convergence without explicit projections under the stated conditions.
  • Problems such as optimization over the sphere gain reliable first-order solvers once the objective meets the manifold LPŁ inequality.
  • Linear rates allow practical stopping criteria based on observed function-value decrease.

Where Pith is reading between the lines

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

  • The same LPŁ condition could be checked numerically on concrete manifold problems to predict whether linear convergence will be observed in practice.
  • Rate comparisons between gradient projection and conditional gradient on identical instances would clarify when one method is preferable.
  • Extensions to other retraction-based methods on manifolds might follow if the LPŁ inequality can be verified after retraction.

Load-bearing premise

The objective function satisfies the Lezanski-Polyak-Lojasiewicz condition on the manifold.

What would settle it

A concrete smooth function on the unit sphere for which the gradient projection iterates fail to approach a stationary point at a linear rate while all other stated assumptions hold.

Figures

Figures reproduced from arXiv: 1906.11580 by Andrey Tremba, Boris Polyak, Maxim Balashov.

Figure 1
Figure 1. Figure 1: Lemma 1. We have (13) f(x1) − f(x0) = f(x1) − f(z) + f(z) − f(x0), kx0 − zk = tkPTx0 f 0 (x0)k, (14) kx1 − zk ≤ R − p R2 − kx0 − zk 2 ≤ kx0 − zk 2 R = t 2kPTx0 f 0 (x0)k 2 R [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Stationary point (0, 0) is not a solution. 5. Appendix Proposition 1. Suppose that the set Q ⊂ R n is proximally smooth with constant R > 0, the function f has the Lipschitz continuous gradient f 0 with constant L1 > 0 and the point x0 is a local minimum in the problem minQ f. Then −f 0 (x0) ∈ N(Q, x0). Proof. If f 0 (x0) = 0 then −f 0 (x0) = 0 ∈ N(Q, x0) by the definition of normal cone. Assume that f 0 (… view at source ↗
Figure 3
Figure 3. Figure 3: Angle ϕ. Hence f(x) − f(x0) ≤ −kf 0 (x0)kε(p, q) + L1 2 ε 2 ≤ −kf 0 (x0)kεC + L1 2 ε 2 < 0 for sufficiently small ε > 0. A contradiction. Acknowledgements The work was supported by Russian Science Foundation (Project 16-11-10015). References [1] P.-A. Absil, R. Mahony, R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Prince￾ton University Press, Princeton and Oxford, 2008. [2] M. V. Balashov, Uni… view at source ↗
read the original abstract

Minimization of a smooth function on a sphere or, more generally, on a smooth manifold, is the simplest non-convex optimization problem. It has a lot of applications. Our goal is to propose a version of the gradient projection algorithm for its solution and to obtain results that guarantee convergence of the algorithm under some minimal natural assumptions. We use the Lezanski-Polyak-Lojasiewicz condition on a manifold to prove the global linear convergence of the algorithm. Another method well fitted for the problem is the conditional gradient (Frank-Wolfe) algorithm. We examine some conditions which guarantee global convergence of full-step version of the method with linear rate.

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 / 2 minor

Summary. The manuscript proposes a gradient-projection algorithm for minimizing a smooth function on a smooth manifold (e.g., the sphere) and proves its global linear convergence under the Lezanski-Polyak-Lojasiewicz condition adapted to the manifold. It additionally examines conditions guaranteeing global linear convergence of the full-step conditional-gradient (Frank-Wolfe) method.

Significance. If the stated proofs hold, the work supplies explicit linear-rate guarantees for two standard first-order methods on manifolds under a single, clearly identified geometric assumption (the manifold LPL inequality). This conditional structure is a strength: the paper does not claim the rates for arbitrary smooth objectives. The result is a modest but useful extension of classical LPL-based analysis to the constrained manifold setting.

minor comments (2)
  1. [Abstract] Abstract: the spelling 'Lezanski-Polyak-Lojasiewicz' should be aligned with the conventional 'Łojasiewicz-Polyak-Łojasiewicz' (or LPL) throughout the manuscript for consistency with the literature.
  2. [Introduction / Main results] The manuscript asserts that 'proofs exist' for the linear rates; the full derivations, error-bound constants, and handling of the tangent-space projection should be cross-checked against the manifold geometry in the main text to ensure the constants are explicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recognition of its conditional linear-rate guarantees under the manifold LPL inequality. The recommendation for minor revision is appreciated.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes convergence of gradient-projection and conditional-gradient methods on manifolds under the explicit hypothesis that the objective satisfies the Lezanski-Polyak-Lojasiewicz inequality on the manifold. Linear rates are derived directly from this stated assumption; the proofs are conditional by design and do not reduce any claimed rate to a fitted parameter, a renamed input, or a self-citation chain. No enumerated circularity pattern appears in the derivation architecture.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard domain assumptions from smooth optimization; no free parameters, invented entities, or ad-hoc axioms are mentioned.

axioms (2)
  • domain assumption The objective function is smooth on the manifold.
    Required for gradients to exist and for the projection step to be well-defined.
  • domain assumption The constraint set is a smooth manifold.
    Stated explicitly as the setting for both algorithms.

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