An efficient gradient projection method for stochastic optimal control problem with expected integral state constraint

Qiming Wang; Wenbin Liu

REVIEW 2 major objections 5 minor 45 references

An efficient gradient projection method for stochastic optimal control problem with expected integral state constraint

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper's central claim is that a gradient projection method with an explicit multiplier achieves first-order convergence for both control and multiplier in stochastic optimal control with an expected integral state constraint.

desk verdict Useful extension of the gradient-projection framework to expected integral state constraints, but the advertised first-order rate for the LSMC scheme rests on an unproved regression-error bound. read the letter →

arxiv 2412.17363 v1 pith:WLKMEYIC submitted 2024-12-23 math.OC

Qiming Wang , Wenbin Liu This is my paper · ORCID

classification math.OC MSC 60H3565K1065C2093E20

keywords stochasticoptimalcontrolexpectedintegralstateconstraintbackwarddifferentialequationgradientprojectionmethodleastsquareMonteCarlofirst-orderconvergenceLagrangemultiplier

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

This paper tackles stochastic optimal control problems in which the expected time-integral of the state is required to stay below a given level $\delta$. The authors derive the first-order optimality system through a Lagrange multiplier and a backward stochastic differential equation, then propose an iterative gradient projection method that enforces the state constraint at every step by constructing the multiplier explicitly. Discretizing the forward-backward system with Euler time stepping and approximating conditional expectations by least-squares Monte Carlo yields a fully computable scheme. The main claim is that both the control and the multiplier converge at first order in the time step, provided the Monte Carlo regression error is controlled. Numerical experiments, including a five-dimensional example, support the predicted rates.

What carries the argument

The load-bearing object is the projection operator with an explicit multiplier. For linear drift, the projection $P$ onto the convex feasible set $U(\delta)$ is realized by shifting the unprojected control along the adjoint direction $\psi(t)b'_u(t)$, where $\psi$ solves the ODE $-d\psi=(1+\psi b'_y(t))dt$, $\psi(T)=0$. The multiplier $\mu$ is selected so that the expected integral state exactly hits the constraint when the tentative control violates it, using the companion ODE $\phi$ whose integral $\int_0^T\phi(t)\,dt$ normalizes the correction. In the fully discrete scheme the same construction is applied to piecewise-constant controls, with the $L^2$ projection onto step functions and with $J'$ replaced by its LSMC approximation $J'_N$; the contraction argument for the iteration then converts the projection error and the gradient approximation error $\varepsilon_N$ into the stated first-order bounds.

What would settle it

Run Example 5.2 with $L=C N^2$ paths and measure $\varepsilon_N = \sup_i \|J'(u^{i,N}) - J'_N(u^{i,N})\|$ directly; if the measured quantity decays slower than $\Delta t$, say like $\Delta t^{1/2}$, then the first-order claim for the LSMC scheme fails because the deferred regression-error bound in Remark 4.8 is not satisfied. Equivalently, compare the convergence slopes with $L=N^2$ and $L=N^3$: if the slope drops below one when $L=N^2$, the missing regression term is the bottleneck.

Watch

Extended reading notes

Core claim

The central discovery is that for linear-drift state dynamics $b(y,u)=b'_y(t)y+b'_u(t)u+m(t)$, the projection onto the feasible control set $U(\delta)$ has an explicit form: a tentative gradient step $u^{i+1/2,N}=u^{i,N}-\rho J'_N(u^{i,N})$ is corrected by subtracting $\rho\,\mu^{i,N}_{h,KL}\sum_n \psi_n b'_u(t_n)\chi_{I_n^N}(t)$, where the multiplier $\mu^{i,N}_{h,KL}$ is chosen by the formula $\max\{E[\int_0^T \hat y^{i+1,N}_t dt]-\delta,0\}/(\rho \int_0^T \tilde\phi(t)\,dt)$. This choice guarantees $E[\int_0^T y^{i+1,N}_t dt]\le\delta$ at every iteration. The paper proves error bounds $\limsup_{i\to\infty}\|u^*-u^{i+1,N}\|\le C(\Delta t+\varepsilon_N)$ and $\limsup_{i\to\infty}|\mu^* - \mu^{i,N}_h|\le C(\varepsilon_N+\Delta t)$, with $\varepsilon_N=\sup_i\|J'(u^{i,N})-J'_N(u^{i,N})\|$, and argues that with $L=O(N^2)$ paths and suitable basis functions $\varepsilon_N=O(\Delta t)$, yielding first-order convergence.

Load-bearing premise

The promised first-order convergence depends on the unproved assumption that the statistical regression error in the least-squares Monte Carlo gradient is no larger than the time step; the paper leaves that bound to earlier work rather than proving it for its algorithm.

Editorial extensions

If this is right

If the error estimate is right, halving the time step halves the asymptotic error in both control and multiplier for linear-drift problems, so doubling $N$ yields slope-one convergence.
The state constraint is satisfied at every iterate by construction, not just in the limit, because the explicit multiplier formula is designed to keep $E[\int_0^T y^{i+1,N}_t\,dt]\le\delta$.
The method works without a space grid, using least-squares Monte Carlo, so it can be applied to higher-dimensional state spaces; a five-dimensional numerical example is included.
The grid-based version of the same algorithm also achieves $\varepsilon_N=O(\Delta t)$ under the cited regularity conditions, giving a first-order alternative when regression bases are hard to choose.
With $L=O(N^2)$ simulated paths and enough basis functions, the gradient approximation error is of order $\Delta t$, so the fully discrete scheme inherits first-order convergence from the temporal discretization.

Reading between the lines

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

A natural extension the paper does not pursue is to replace the single scalar constraint by a vector of expected integral constraints; the same shift-along-adjoint idea would need one multiplier per constraint, with a vector-valued normalization matrix replacing $\int_0^T\phi\,dt$.
The paper's error analysis separates time discretization error from projection error and Monte Carlo regression error; measuring those three terms separately on the paper's own examples would tell practitioners how to allocate $N$, $L$, and basis size.
The strong monotonicity and Lipschitz assumptions on $J'$ are likely to limit the method to convex-like problems; testing it on the nonconvex benchmark without exact solution in Example 5.3 with larger step sizes would reveal whether the contraction constant remains inside the unit interval.
If the deferred LSMC regression bound $O(\Delta t)$ fails for some practical basis choice, one could still retain first-order control convergence by using a higher-order accurate regression estimator or by coupling LSMC with a variance-reduction strategy; this is an inference, not a claim of the paper.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new thing is the explicit multiplier construction for the expected integral state constraint. That construction, Theorem 3.1, is clever and mostly correct, and it turns an implicit constraint into a computable gradient projection update. The paper is also the first numerical analysis for this class of SOCPs; the grid-method variant rests on the established estimate from [11] and looks solid. I would not fault the self-citation there, since the cited result is a published theorem.

The soft spot is the advertised first-order convergence of the LSMC scheme. Theorem 4.7 decomposes ε_N into temporal, projection, and empirical regression errors, but the two regression terms — the max over n of E[(P̂^K_n − P̂^{K,L}_n)^2] and the sum of Δt E[(Q̂^{K̃}_n − Q̂^{K̃,L}_n)^2] — are not controlled by any theorem in the paper. Remark 4.8 explicitly says the analysis is 'rather involved and technical' and defers to [27,28]. Since Theorems 4.2 and 4.3 both contain ε_N additively, the claim that Algorithm 1 converges at first order for the control and multiplier is not established. The numerical experiments are consistent with first order, but an experiment is not a proof. This is a load-bearing gap for the LSMC part of the paper, though not for the grid-method variant.

Two smaller points. In the proof of Theorem 4.2, the term α^{i+1}‖u∗,N − u0,N‖ is replaced by Δt times the same norm as i→∞; it should vanish. The conclusion is unaffected, but the displayed inequality is wrong as written. Also, the abstract says the method 'admits first order convergence' without the caveat that the LSMC result depends on an unproved bound; that should be qualified.

My recommendation is to send it to serious peer review. The core algorithmic idea is sound and the error-anatomy framework is useful; the missing regression bound is real but seems fixable, either by supplying the bound from the cited literature or by restating the LSMC result as first order up to an unquantified regression error. A referee with BSDE/LSMC background can push on this. If the gap is closed, this is a solid paper.

Referee Report

2 major / 5 minor

Summary. The paper develops a gradient projection method for stochastic optimal control problems with an expected integral state constraint under a linear drift assumption. It derives the first-order optimality system with a Lagrange multiplier, constructs an explicit projection by choosing the multiplier to enforce the state constraint, and then discretizes the resulting FBSDEs by the Euler scheme in time and by least-squares Monte Carlo in space. Error estimates for the control and the multiplier are stated, and the authors claim first-order convergence for both the LSMC scheme and the grid-method variant, supported by numerical experiments.

Significance. If the first-order claim were fully established, this would be a useful contribution: the explicit multiplier construction is elegant and self-contained, and extending efficient gradient projection to expected integral state constraints is nontrivial. The grid-method variant inherits a first-order estimate from the cited literature [11], and the numerical experiments are consistent with first-order rates for both the control and the multiplier. However, the genuinely new LSMC convergence claim depends on an empirical-regression error bound that is not proved, so the significance of the paper is currently conditional on closing that gap.

major comments (2)

[§4.2, Theorem 4.7 and Remark 4.8] The advertised first-order convergence of the fully discrete LSMC scheme is not established. Theorem 4.7 bounds ε_N^2 by a temporal term, two projection terms, and two empirical regression terms, namely max_n E[(P̂^K_n(y_n) − P̂^{K,L}_n(y_n))^2] and Σ_n Δt E[(Q̂^{K̃}_n(y_n) − Q̂^{K̃,L}_n(y_n))^2]. No theorem in the paper controls these two terms: Lemma 4.5 concerns only the projection error between P̂ and P̂^K, and the sentence after Remark 4.8 merely asserts that enough paths and basis functions make ε_N first-order. Remark 4.8 explicitly defers the analysis of the simulation step to [27,28]. Since Theorems 4.2 and 4.3 contain ε_N additively, the claimed O(Δt) convergence for Algorithm 1 is not proven as stated. The grid-method variant, whose ε_N=O(Δt) is cited from [11], is not affected by this gap.
[§4.2, Eq. (4.37)] The claimed Monte Carlo bound O(L^{-1/2}) in (4.37) treats the LSMC approximations P̂^{K,L}_n and Q̂^{K̃,L}_n as fixed functions while the empirical average is taken over the L paths. In the actual scheme these regression functions are estimated from the same L paths, so a valid estimate must include the regression-estimation error, which is exactly one of the terms left uncontrolled in Theorem 4.7. Thus the decomposition in Theorem 4.7 does not reduce the empirical regression error to a standard Monte Carlo error; it only relocates the difficulty. This is a load-bearing point for the paper's central claim, not a presentation issue.

minor comments (5)

[§4.2, Eq. (4.37) and surrounding text] The notation J′_N and J'_N is used for both the expectation-based estimator and the Monte Carlo estimator; the two objects should be distinguished by different symbols to make the proof readable.
[§4.1, Theorem 4.1 proof] In Eq. (4.14), the factor (1 + C_b/N)^N should read (1 + C_b T/N)^N since Δt = T/N; the displayed estimate loses T.
[§4.2, Eq. (4.15)] After 'i → +∞', terms such as α^{i+1}‖u^{*N} − u^{0,N}‖ vanish; they should not be bounded below by Δt‖u^{*N} − u^{0,N}‖. The intended statement is a limsup estimate, and the current display is confusing.
[§4.2, Eq. (4.30)] The constant C_ψ is used in (4.30) but never defined; it should be introduced as the constant from Theorem 4.1 controlling max_n |ψ(t_n) − ψ_n|.
[§5, numerical experiments] The numerical section does not report the number of basis functions K and K̃ used for the HC and VP bases. Since the unproved regression error depends on these parameters, reporting them would help the reader assess the gap between theory and experiment.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the LSMC first-order claim rests on an unproved empirical-error bound, which is a correctness gap, not a circular reduction.

full rationale

The derivation is self-contained in the relevant sense. Theorem 2.2 derives the optimality system from the Lagrangian and Robinson's regularity condition; Theorem 3.1 constructs the multiplier µ^{i,*} by formula (3.9) from the state-constraint violation and then verifies the projection inequality (3.4), so the projection is not fitted to the unknown solution. Theorems 4.2 and 4.3 are conditional estimates: they bound control and multiplier error by C(Δt+ε_N), where ε_N is the gradient approximation error. Theorem 4.7 splits ε_N into temporal, projection, and empirical regression terms. The empirical regression terms are not controlled by any theorem in the paper: Remark 4.8 explicitly defers this analysis to [27,28], and the closing sentence of §4.2 asserts first order by choosing L=O(N^2) and enough basis functions. That is an omitted proof or unsupported assumption, but not a circularity: the missing O(Δt) bound is neither assumed as the conclusion nor fitted from data, and [27,28] are external works. The self-citations to [11] and [12] provide a standard projection lemma (Lemma 2.3) and a grid-method error estimate used in Section 5; both are published, parameter-free results with stated assumptions and do not contain the target result, so they are real evidence and do not raise the circularity score.

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

No parameter is fitted to data; method constants ρ, L, N, ε0 are algorithmic choices, and theoretical constants C1,C2,cb are assumptions, not fitted values. No new physical or mathematical entities are postulated beyond the standard Lagrange multiplier and auxiliary ODE solutions ψ, φ, which are derived rather than invented.

assumptions (5)

domain assumption Robinson's regularity condition holds at the optimal control (Definition 2.1, eq. (2.4)).
Needed to guarantee existence of the Lagrange multiplier μ* via Theorem 1.56 of [40] in the proof of Theorem 2.2. It is assumed at the optimum and not verified for the problem class.
domain assumption Drift coefficient is affine in (y,u) with deterministic coefficient functions b'_y(t), b'_u(t), m(t) as in (3.1).
Restricts the method to linear drift; needed for U(δ) convexity and for the explicit projection formula in Theorem 3.1.
ad hoc to paper J'(·) is Lipschitz and uniformly monotone with constants C1,C2; u* and J'(u*) are Lipschitz.
Assumed in Theorem 4.2 for contraction of the gradient iteration; not derived from Assumption 2.1 and may fail for non-convex objectives.
domain assumption 0 < cb ≤ |b'_u| and |b''_{y,t}| + |b''_{u,t}| ≤ Ct (Assumption 4.1).
Used for positivity of the projection denominator and for first-order estimates of ψ and b'_u.
ad hoc to paper The LSMC projection error bounds of [25,27,32] apply, and the empirical regression error can be made O(Δt) by suitable basis dimension and path count.
Invoked to assert ε_N=O(Δt); no theorem in the paper quantifies the empirical regression term, and Remark 4.8 defers to the literature.

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Cite this review

Pith. "Pith review of An efficient gradient projection method for stochastic optimal control problem with expected integral state constraint." pith.science (2026). https://pith.science/paper/WLKMEYIC

@misc{pith2026241217363,
  author       = {Pith},
  title        = {Pith review of: An efficient gradient projection method for stochastic optimal control problem with expected integral state constraint},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WLKMEYIC}},
  note         = {Machine review of arXiv:2412.17363}
}

read the original abstract

In this work, we present an efficient gradient projection method for solving a class of stochastic optimal control problem with expected integral state constraint. The first order optimality condition system consisting of forward-backward stochastic differential equations and a variational equation is first derived. Then, an efficient gradient projection method with linear drift coefficient is proposed where the state constraint is guaranteed by constructing specific multiplier. Further, the Euler method is used to discretize the forward-backward stochastic differential equations and the associated conditional expectations are approximated by the least square Monte Carlo method, yielding the fully discrete iterative scheme. Error estimates of control and multiplier are presented, showing that the method admits first order convergence. Finally we present numerical examples to support the theoretical findings.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [11]

Gong, B., Liu, W.B., Tang, T., Zhao, W.D., Zhou, T.: An eﬃc ient gradient projection method for stochastic optimal control problems, SIAM J. Numer. Anal. 55(6), 2982- 3005 (2017)

work page 2017

[2] [1]

Korn, R.: Optimal Portfolios: Stochastic Models for Opti mal Investment and Risk Management in Continuous Time, W orld Scientiﬁc, Singapore (1997)

work page 1997

[3] [2]

Control Optim

Korn, R., Kraft, H.: A stochastic control approach to port folio problems with stochastic interest rates, SIAM J. Control Optim. 40(4), 1250-1269 (2001)

work page 2001

[4] [3]

Xu, W.S.: Maximum principle for a stochastic optimal cont rol problem and application to portfolio/consumption choice, J. Optimiz. Theory App. 98(3), 719-731 (1998). 30 Qiming W ang, W enbin Liu

work page 1998

[5] [4]

Sethi, S.P., Zhang, Q.: Hierarchical Decision Making in S tochastic Manufacturing Systems, Birkhauser, Boston (1994)

work page 1994

[6] [5]

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