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arxiv: 2604.04602 · v1 · submitted 2026-04-06 · 📡 eess.SY · cs.SY· math.OC

Stochastic Model Predictive Control with Online Risk Allocation and Feedback Gain Selection

Filipe Marques Barbosa , Johan L\"ofberg This is my paper

Pith reviewed 2026-05-10 19:36 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords stochastic model predictive controlchance constraintsrisk allocationfeedback gain selectionmixed-integer conic programmingprobit approximationpath planning
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The pith

Stochastic model predictive control with joint optimization of risk allocation and feedback policies can be reformulated as a mixed-integer conic program.

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

The paper seeks to make stochastic model predictive control tractable when both risk allocation and feedback policies must be chosen online. Normally this produces nonconvex problems because chance constraints contain products of the feedback law with the risk variables and because they embed the nonconvex probit function. By deriving disjunctive convex chance constraints, restricting the feedback law to a finite precomputed set of candidates, and replacing probit compositions with power- and exponential-cone representable approximations, the authors obtain a mixed-integer conic program. This program can be solved with standard off-the-shelf solvers and extends to general chance constraints that mix exclusive disjunctions with Gaussian variables. The approach is illustrated on a path-planning task.

Core claim

The paper establishes that disjunctive convex chance constraints together with power- and exponential-cone approximations to the probit function allow the simultaneous optimization of risk allocation and feedback gains in stochastic MPC to be cast as a mixed-integer conic optimization problem that is solvable by existing software, while preserving the original probabilistic guarantees up to the accuracy of the chosen approximations.

What carries the argument

Disjunctive convex chance constraints that replace nonconvex products of feedback laws and risk allocations, combined with power- and exponential-cone representable approximations to the probit function, yielding a mixed-integer conic program.

If this is right

  • The stochastic MPC problem becomes solvable in real time with standard conic solvers rather than requiring custom nonconvex algorithms.
  • The same reformulation applies directly to any chance constraint that contains products of exclusive disjunctive variables and Gaussian random variables.
  • Feedback policies can be adapted online while risk is allocated at each step without sacrificing convexity.
  • Path-planning problems with probabilistic obstacle avoidance become practical instances of the new formulation.

Where Pith is reading between the lines

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

  • The method could support safety-critical applications where risk levels must be adjusted on the fly according to changing conditions or priorities.
  • If the candidate feedback set is generated by a learning procedure, the overall scheme could combine data-driven policy selection with formal probabilistic guarantees.
  • Similar cone approximations might be developed for other quantile functions, opening the approach to non-Gaussian uncertainties.
  • The computational scaling with the number of candidate feedback laws could be studied to determine how large the precomputed set can be before real-time feasibility is lost.

Load-bearing premise

The power- and exponential-cone approximations to the probit function are accurate enough that the resulting constraints still deliver the intended probabilistic guarantees, and that restricting the feedback law to a finite precomputed set does not materially degrade closed-loop performance.

What would settle it

Run the closed-loop system on the path-planning example, measure the empirical frequency of constraint violations, and check whether it stays at or below the prescribed risk levels; separately compare solution times and achieved cost against a two-stage nonconvex solver on identical instances.

Figures

Figures reproduced from arXiv: 2604.04602 by Filipe Marques Barbosa, Johan L\"ofberg.

Figure 1
Figure 1. Figure 1: The curves of the nonconvex function compositions resulting from the formulations in Sec. IV-A (solid) and their corresponding exponential cone-representable approximations (dashed). In this way, a lower power cone-representable approxima￾tion Ψ inv(γ) ≈ 1 Φ−1(1 − γ) (46) can be constructed using a fractional power function combined with compensation terms that preserve convexity and tractabil￾ity as Ψ inv… view at source ↗
Figure 2
Figure 2. Figure 2: A polyhedron representing a stay-in region I encoded as a conjunction of linear equality constraints. Finally, the nonconvex functions in (36), (39), and (43) can be removed from the constraints and their correspond￾ing power and exponential cone-representable approximations used instead. These approximations enable expressing the problems as mixed-integer cone optimization programs and solving them effici… view at source ↗
Figure 4
Figure 4. Figure 4: 1000 Monte Carlo simulations of the state predictions at the first sampling instant, obtained using the best and worst feasible integer solutions. The plots show the prediction envelopes of the best integer solution Xmin (gray) and the worst integer solution Xmax (magenta). The results in each subplot correspond to different parameter ranges used to construct M(δ). −1 −0.5 0 0.5 −0.4 −0.2 0 0.2 x(m) y(m) T… view at source ↗
Figure 5
Figure 5. Figure 5: 100 Monte Carlo simulations of the performed trajectories in Case 1 (without obstacles). The vehicle navigates inside I (light green), from the initial state x0 = [0, −1.18, 0, 0.16]⊤, to T (dark green). As the vehicle’s initial position is close to the boundaries of I, a safer trajectory toward T involves moving away from these limits. of keeping a safe distance from the boundaries of I and O, which conse… view at source ↗
Figure 6
Figure 6. Figure 6: 100 Monte Carlo simulations of the performed trajectories in Case 2 (with obstacles). The vehicle navigates within I (light green), from the initial state x0 = [0, −1, 0, 0]⊤, toward T (dark green), while avoiding O (red). Larger weights on the risk allocation (S = diag (10, 10)) produce the trajectories shown with dark lines. Lower weights on the risk allocation (S = diag (1, 1)) produce the trajecto￾ries… view at source ↗
Figure 7
Figure 7. Figure 7: The time to solve the optimization problem at every sampling time in both cases. The solution times of the Inverse (blue), Root (yellow), and Logarithm (orange) probit approaches are compared to the exhaustive search method (dashed gray). Note the logarithm scale for the solution times in Case 2. constraints involving mutually exclusive binary variables mul￾tiplying Gaussian stochastic variables, that is, … view at source ↗
read the original abstract

Stochastic Model Predictive Control addresses uncertainties by incorporating chance constraints that provide probabilistic guarantees of constraint satisfaction. However, simultaneously optimizing over the risk allocation and the feedback policies leads to intractable nonconvex problems. This is due to (i) products of functions involving the feedback law and risk allocation in the deterministic counterpart of the chance constraints, and (ii) the presence of the nonconvex Gaussian quantile (probit) function. Existing methods rely on two-stage optimization, which is nonconvex. To address this, we derive disjunctive convex chance constraints and select the feedback law from a set of precomputed candidates. The inherited compositions of the probit function are replaced with power- and exponential-cone representable approximations. The main advantage is that the problem can be formulated as a mixed-integer conic optimization problem and efficiently solved with off-the-shelf software. Moreover, the proposed formulations apply to general chance constraints with products of exclusive disjunctive and Gaussian variables. The proposed approaches are validated with a path-planning application.

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

3 major / 2 minor

Summary. The paper claims that simultaneous optimization of risk allocation and feedback policies in stochastic MPC leads to intractable nonconvex problems due to products involving feedback laws and the nonconvex probit function in chance constraints. It addresses this by deriving disjunctive convex chance constraints, restricting feedback selection to a finite set of precomputed candidates, and replacing probit compositions with power- and exponential-cone representable approximations. The resulting problem is cast as a mixed-integer conic program solvable with off-the-shelf solvers, with extensions to general chance constraints involving exclusive disjunctions and Gaussian variables, and validation on a path-planning application.

Significance. If the cone approximations are shown to be conservative or equipped with explicit error bounds that can be absorbed into the risk allocation, and if the finite feedback candidate set does not materially degrade closed-loop performance, the approach would provide a tractable, software-friendly method for online risk allocation in SMPC. This could be significant for applications requiring probabilistic guarantees under uncertainty, as it avoids two-stage nonconvex optimization while extending to broader classes of disjunctive-Gaussian chance constraints.

major comments (3)
  1. [Reformulation of chance constraints and cone approximations] The central claim relies on the probit approximations preserving the original chance-constraint violation probabilities. However, the manuscript provides no derivation of error bounds, no proof of outer-approximation conservatism, and no quantitative assessment of how approximation errors propagate into the risk allocation (see the reformulation steps following the derivation of disjunctive convex constraints). Without this, the solved MICP policies may violate the allocated risk levels in closed loop.
  2. [Feedback gain selection from precomputed candidates] Restricting the feedback law to a finite precomputed candidate set is presented as enabling convexity, but the paper does not quantify the resulting suboptimality gap relative to the unrestricted case or demonstrate that the candidate set is rich enough to achieve near-optimal performance for the path-planning example (see the section on feedback gain selection).
  3. [Numerical validation] The validation on the path-planning application reports successful solution times but provides no Monte Carlo closed-loop violation probability estimates or comparison against the true (non-approximated) chance-constraint satisfaction rates. This leaves the probabilistic guarantees unverified numerically.
minor comments (2)
  1. [Notation and problem formulation] Notation for the disjunctive variables and the mapping from feedback candidates to the chance-constraint reformulation could be clarified with an explicit table or diagram.
  2. [Abstract and introduction] The abstract states that the formulations 'apply to general chance constraints with products of exclusive disjunctive and Gaussian variables,' but the manuscript should include a brief remark on whether this requires additional assumptions on the disjunction structure.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough and constructive review of our manuscript. We address each major comment point by point below, indicating where revisions will be made to strengthen the presentation and analysis.

read point-by-point responses

  1. Referee: The central claim relies on the probit approximations preserving the original chance-constraint violation probabilities. However, the manuscript provides no derivation of error bounds, no proof of outer-approximation conservatism, and no quantitative assessment of how approximation errors propagate into the risk allocation (see the reformulation steps following the derivation of disjunctive convex constraints). Without this, the solved MICP policies may violate the allocated risk levels in closed loop.

    Authors: We agree that the manuscript would benefit from an explicit treatment of approximation errors. The power- and exponential-cone approximations are constructed as conservative outer approximations of the probit function, which ensures that satisfaction of the approximated constraints implies satisfaction of the original chance constraints at the allocated risk level. In the revised manuscript we will add a dedicated subsection deriving the approximation error bounds and demonstrating how these bounds can be absorbed into the risk allocation by a conservative adjustment of the risk parameters. This will provide a rigorous guarantee that the MICP solutions remain probabilistically safe. revision: yes

  2. Referee: Restricting the feedback law to a finite precomputed candidate set is presented as enabling convexity, but the paper does not quantify the resulting suboptimality gap relative to the unrestricted case or demonstrate that the candidate set is rich enough to achieve near-optimal performance for the path-planning example (see the section on feedback gain selection).

    Authors: The finite candidate set is introduced to restore convexity while retaining the ability to optimize over feedback policies online. The candidates are drawn from standard linear feedback structures commonly used in stochastic control. For the path-planning example the resulting closed-loop trajectories satisfy all constraints and exhibit good performance. To address the suboptimality concern we will augment the revised manuscript with a quantitative comparison of closed-loop cost and feasibility against an expanded candidate set, thereby illustrating that the chosen set is sufficiently rich for the considered application. revision: yes

  3. Referee: The validation on the path-planning application reports successful solution times but provides no Monte Carlo closed-loop violation probability estimates or comparison against the true (non-approximated) chance-constraint satisfaction rates. This leaves the probabilistic guarantees unverified numerically.

    Authors: We concur that empirical verification of the probabilistic guarantees is important. In the revised version we will add Monte Carlo closed-loop simulations that estimate empirical violation probabilities for the obtained policies and compare these rates to the allocated risk levels. Where tractable, we will also include comparisons against the non-approximated (non-conic) formulation to quantify the effect of the cone approximations on realized safety. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper derives disjunctive convex chance constraints by reformulating products involving feedback laws and risk allocations, then replaces probit compositions with power- and exponential-cone approximations to obtain a mixed-integer conic program. These steps rely on standard convex relaxation techniques and cone representability results external to the paper; the finite candidate set for feedback gains is precomputed independently and does not reduce the claimed probabilistic guarantees to a tautology or fitted parameter defined by the authors' own prior outputs. No load-bearing self-citations, self-definitional loops, or renaming of known results appear in the derivation chain. The formulation is presented as a general method validated on a path-planning example, confirming independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard domain assumptions in stochastic control but introduces new reformulation techniques whose correctness depends on approximation quality.

axioms (1)
  • domain assumption Disturbances follow a Gaussian distribution.
    Implicit in the use of the probit function and chance constraints throughout the abstract.

pith-pipeline@v0.9.0 · 5475 in / 1353 out tokens · 40434 ms · 2026-05-10T19:36:26.254890+00:00 · methodology

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Reference graph

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