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A Discretization Approach for Bilevel Optimization with Low-Dimensional and Non-Convex Lower-Level

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Pith's one-line read The paper proposes replacing the lower-level value function of a bilevel problem with a convexified discretized surrogate, and proves that solving the resulting penalized problem yields approximate global solutions of the relaxed bilevel…

desk verdict The discretization idea is genuinely new and worth engaging, but the advertised reach exceeds the proven results: Assumption 3.1 does a lot of work and should be front and center. read the letter →

arxiv 2505.10830 v1 pith:RDJ5QUTJ submitted 2025-05-16 math.OC

Xiaotian Jiang , Ioannis Tsaknakis , Prashant Khanduri , Mingyi Hong This is my paper · ORCID
classification math.OC MSC 90C2690C3090C46
keywords bileveloptimizationvaluefunctionapproximationdiscretizationnon-convexlower-levelpenaltymethodprojectedgradientdescentlow-dimensionalsimplexreformulation
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

Bilevel optimization usually becomes intractable when the lower-level problem is constrained and non-convex: the lower-level value function is then non-differentiable and effectively impossible to evaluate. The paper's approach is to discretize the lower-level feasible set, represent the lower-level solution as a probability vector over sampled points, and replace the value function by a strongly convex simplex minimization whose solution is a projection. It proves that, under a convexity-and-growth condition on lower-level sublevel sets, global solutions of the discretized problem are $\delta$-global solutions of the standard value-function relaxation, with the approximation error controlled by the sampling radius and the regularization parameter. It then shows that a quadratic penalty reformulation shares global minima, local minima, and KKT points with the discretized problem, and gives a projected gradient algorithm with $O(1/T)$ convergence. A sympathetic reader would care because this is a route to constrained non-convex lower-level problems with provable approximation and convergence guarantees.

What carries the argument

The load-bearing object is the pair $(\tilde V_\lambda, \text{(BLO-DISC)})$: the approximate value function $\tilde V_\lambda$ converts a non-convex minimization over $y$ into a strongly convex minimization over the simplex, making the bilevel constraint smooth, and the discretized problem reparameterizes $y$ as $\sum_i p_i y^{(i)}$. The argument runs through a chain of equivalences. Lemma 3.1 uses Assumption 3.1 and the covering condition to show that a feasible weighted combination of samples is feasible for the value-function relaxation; Theorem 3.1 converts that into $\delta$-global solution equivalence in both directions. Lemma 3.2 bounds the penalty feasibility gap at KKT points by $5L_fD/(2\gamma)$, and Proposition 3.1 transfers KKT, local, and global solution status from the penalty problem to the discretized problem. On the computational side, the identity that $p^*(x)$ is a Euclidean projection onto the unit simplex makes each iteration of DIVIDE-BLO cheap, and the derived Lipschitz constant drives the $O(1/T)$ projected-gradient convergence bound.

What would settle it

Construct a lower-level problem with two narrow wells of nearly equal depth where one well is smaller than the grid spacing, so Assumption 3.1 fails. Run DIVIDE-BLO with a grid that misses the small well; if the returned induced point is not a $\delta$-global solution of the value-function relaxation because a better $f$-value exists inside the missed well, that observation would refute the equivalence claim outside its stated assumption.

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Extended reading notes

Core claim

At the center of the paper is the discretized value function $\tilde V_\lambda(x)=\min_{p\in\Delta}\sum_{i=1}^k p_i g(x,y^{(i)})+\frac{\lambda}{2}\|p\|^2$, where $\{y^{(i)}\}$ is a covering of the lower-level set $Y$ of radius $r$ and $\Delta$ is the unit simplex. This function is differentiable and has a closed-form gradient through the argmin $p^*(x)$, which is the projection of $(-g(x,y^{(1)})/\lambda,\dots,-g(x,y^{(k)})/\lambda)$ onto the simplex. The paper's main equivalence theorem states that if $(x^*,p^*)$ is a global solution of the discretized problem with relaxation coefficient $\epsilon_1\le[\epsilon_0/(1+4DL_g)]^2$, then $(x^*, \sum_i p_i^* y^{(i)})$ is a $\delta$-global solution of the value-function relaxation with coefficient $(1+4DL_g)\sqrt{\epsilon_1}$, under Assumption 3.1; a converse direction holds as well. The same section proves that the penalty problem (BLO-PEN) and the discretized problem (BLO-DISC) have identical global minima, local minima, and KKT points, so solving the smooth penalized problem is a valid surrogate for the discretized problem.

Load-bearing premise

Everything rests on Assumption 3.1: the lower-level sublevel sets just above the minimum must be convex for small gaps, and any near-optimal $y$ must lie within a power-law distance $C\,|g(x,y)-V(x)|^\beta$ of the optimal set; without this, a finite grid can miss the true feasible region and the claimed equivalence between the discretized and relaxed problems collapses.

Editorial extensions

If this is right

  • For any prescribed accuracy, taking the sampling radius and $\lambda$ small enough makes the discretized problem track the value-function relaxation at the $\delta$-global level, so the approximation error is controllable by the number of sample points $k$ and by $\lambda$.
  • Every KKT point obtained by the penalty method is a KKT point of the discretized problem for some relaxation coefficient $\epsilon\le 5L_fD/(2\gamma)$; increasing the penalty parameter $\gamma$ forces the solutions closer to feasibility.
  • Projected gradient descent on the penalized objective converges in finite time to a stationary point at rate $1/T$, and any convergent subsequence reaches a KKT point of both the penalized and discretized problems.
  • In the numerical study, on a non-convex synthetic lower-level problem the method consistently reached zero lower-level violation and zero total gap, while two comparison algorithms sometimes stalled at infeasible local optima; on the ensemble-learning application the weighted model outperformed every base model.

Reading between the lines

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

  • The method's practical reach is governed by the covering number bound $k=O((D\sqrt{m}/r)^m)$, so the low-dimensional lower-level assumption is doing real work; a fair extension would be an explicit sample-complexity statement linking $r$ to $\epsilon_1$ and $\delta$.
  • Because Assumption 3.1 is exactly what rules out the missed-region failure in the paper's own Figure 2, a cheap empirical diagnostic is to rerun DIVIDE-BLO with two independent grids of the same radius and compare the induced solutions; disagreement would signal that the convex-sublevel-set condition is close to failing.
  • The convex-hull-of-samples construction is a general template: any non-convex lower-level problem whose near-optimal sublevel sets are convex could be attacked by the same discretize-and-convexify scheme, and it suggests trying adaptive or random sampling schemes with variance-reduced estimators.
  • The $\delta$ in the equivalence grows like $\epsilon_1^{\beta/2}$ through the growth exponent $\beta$; testing problems with $\beta=1$ versus $\beta<1$ would show how the approximation guarantee degrades in practice.
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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

4 major / 3 minor

Summary. The paper proposes a discretization approach for optimistic bilevel programs with possibly nonconvex and constrained lower-level problems. It approximates the lower-level value function V(x)=min_{y∈Y} g(x,y) by Vλ(x)=min_{p∈Δ} Σ_i p_i g(x,y^{(i)}) + (λ/2)||p||² over a covering set {y^{(i)}}, and studies three reformulations: the relaxed value-function problem (BLO-VF), the discretized problem (BLO-DISC), and a penalty problem (BLO-PEN). The central theoretical result is a δ-global solution equivalence between (BLO-VF) and (BLO-DISC) under Assumption 3.1, a one-directional KKT/local/global solution correspondence between (BLO-PEN) and (BLO-DISC), and a projected-gradient algorithm (DIVIDE-BLO) with a O(1/T) stationarity convergence guarantee. Numerical experiments on a one-dimensional nonconvex lower-level problem and on an ensemble-learning hyperparameter-selection problem illustrate the approach.

Significance. If the results hold, the paper offers a genuinely different surrogate for constrained nonconvex lower-level bilevel problems: the regularized discretized value function is smooth and easy to evaluate, in contrast to the nonsmooth original value function. The paper is largely self-contained, with complete proofs of Proposition 2.1, Lemma 3.2, Proposition 3.1, and Theorem 4.1, and the DIVIDE-BLO algorithm is clearly specified and reproducible in structure. The main limitation is that the equivalence between (BLO-DISC) and the relaxed problem (BLO-VF) depends on Assumption 3.1, which requires convex near-optimal sublevel sets and a Hölderian distance-to-argmin bound; this restriction is not disclosed in the abstract and substantially narrows the advertised 'non-convex' scope. The algorithm itself and Theorem 4.1 do not use Assumption 3.1, so the method may still be useful beyond the equivalence guarantee, but the paper should say so explicitly.

major comments (4)
  1. [Abstract and Section 3.1, Assumption 3.1] The abstract and Contributions 1–2 present the method as handling non-convex and constrained lower-level problems, but the only result connecting (BLO-DISC) to (BLO-VF), Theorem 3.1, relies on Assumption 3.1, which requires convex sublevel sets for all ε≤ε0 and a Hölderian distance-to-argmin bound. This is a substantial restriction: functions such as g(y)=(||y||²-1/4)² on the unit disk have disconnected sublevel sets and violate Assumption 3.1(1), and the paper's own Figure 2 shows that without the convexity, grid sampling can miss the feasible region. The abstract and contribution list should state this assumption explicitly, for example by saying the equivalence holds for nonconvex lower-level problems whose near-optimal sublevel sets are convex and which satisfy a Hölderian error bound, rather than implying general nonconvex lower levels.
  2. [Appendix A.2, proof of Theorem 3.1 Part 1] In the feasibility check for (x',p') in the proof of Theorem 3.1 Part 1, the chain of inequalities uses |V(x')-Vλ(x')| ≤ λ/2, but Proposition 2.1(b) gives |V(x')-Vλ(x')| ≤ 2LgD√m/k^{1/m} + λ/2, and the covering term is not negligible in general. The intended conclusion can be recovered because Vλ≥V, as shown in the proof of Proposition 2.1, so that g(x',y^{(i)})-Vλ(x') ≤ g(x',y^{(i)})-V(x') suffices. As written, however, the displayed absolute-value bound is not valid and the proof should be corrected.
  3. [Section 3.2 / Contribution 2] Contribution 2 states that (BLO-PEN) and (BLO-DISC) 'share the same set of local minima, global minima, and KKT points', but Proposition 3.1 only establishes the forward direction: a KKT/local/global solution of (BLO-PEN) is also a corresponding solution of (BLO-DISC) for an ε that depends on the point and on γ. No converse is proved. This one-directional statement is sufficient for using (BLO-PEN) as a surrogate, but the 'same set' phrasing overstates the result and should be revised to reflect the actual direction established.
  4. [Lemma 3.1 and Theorem 3.1] The proofs of Lemma 3.1 and Theorem 3.1 repeatedly use the step that a sample point y^{(i)} exists with ∥y^{(i)}-y*∥≤ε1/(2Lg), citing the condition k≥(4LgD√m/ε1)^m. However, Assumption 2.2 only says eY is a covering of some radius r; a set of k points does not automatically have radius ≤ε1/(2Lg merely because k meets the lower bound for the minimal covering number. The statements should either explicitly require r≤ε1/(2Lg) (or that eY is chosen with this radius), or the proofs should be revised to state this additional condition, otherwise the key covering step does not follow from the assumptions as written.
minor comments (3)
  1. [Section 3, first paragraph] The sentence 'we will establish the ... global/local/KKT solution equivalence between (BLO-VF) and (BLO-DISC) in Section 3.2' appears to be a typo: Section 3.2 actually treats (BLO-DISC) and (BLO-PEN), and Figure 1 confirms this.
  2. [Lemma 4.1, Eq. (4.1)] The expression '(Lg+Lg)' in the formula for Lγ is confusing even though it is consistent with the proof; it would be clearer to write 2Lg or to define two different Lipschitz constants for ∇g and g.
  3. [Proof of Proposition 2.1(b)] In the proof of part (b), the line 'The equality in (a)' refers to the inequality just derived in part (b), not to part (a); this cross-reference should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the discretized value-function and penalty equivalences are proved from explicit Lipschitz/covering assumptions; the only flagged issue is a non-circular overclaim about the one-directional penalty equivalence.

full rationale

The derivation chain is self-contained. The approximate value function Ṽ_λ(x) is defined directly from the sampled values g(x,y^(i)) in (DISC-VF), and Proposition 2.1 bounds its error via the covering radius and Lipschitz constants rather than by assuming the target result. Theorem 3.1's δ-global equivalence between (BLO-VF) and (BLO-DISC) is an explicit proof using Lemma 3.1, Assumption 3.1 (convex sublevel sets and a Hölderian error bound), and the covering radius; it is not a restatement of the definition. The penalty result in Proposition 3.1 is one-directional and chooses ε=G(x*,p*) to satisfy complementary slackness, which is the standard exact-penalty construction and is transparently stated; the Contribution 2 phrase 'share the same set' overstates this ε-dependent, one-directional result, but that is a soundness/framing issue rather than a circular reduction. Self-citations ([10], [12], [19], [27], [28]) are used only as related work, baseline algorithms, or standard background and are not load-bearing for the new theorems. The main caveat—Assumption 3.1's convex-sublevel-set condition is absent from the abstract's 'non-convex lower-level' claim—concerns scope and correctness transparency, not circularity. No fitted parameters are renamed as predictions.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The central claims rest on standard smoothness/compactness assumptions, the existence of a covering of the lower-level feasible set, and the strong regularity condition in Assumption 3.1. The algorithm parameters k, λ, γ, and ε are user-chosen hyperparameters, not fitted to data. No new physical or mathematical entities are postulated; Ṽ_λ is a constructed mathematical object rather than an invented entity with independent evidence requirements.

free parameters (4)
  • k (number of discretization points)
    User-chosen. The approximation error bound in Proposition 2.1(b) and the sufficient conditions in Theorem 3.1 depend on k. Larger k improves accuracy at higher computational cost.
  • λ (quadratic regularization in Ṽ_λ)
    User-chosen. It must satisfy λ ≤ ε1 in Theorem 3.1 and λ ≤ L_f D / γ in Lemma 3.2. The approximation error includes a λ/2 term, so smaller λ gives better approximation.
  • γ (penalty parameter)
    User-chosen. The feasibility gap for KKT points of (BLO-PEN) relative to (BLO-DISC) is bounded by O(1/γ) in Lemma 3.2. Larger γ improves feasibility but may affect conditioning.
  • ε1 or ε (relaxation coefficient)
    User-chosen small constant in (BLO-VF), (BLO-DISC), and the stopping criterion. The δ-global equivalence in Theorem 3.1 depends on ε1.
assumptions (3)
  • domain assumption Assumption 2.1: f and g are continuously differentiable; X and Y are convex and compact with bounded norms; f and g have Lipschitz continuous and bounded gradients; the lower-level problem is feasible for every x.
    Standard regularity conditions for bilevel optimization, cited to prior works such as [10, 11, 37]. They ensure the value function approximation and gradient computations are well-defined.
  • domain assumption Assumption 2.2: the sampled set Ỹ = {y^(1),...,y^(k)} forms a covering of Y with radius r.
    Needed for the approximation error bound on Ṽ_λ. The paper acknowledges that for complex constraint sets random sampling cannot be provably guaranteed to produce a covering, so this is a practical assumption that may fail in general.
  • ad hoc to paper Assumption 3.1: for every x, the level sets {y ∈ Y : g(x,y) - V(x) ≤ ε} are convex for ε ≤ ε0, and dist(y, arg min g(x,·)) ≤ C |g(x,y) - V(x)|^β for all y in the ε0-level set.
    Introduced specifically for the equivalence between (BLO-VF) and (BLO-DISC) in Theorem 3.1. It is not standard in bilevel optimization literature and substantially restricts the class of lower-level problems, effectively requiring quasiconvex-like structure near the optimum.

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Cite this review

Pith. "Pith review of A Discretization Approach for Bilevel Optimization with Low-Dimensional and Non-Convex Lower-Level." pith.science (2026). https://pith.science/paper/RDJ5QUTJ

@misc{pith2026250510830,
  author       = {Pith},
  title        = {Pith review of: A Discretization Approach for Bilevel Optimization with Low-Dimensional and Non-Convex Lower-Level},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RDJ5QUTJ}},
  note         = {Machine review of arXiv:2505.10830}
}
read the original abstract

Bilevel optimization (BLO) problem, where two optimization problems (referred to as upper- and lower-level problems) are coupled hierarchically, has wide applications in areas such as machine learning and operations research. Recently, many first-order algorithms have been developed for solving bilevel problems with strongly convex and/or unconstrained lower-level problems; this special structure of the lower-level problem is needed to ensure the tractability of gradient computation (among other reasons). In this work, we deal with a class of more challenging BLO problems where the lower-level problem is non-convex and constrained. We propose a novel approach that approximates the value function of the lower-level problem by first sampling a set of feasible solutions and then constructing an equivalent convex optimization problem. This convexified value function is then used to construct a penalty function for the original BLO problem. We analyze the properties of the original BLO problem and the newly constructed penalized problem by characterizing the relation between their KKT points, as well as the local and global minima of the two problems. We then develop a gradient descent-based algorithm to solve the reformulated problem, and establish its finite-time convergence guarantees. Finally, we conduct numerical experiments to corroborate the theoretical performance of the proposed algorithm.

Figures

Figures reproduced from arXiv: 2505.10830 by the authors.

Figure 1
Figure 1. Equivalence between different problems. More specifically, we will establish the δ-global solution equivalence between (BLO-VF) and (BLO-DISC) in Section 3.1, and the global/ local/ KKT solution equivalence between (BLO-VF) and (BLO-DISC) in Section 3.2; see [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 3
Figure 3. This figure depicts the con￾straint set Y and its discretization; the sample points are the grid intersections. When ϵ is sufficiently small, the level set {y ∈ Y : g(x, y) − V (x) ≤ ϵ} is convex. Therefore, the convex hull of the top-left and bottom-right regions (delineated by the blue lines) is part of the level set, and our sample points are able to capture the entire level set. Lemma 3.1. Suppose Assumptions 2.… view at source ↗
Figure 4
Figure 4. The top two figures and the bottom left figure respectively show the lower [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

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