Nonconvex Bilevel Programming

• Nonconvex bilevel programming is a hierarchical optimization framework characterized by a nonconvex lower-level problem yielding multiple, discontinuous solutions.
• Advanced stationarity concepts and relaxed constraint qualifications like RCPLD are essential for ensuring feasible and sharp optimality conditions.
• Global solution methods such as SOS relaxations, cutting-plane techniques, and discretization schemes provide practical pathways for achieving convergence.

Nonconvex bilevel programming concerns hierarchical optimization where the lower-level (LL) subproblem is nonconvex—often resulting in nonunique, potentially discontinuous, and structurally intricate solution mappings. Such problems arise in hyperparameter optimization, adversarial learning, meta-learning, equilibrium modeling, robust control, and many other domains. The nonconvexity of the LL problem presents formidable theoretical and computational challenges distinct from convex or strongly convex bilevel programs: standard reduction techniques, stationarity concepts, constraint qualifications, and algorithmic guarantees often break down or require profound generalization.

1. Mathematical Formulation and Fundamental Obstacles

A general nonconvex bilevel problem is given by: $$\min_{x \in \mathcal{X} \subset \mathbb{R}^n} f(x, y^*(x)), \quad\text{where}\quad y^*(x) \in \arg\min_{y \in \mathcal{Y} \subset \mathbb{R}^m} g(x, y)$$ with $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$ and $g:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$ smooth but nonconvex in $y$ (Jiang et al., 1 Sep 2025, Bi et al., 2022, Jeyakumar et al., 2015, Ye, 2019). The feasible set for the upper level is implicitly determined by the (possibly set-valued, discontinuous, and non-singleton) multivalued mapping $S(x) = \arg\min_{y \in \mathcal{Y}} g(x, y)$.

Key issues:

• Nonuniqueness and discontinuity of lower-level solutions: Unlike strongly convex LL problems (which guarantee single-valued, smooth solution maps $y^*(x)$), here $S(x)$ may have multiple isolated or connected components that can change discontinuously with $x$ (Jiang et al., 1 Sep 2025).
• Ill-posedness of the reduced problem: Plugging $y^*(x)$ into $f(x,y)$ is generally not well-defined, since set-valuedness and discontinuity prevent application of the classical chain rule or implicit function theory.
• Failure of KKT-based reformulations: In nonconvex LL settings, the LL KKT conditions are only necessary, potentially yielding extra (spurious) stationary points not corresponding to globally optimal LL solutions (Ye, 2019).

2. Stationarity, Constraint Qualifications, and Optimality Conditions

Standard first-order stationarity concepts must be carefully generalized.

• Combined (single-level) reformulations: (Ye, 2019) advocates "combined program" (CP) approaches: augment the problem with value-function constraints $f(x, y) - V(x) \leq 0$, where $V(x) = \inf_{y} g(x, y)$, combined with the LL stationarity system for candidate $y$.
• Relaxed CQs: The relaxed constant positive linear dependence (RCPLD) condition is tailored for these nonsmooth, nonconvex formulations. RCPLD tracks limiting linear dependencies among all active constraint gradients—including those from nonsmooth penalties—permitting sharper, checkable necessary stationarity conditions (M-stationarity) under mild regularity (Ye, 2019).

CQ	Scope	Features
MFCQ	Classical NLP/MPCC	Requires Slater-type directions; too strong for nonconvex LL
CPLD	Nonconvex NLP	Weaker, but still requires constant dependence under perturb
RCPLD	Nonconvex bilevel	Allows limiting dependencies, includes LL value constraint

• Single-level versus KKT reformulations: KKT-based MPEC and MPCC formulations under nonconvex LL are only necessary. Augmenting with value-function inequalities is essential to exclude spurious non-bilevel-feasible solutions (Ye, 2019). Stationarity conditions derived under RCPLD (and additional calmness/partial penalization) are thus substantially sharper and more practical than MPCC/MPEC CQs in the nonconvex context.

3. Global Solution Techniques: SDP Relaxations and Constraint Generation

When the LL is polynomial (or broadly, semi-algebraic), several global solution methods exist:

• SOS/SDP relaxations: For problem classes with all polynomials, a convergent joint–marginal Sum-of-Squares (SOS) hierarchy can be constructed. The lower-level value function $J(x) = \min_{\{w : h_j(w) \le 0\}} G(x, w)$ is underapproximated by a sequence of SOS relaxations, building increasingly tight polynomial underestimators, which are embedded into single-level relaxations with moment constraints. Under an Archimedean compactness assumption, the upper bound sequence converges to the global bilevel optimum as both relaxation order and LL approximation tolerance go to zero (Jeyakumar et al., 2015).
• Constraint-generation/cutting-plane (semi-infinite) approaches: For general (potentially nonsmooth or discrete) LLs, the Nash equilibrium feasibility or "minimum disequilibrium" can be formulated as a semi-infinite program. Algorithms maintain a finite subset of constraint indices and iteratively solve master problems, refining the subset with new violating followers' responses. Under compactness and continuity, finite termination with certified near-optimality is possible (Harwood et al., 2021).
• Discretization in low-dimensional LLs: When the LL is low dimensional and nonconvex with constraints, a discretization of the feasible set followed by convexification of the LL value function allows construction of penalty functions for the BLO problem. These surrogates can be addressed by deterministic descent methods with finite-time convergence guarantees, though full technical details are pending the release of complete source texts (Jiang et al., 16 May 2025).

4. First-Order Algorithms for Nonconvex-NPL Bilevel Problems

Recent advances outline several algorithmic paradigms:

• KKT-based penalty and projection: Nonconvex, nonsmooth bilevel problems with LL stationarity constraints can be tackled by hybrid schemes—combining momentum-accelerated subgradient descent with penalty terms enforcing LL stationarity, and feasibility-restoration steps that project iterates closer to the LL stationary-manifold. Careful two-timescale step-size control and alternating strategies allow global convergence of cluster points to feasible first-order stationary solutions, using only first-order oracles for both objectives (Xiao et al., 28 May 2025).
• Stationarity-based single-level reformulations: For unconstrained, nonconvex LL problems, the optimistic-reformulation $$\min_{x,y} f(x, y), \ \text{s.t.} \ \nabla_y g(x,y) = 0$$ gives rise to an equivalent single-level KKT-based constraint system. Relaxed-gradient-flow and projection-based algorithms, notably involving explicit quadratic subproblems with closed-form solutions, achieve convergence to ε-KKT points in $O(\epsilon^{-1.5})$ iterations under a mild PL regularity on the LL gradient map, and $O(\epsilon^{-3})$ in the fully general nonconvex case (Abolfazli et al., 24 Apr 2025).
• Discretization-approximation approaches: For low-dimensional nonconvex LLs, sampling-based approximations of the value function and subsequent convexification permit direct penalty construction and gradient-based upper-level descent, with structural guarantees linking the KKT points and local/global minima of the surrogate and original problems [(Jiang et al., 16 May 2025), abstract].

5. Complexity and Hardness Results

The complexity of nonconvex bilevel programming reflects both the nonconvexity of the LL and the hierarchical structure:

• Lower bounds in smooth nonconvex–strongly convex settings: Any deterministic zero-respecting first-order algorithm must perform at least $\Omega(\kappa^{3/2} \epsilon^{-2})$ queries (with $\kappa=L_g/\mu$, the LL condition number) to reach an ε-stationary point. The stochastic oracle complexity is $\Omega(\kappa^{5/2} \epsilon^{-4})$, indicating that even in simplified quadratic LL regimes, bilevel programming is strictly harder than both single-level nonconvex and min-max problems. Large gaps remain between these bounds and current upper bounds, highlighting the complexity of hierarchical nonconvex optimization (Ji, 24 Nov 2025).
• Computational pathology under ε-relaxation: For continuous, nonconvex LL with even a unique LL minimizer and Slater's condition, any ε-relaxation—even with arbitrarily small ε—can induce arbitrarily large errors in the outer variable and objective. This fundamental pathology (absent from linear bilevels) highlights the necessity of exact or certified LL stationarity feasibility, as standard nonlinear global-optimization methods (relaxing constraints with tolerance) can fail catastrophically (Beck et al., 2022).

6. Interpretive Approaches, Regularization, and Modeling Strategies

• Correspondence-driven leader-follower modeling: The "classical hyperfunction" assumption—namely that the LL always attains a global minimizer—is often unrealistic. A correspondence-driven hyperfunction models the follower as an algorithmic, bounded-rational agent, whose behavior is generated by a fixed algorithm, initialization, and step-size schedule. The resulting effective objective is generally discontinuous in $x$, exhibiting bifurcation phenomena at degenerate stationary points of $g(x,\cdot)$. Gaussian smoothing and stochastic projected gradient descent with cubic-regularized LL solvers can sidestep discontinuity, offering pointwise and gradient convergence to the smoothed leader objective, with explicit oracle complexity bounds under fold bifurcation geometric structure (Jiang et al., 1 Sep 2025).
• Implications for algorithm design: The prevalence of bifurcation sets (of measure zero but nontrivial dimension) implies that almost everywhere the LL problem is Morse and that algorithmic behavior is locally stable, but near these sets, slowdowns and discontinuity are inevitable. Algorithmic complexity is sharply increased in the presence of degenerate LL stationary points, with the number of LL first- and second-order oracle calls scaling as $\widetilde O(\epsilon^{-10})$ to reach ε-stationarity (Jiang et al., 1 Sep 2025).

7. Broader Impact and Future Directions

Nonconvex bilevel programming theory now encompasses a range of stationarity frameworks, constraint qualifications, global-search hierarchies, and algorithmic paradigms suited to the demands of general machine learning, game theory, and engineering settings. Key open problems and future directions include:

• Further bridging the complexity gap between lower and upper bounds for first-order (and higher-order) methods.
• Development of practical algorithms for large-scale bilevel instances where the LL is nonconvex, possibly non-smooth, and lacks global regularity.
• Incorporation of stochasticity, decentralization, and non-Euclidean geometry in bilevel solvers, notably for PL-type LL objectives under heavy-tailed noise (Zhang et al., 19 Sep 2025).
• Fundamental investigations of modeling validity for the leader-follower structure under computationally realizable LL responses.

Nonconvex bilevel programming thus remains a vibrant field whose theoretical, algorithmic, and modeling frontiers are closely connected and rapidly evolving.