Bilevel MPC for Linear Systems: A Tractable Reduction and Continuous Connection to Hierarchical MPC

Abstract: Model predictive control (MPC) has been widely used in many fields, often in hierarchical architectures that combine controllers and decision-making layers at different levels. However, when such architectures are cast as bilevel optimization problems, standard KKT-based reformulations often introduce nonconvex and potentially nonsmooth structures that are undesirable for real-time verifiable control. In this paper, we study a bilevel MPC architecture composed of (i) an upper layer that selects the reference sequence and (ii) a lower-level linear MPC that tracks such reference sequence. We propose a smooth single-level reduction that does not degrade performance under a verifiable block-matrix nonsingularity condition. In addition, when the problem is convex, its solution is unique and equivalent to a corresponding centralized MPC, enabling the inheritance of closed-loop properties. We further show that bilevel MPC is a natural extension of standard hierarchical MPC, and introduce an interpolation framework that continuously connects the two via move-blocking. This framework reveals optimal-value ordering among the resulting formulations and provides inexpensive a posteriori degradation certificates, thereby enabling a principled performance-computational efficiency trade-off.

• The paper introduces a tractable single-level convex reduction that transforms the nonconvex bilevel MPC formulation into a complementarity-free optimization problem.
• The methodology guarantees equivalence to centralized MPC, preserving closed-loop stability and unique optimal solutions under strong convexity.
• A move-blocking strategy with certified performance bounds is proposed, achieving a reliable trade-off between computational efficiency and optimal control performance.

Bilevel MPC for Linear Systems: Reduction, Equivalence, and Hierarchical Connections

Introduction and Problem Formulation

The paper addresses bilevel model predictive control (MPC) for discrete-time linear systems within hierarchical control architectures. The classical hierarchical MPC (HMPC) configuration separates an upper layer responsible for steady-state or reference signal selection, and a lower layer executing a constrained tracking MPC. However, hierarchically structured bilevel MPC, when directly cast as a bilevel optimization, typically leads to nonconvex formulations via Karush-Kuhn-Tucker (KKT) embedding or mixed-integer programming. This undermines tractability, induces nonuniqueness, and impedes real-time feasibility.

The focus is on systems of the form $x_{k+1} = A x_k + B u_k$ and a decomposition where the upper level selects a horizon-wise reference parameter $\Theta$ (constructed to parameterize steady-state pairs in the plant), while the lower level implements a strongly convex quadratic program (QP) tracking MPC.

Tractable Reduction to a Single-Level Formulation

A core contribution of the work is the proposal and analysis of a tractable single-level reduction for bilevel MPC problems. The classical bilevel formulation (Problem $\mathbf{P}_1$) enforces the inter-level coupling via the solution set of a lower-level tracking MPC QP for a fixed reference. The prevailing approach introduces complementarity constraints via the KKT system, resulting in a nonconvex mathematical program with complementarity constraints (MPCC).

The authors propose a reduction (Problem $\mathbf{P}_2$) by lifting all lower-level constraints to the upper problem, imposing only the lower-level stationarity condition $\nabla_U F_l(U; \Theta, x_0) = 0$. This makes the problem convex (when all upper-level objectives/constraints are convex) and complementarity-free. They rigorously establish that under a block-structured nonsingularity condition (i.e., $\bar{\Gamma}$, a function of stage costs and plant matrices, is invertible), the reduced problem is equivalent to the original MPCC: solution-sets and optimal values coincide, and uniqueness is guaranteed when $F_u$ is strongly convex.

This reduction positions the bilevel MPC as theoretically and practically on par with a centralized MPC (Problem $\mathbf{P}_3$), obviating the difficulties of solving nonconvex MPCCs:

(Figure 1)

Figure 1: Numerical comparison of P_{#1{0}, P_{#1{1}, and P_{#1{2}}. P_{#1{2}} reliably reproduces the centralized solution (dotted lines), whereas P_{#1{1}} does so only under favorable initialization.

Furthermore, the cascade structure (upper-level reference selector plus lower-level tracking MPC) preserves closed-loop stability and uniqueness properties of the centralized solution, as the equivalent optimizer in $U$ is uniquely realizable in all three formulations.

Hierarchical MPC, Move-Blocking, and Performance-Complexity Trade-Offs

The manuscript formalizes the relationship between standard HMPC and bilevel MPC by considering structured subspaces of the reference parametric trajectory $\Theta$—a move-blocking approach. The simplest case (one-block HMPC) corresponds to a constant reference across the horizon; more refined blocking strategies expand the subspace in which $\Theta$0 resides, interpolating between HMPC and full bilevel MPC. This framework accommodates the continuous transition between computational efficiency (low-dimensional blocked subspaces) and closed-loop performance (full-dimensional reference optimization).

A comprehensive optimal-value ordering is proven: as the subspace expands, performance monotonically improves, establishing $\Theta$1 where $\Theta$2 is HMPC, $\Theta$3/$\Theta$4 are the blocked bilevel values, and $\Theta$5 is the (unblocked) bilevel optimum.

A constructive algorithm is presented for building low-rank blocking matrices $\Theta$6 from sampled initial conditions and control data that guarantee $\Theta$7 over those regimes, with the practical effect that blocking can be heavily reduced (leading to computational savings) without exceeding the steady-state suboptimality of HMPC.

(Figure 2)

Figure 2: Performance degradation relative to P_{#1{2}} for move-blocked problems and HMPC. As the blocking structure is relaxed (higher $\Theta$8), performance converges to the unblocked solution; upper bounds effectively track the attainable degradation.

Certified Bounds on Blocking-Induced Degradation

The introduction of certified, a posteriori, performance bounds quantifies the suboptimality incurred by move-blocking (or by HMPC). These bounds are constructed by leveraging strong convexity, yielding a dual-based certificate $\Theta$9. Importantly, these bounds can be computed without explicitly solving the higher-dimensional (less-blocked) bilevel programs, enabling efficient "stop-or-refine" design decisions without exhaustive optimization over all hierarchical blockings.

The bounds hold under mild constraint qualification conditions and are efficiently computable (as nonnegative least squares in the number of active constraints at the evaluation point).

Numerical Results

A series of numerical experiments, including a benchmark integrator-containing toy model and a data-driven quadrotor example via Koopman EDMD lifts, validate the theoretical claims. The key observations are:

• The single-level reduction $\mathbf{P}_1$0 deterministically converges to the centralized optimal sequence, while the original nonconvex bilevel program $\mathbf{P}_1$1 displays solution sensitivity to initialization and is prone to local suboptimality.
• Move-blocked formulations exhibit performance monotonicity as blocking is relaxed; a-priori constructed low-rank $\mathbf{P}_1$2 matrices outperform HMPC with minimal computational overhead.
• The certified upper-bounds are effective in quantifying the remaining performance gap and support principled, sample-efficient controller synthesis.
• In nonlinear settings handled via Koopman-based linearization, the reduced bilevel MPC provides better transient constraint handling and convergence speed than HMPC, with attainable computational cost, making real-time deployment feasible.

Implications and Future Directions

The tractable reduction bridges hierarchical and centralized optimal control, enabling bilevel architectures to inherit closed-loop guarantees and uniqueness properties previously restricted to centralized MPC. The continuous connection via move-blocking unifies design methodologies across hierarchical architectures, supporting systematic performance-complexity trade-off quantification. The results lower the barrier for safety/verification-critical deployments of hierarchical controllers, particularly in application domains historically reliant on ad hoc or purely steady-state reference policies.

Theoretical extensions could address robustification to model mismatch, adaptive/learning-based upper-level controllers, or nonlinear (non-quadratic) settings. Future practical work may explore fast custom solvers leveraging this reduction, or distributed/networked extensions for large-scale systems.

Conclusion

The paper systematically analyzes bilevel MPC architectures for linear systems, providing an exact and tractable convex reduction that circumvents the limitations of nonconvex MPCC formulations. The proposed approach both recovers and unifies centralized and hierarchical MPC under verifiable algebraic conditions and extends to practical, certificate-backed trade-off design via move-blocking. The established methodology facilitates both practical deployment (including in data-driven settings) and theoretical analysis for hierarchical closed-loop systems.