Over-Approximating Minimizer Sets of Constrained Convex Programs with Parametric Uncertainty via Reachability Analysis

Abstract: We study the set of solutions to a parameterized, strongly convex optimization problem whose cost depends on uncertain, bounded parameters. We compute a certified outer approximation of the corresponding set of optimizers, using convergence properties of the projected gradient descent (PGD) algorithm for convex programs. Concretely, by treating the cost parameter as constant but unknown, we interpret the PGD iterates as an uncertain dynamical system and analyze its forward reachable sets. Since PGD converges exponentially to the unique optimizer for each fixed parameter, these reachable sets provide outer approximations of the optimizer set, with an explicit error bound that decays exponentially with the iteration count. We apply system-level synthesis (SLS) on the PGD dynamics to optimize the step-size sequence and obtain reachable-set over-approximations. Our method outperforms existing baselines in over-approximating, with low conservativeness, the minimizer sets of convex programs with uncertain costs and high-dimensional decision variables.

• The paper proposes a novel SLS reachability method to over-approximate minimizer sets in convex programs with bounded parametric uncertainty.
• It reformulates projected gradient descent as an uncertain dynamical system to compute forward reachable sets with exponential convergence guarantees.
• Empirical evaluations demonstrate that the SLS approach yields tighter, less conservative bounds than traditional sensitivity methods, even in high-dimensional settings.

SLS-Based Over-Approximation of Minimizer Sets under Parametric Uncertainty

Problem Formulation and Motivation

The paper "Over-Approximating Minimizer Sets of Constrained Convex Programs with Parametric Uncertainty via Reachability Analysis" (2604.27355) addresses the challenge of computing certified outer approximations for minimizer sets arising from convex programs with cost functions affected by unknown, bounded parameters. The practical motivation arises from contexts such as robust control and robotics, where safe planning must account for uncertainty in intent, physical properties, or environmental data. While traditional robust optimization focuses on worst-case optimality, this work aims to characterize the entire set of minimizers for all admissible parameter values, thus facilitating a safer and more expressive downstream decision-making pipeline.

Methodological Framework

Projected Gradient Descent as an Uncertain Dynamical System

The approach leverages the exponential convergence properties of Projected Gradient Descent (PGD) in strongly convex settings. By treating the unknown parameter as constant but uncertain, PGD updates are interpreted as the trajectories of a dynamical system under uncertainty. This reformulation is critical: outer approximations of the minimizer set can be constructed by analyzing the forward reachable sets of these PGD dynamics.

System-Level Synthesis for Reachability

System-Level Synthesis (SLS) is adopted to optimize step-size sequences and compute reachable-set over-approximations for the PGD system. SLS frameworks allow tractable computation of forward reachable sets even in high-dimensional settings and under nonlinearities, providing less conservative bounds compared to classical sensitivity analysis and fixed-step PGD schemes.

For differentiable PGD dynamics (e.g., unconstrained or affine constraints), linear time-varying error systems are derived and robust feedback policies are synthesized using SLS. For non-differentiable constraints, as in polyhedral projections, a smoothing mechanism based on sampled convolutions is applied, enabling SLS methods to function as proxies with quantifiable approximation error.

Theoretical Guarantees

The convergence analysis formalizes that reachable sets of PGD iterates exponentially contract to the single optimizer for any fixed parameter. Explicit bounds on the Hausdorff distance between the minimizer set and the reachable set approximation are given, with decay rates independent of the parameter value. Auxiliary error bounds are imposed to maintain constraint satisfaction and robustness across all parameter realizations.

Empirical Evaluation

Several numerical examples substantiate the theoretical constructs:

1. Unconstrained Quadratic Cost: Over-approximations for single-variable quadratic minimization under scalar parameter uncertainty exhibit conservative containment with exponentially shrinking error bounds. SLS-derived outer approximations are demonstrably tighter than those from worst-case sensitivity baselines.
2. High-Dimensional LQR with Intent Uncertainty: The method scales effectively to 64-dimensional optimization, with step-size adaptation yielding non-conservative bounds on position trajectories. Classical sensitivity via implicit differentiation yields much looser bounds compared to SLS-based reachability formulations.
3. Constrained Quadratic Cost (Nonsmooth Case): Smoothed PGD dynamics enable approximation despite non-differentiability. Conservativeness increases as trajectories approach constraint boundaries, consistent with additional error terms from smoothing.

Strong numerical outcomes are highlighted: in high-dimensional settings, the SLS-based method yields significantly less conservative approximations than sensitivity-based baselines. The approach is tractable for up to 64-dimensional decision variables, with computation runtimes scaling efficiently.

Implications and Future Directions

The results enable rigorous safety guarantees for robust planning under parametric uncertainty, directly applicable to safe robotics, economic optimization, and forecast-based control. The methodology accommodates uncertainty propagation in adaptive and online estimation contexts, offering practical strategies to fuse optimization and reachability analysis.

On the theoretical side, the paper establishes a link between convex optimization, nonlinear dynamics, and reachability analysis, suggesting new avenues for global solution set characterization beyond traditional perturbation and sensitivity analysis. The use of SLS in this context extends its utility from robust control to optimization uncertainty quantification.

Potential future developments include integrating the method with real-time adaptive control, wherein parameter sets are dynamically contracted from streaming observations. Extension to general nonlinear or nonconvex settings, potentially with probabilistic guarantees, is also suggested.

Conclusion

This work provides a methodologically rigorous, computation-efficient framework for over-approximating minimizer sets of convex programs with uncertain cost parameters, leveraging PGD convergence and system-level synthesis-based reachability analysis to produce tight, certified bounds. The theoretical guarantees and empirical validations indicate superior performance over classical sensitivity approaches, offering robust tools for safe decision-making in uncertain optimization landscapes (2604.27355).