$\mathcal{L}_{1}$ Adaptive Optimizer for Online Time-Varying Convex Optimization

TL;DR

The paper tackles online time-varying convex optimization where online prediction models can be imperfect. It introduces $\mathcal{L}_{1}$ adaptive optimization ($\mathcal{L}_{1}$-AO), which augments a baseline TV optimizer with an adaptive law and a low-pass filter to compensate for prediction inaccuracies, yielding provable bounds on the gradient and optimality gap. The approach reframes TV optimization as an uncertain gradient regulation problem and provides a rigorous analysis showing uniform bounds and uniform ultimate boundedness for the gradient and decision trajectory. Numerical simulations on constrained TV problems and robot navigation demonstrate that $\mathcal{L}_{1}$-AO achieves reliable, real-time tracking of the optimal solution while respecting safety constraints, outperforming non-adaptive TV methods under prediction error.

Abstract

We propose an adaptive method for online time-varying (TV) convex optimization, termed $\mathcal{L}_{1}$ adaptive optimization ($\mathcal{L}_{1}$-AO). TV optimizers utilize a prediction model to exploit the temporal structure of TV problems, which can be inaccurate in the online implementation. Inspired by $\mathcal{L}_{1}$ adaptive control, the proposed method augments an adaptive update law to estimate and compensate for the uncertainty from the prediction inaccuracies. The proposed method provides performance bounds of the error in the optimization variables and cost function, allowing efficient and reliable optimization for TV problems. Numerical simulation results demonstrate the effectiveness of the proposed method for online TV convex optimization.

Figures (4)

• Figure 1: (a) Time-varying (TV) optimization methods update the optimization variable continuously using a prediction model to track the optimal solution. (b) In the online implementation, the prediction model may be inaccurate, leading to performance degradation. The proposed $\mathcal{L}_{1}$ adaptive optimization ($\mathcal{L}_{1}$-AO) compensates for the uncertainty from prediction inaccuracies and provides predictable robustness certification for TV convex optimization problems. The heatmap indicates the TV cost function.
• Figure 2: Architecture comparison between $\mathcal{L}_{1}$ adaptive control ($\mathcal{L}_{1}$-AC) and the proposed $\mathcal{L}_{1}$ adaptive optimization ($\mathcal{L}_{1}$-AO). Both architectures are designed to compensate for the uncertainty to recover the performance of the baseline controller or optimizer. (a) In $\mathcal{L}_{1}$-AC, the state predictor predicts the state of the uncertain (physical) system. (b) In $\mathcal{L}_{1}$-AO, the gradient system (a virtual system) is considered. The rate of optimization variables, $\dot{v}$, is regarded as the control input, and the gradient predictor is used to predict the gradient of the cost function.
• Figure 3: Example 1: Simulation result. The gray area denotes the TV constraint. The heatmap indicates the cost function. (a) The modified PCIP with $P=10$ violated the TV constraint (magenta star). The modified PCIP with a high gain $P=10^{3}$ induced a huge update rate at the initial phase, undesirable for numerical stability. (b) The $\mathcal{L}_{1}$-AO updates the optimization variable without an abrupt update. The optimization variable is confined within a green tube, preventing constraint violation.
• Figure 4: Example 2: (a)-(d) 2D trajectories of a robot. For visualization, only the results of PCIP and $\mathcal{L}_{1}$-AO with the baseline of PCIP are depicted. (e)-(f) Error in optimization variable: (e) PCIP vs $\mathcal{L}_{1}$-AO (baseline: PCIP), (f) Modified PCIP vs $\mathcal{L}_{1}$-AO (baseline: Modified PCIP).

Theorems & Definitions (12)

• Definition 1
• Theorem 1
• proof
• Corollary 1
• proof
• Remark 1
• Lemma 1
• proof
• Lemma 2
• proof