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

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.