Time-Varying Convex Optimization with $O(n)$ Computational Complexity

Abstract: In this article, we consider the problem of unconstrained time-varying convex optimization, where the cost function changes with time. We provide an in-depth technical analysis of the problem and argue why freezing the cost at each time step and taking finite steps toward the minimizer is not the best tracking solution for this problem. We propose a set of algorithms that by taking into account the temporal variation of the cost aim to reduce the tracking error of the time-varying minimizer of the problem. The main contribution of our work is that our proposed algorithms only require the first-order derivatives of the cost function with respect to the decision variable. This approach significantly reduces computational cost compared to the existing algorithms, which use the inverse of the Hessian of the cost. Specifically, the proposed algorithms reduce the computational cost from $O(n^3)$ to $O(n)$ per timestep, where $n$ is the size of the decision variable. Avoiding the inverse of the Hessian also makes our algorithms applicable to non-convex optimization problems. We refer to these algorithms as $O(n)$-algorithms. These $O(n)$-algorithms are designed to solve the problem for different scenarios based on the available temporal information about the cost. We illustrate our results through various examples, including the solution of a model predictive control problem framed as a convex optimization problem with a streaming time-varying cost function.

• The paper introduces algorithms that lower computational complexity from O(n^3) to O(n) using first-order derivative methods.
• It employs a prediction-update approach to dynamically track optimal solutions with reduced steady-state error.
• The methods extend to real-world scenarios like MPC and autonomous systems, offering scalable, resource-efficient solutions.

Overview of Time-Varying Convex Optimization with Reduced Computational Complexity

This paper tackles the significant challenge of unconstrained time-varying convex optimization, a rapidly growing field aimed at real-time decision-making where cost functions vary with time. The standard approach of periodically freezing the cost function and iteratively seeking a local minimum proves inefficient, particularly when computational resources are limited or when the system must adapt swiftly to changes. Therefore, the authors propose novel algorithms that utilize first-order derivatives to achieve optimal tracking with reduced computational cost.

The central contribution lies in transforming the computational complexity from $O(n^3)$, typical of methods requiring Hessian inversions, to $O(n)$. This innovation becomes feasible by employing only the first-order derivatives, specifically the gradients of the cost function concerning decision variables. As a result, not only do these algorithms promise computational efficiency, but they also extend applicability to non-convex optimization scenarios, where traditional second-order methods struggle due to the possible inexistence of Hessians.

Key Findings and Algorithmic Innovation

The paper presents algorithms combining predictions and updates to harness information about cost functions' temporal dynamics effectively. The first algorithm, characterized by straightforward implementation, predicates on first-order derivatives to adjust the trajectory per timestep, maintaining computational demands strictly at $O(n)$. Another notable variant is the hybrid algorithm that transitions to a second-order gradient tracking approach in situations where the gradient is close to zero, alleviating numerical instability inherent in first-order methods.

The numerical results showcase the algorithms' ability to track the optimal trajectory with significantly reduced steady-state error compared to traditional methods. Two examples affirm these findings: one on a theoretical cost function with explicit temporal derivative availability and another on an MPC problem illustrating real-world applicability.

Implications and Future Directions

This work notably impacts areas such as autonomous systems, power grids, and machine learning, where real-time adaptive control is paramount, and computational affordances are stringent. The applicability to non-convex optimization extends the relevance across more complex decision-making landscapes, including those governed by streaming data without predefined cost structures.

Future research could further explore the integration of these algorithms within distributed systems, leveraging their reduced computational footprint. Additionally, bridging first-order methods' limitations with the precision of second-order alternatives using hybrid strategies remains a fertile ground for future exploration.

The authors' rigorous analysis and approach serve as a critical step towards agile, resource-efficient optimization, paving the path for more responsive and robust decision-making processes in dynamic environments.