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

M. Rostami, S. S. Kia

TL;DR

This article proposes 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 and significantly reduces computational cost compared to the existing algorithms, which use the inverse of the Hessian of the cost.

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.

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

TL;DR

This article proposes 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 and significantly reduces computational cost compared to the existing algorithms, which use the inverse of the Hessian of the cost.

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 to per timestep, where 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 -algorithms. These -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.

Paper Structure

This paper contains 11 sections, 7 theorems, 51 equations, 7 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Objective statement
  4. $O(n)$ algorithms with first-order derivatives
  5. An $O(n)$ algorithm without explicit knowledge of variation of cost across time
  6. An $O(n)$ algorithm that uses the time derivative of the gradient
  7. A hybrid first- and second-order algorithms
  8. Simulation results
  9. A case of time-varying cost function whose time derivatives are available explicitly
  10. Model Predictive Control: a case of a time-varying cost whose temporal derivatives are not available explicitly
  11. Conclusion

Key Result

Lemma 2.1

(Bound on the cost difference of the optimizer): Consider the optimization problem eq::opt under Assumptions asm:str_convexity, asm:M_Lip and asm:bound_dfstar. Then, where $\psi=\delta (K_1 + \frac{\delta}{2}K_3) + \frac{K^2_2 \delta^2}{2m} (\frac{M\delta}{m} + 2)$, with $\delta=t_{k+1}-t_k\in{\mathbb{R}}_{>0}$ being the sampling timestep of the optimal cost function across time.

Figures (7)

  • Figure 1: A time-varying $f(\boldsymbol{\mathbf{x}},t)$ vs. $\boldsymbol{\mathbf{x}}$ and $t$ (gray plot) and the trajectory of $f(\boldsymbol{\mathbf{\mathsf{x}}}^\star_t,t)$ vs. $\boldsymbol{\mathbf{\mathsf{x}}}^\star(t)$ and $t$ (red curve). The darker contours are added to improve visualization of the function and does not signify any particular property of the function.
  • Figure 2: An example case that demonstrates the role of the prediction step (line 3) of Algorithm \ref{['Alg::1']}. As we can see in this example, for both cases of $\nabla_t f(\boldsymbol{\mathbf{x}}_k,t_k) \geq 0$ (plots in the left column) and $\nabla_t f(\boldsymbol{\mathbf{x}}_k,t_k) < 0$ (plots in the right column) $f^-_1(t_{k+1})\leq f^-_{\text{g}}(t_{k+1})$.
  • Figure 3: An example case that demonstrates the role of the prediction step (line 3) of Algorithm \ref{['Alg::3']}. As we can see in this example, for both cases of $\nabla_t f(\boldsymbol{\mathbf{x}}_k,t_k) \geq 0$ and $\nabla_t f(\boldsymbol{\mathbf{x}}_k,t_k) < 0$ the statement of Lemma \ref{['lem::Alg3_vs_Alg1']} holds, i.e., $f^-_3(t_{k+1})\leq f^-_1(t_{k+1})\leq f^-_{\text{g}}(t_{k+1})$.
  • Figure 4: The trajectories of the proposed Algorithms \ref{['Alg::1']}, \ref{['Alg::3']} and the second-order Hessian-based algorithm of AS-AM-AK-GL-AR:16 compared to the trajectory generated by the gradient descent algorithm shown in time interval $t\in[0.4, 1.4]$ for the cost function \ref{['num1_cost']}.
  • Figure 5: Log error of the performance of Algorithm \ref{['Alg::1']}, 3 and \ref{['Alg::4']} versus gradient descent algorithm and the Hessian based algorithm proposed in AS-AM-AK-GL-AR:16 with respect to the sampling time $t_k$.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Lemma 2.1
  • Lemma 4.1
  • Theorem 4.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Theorem 4.2
  • Lemma 4.2
  • Theorem 4.3
  • Remark 4.4
  • ...and 1 more