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CBO algorithm with average drift and applications to portfolio optimization

Hyeong-Ohk Bae, Seung-Yeal Ha, Chanho Min, Jane Yoo, Jaeyoung Yoon

TL;DR

The proposed consensus based optimization algorithm with average drift (in short Ad-CBO) exhibits higher searching speed, lower tracking errors and regret bound than the CBO without stochastic diffusion.

Abstract

We propose a consensus based optimization algorithm with average drift (in short Ad-CBO) and provide a theoretical framework for it. In the theoretical analysis, we show that particle solutions to Ad-CBO converge to a global minimizer. In numerical simulations, we examine Ad-CBO's performance in optimizing static and dynamic objective functions. As a real-time application, we test the efficiency of Ad-CBO to find the optimal portfolio given stochastically evolving multi-asset prices in a financial market. The proposed Ad-CBO exhibits higher searching speed, lower tracking errors and regret bound than the CBO without stochastic diffusion

CBO algorithm with average drift and applications to portfolio optimization

TL;DR

The proposed consensus based optimization algorithm with average drift (in short Ad-CBO) exhibits higher searching speed, lower tracking errors and regret bound than the CBO without stochastic diffusion.

Abstract

We propose a consensus based optimization algorithm with average drift (in short Ad-CBO) and provide a theoretical framework for it. In the theoretical analysis, we show that particle solutions to Ad-CBO converge to a global minimizer. In numerical simulations, we examine Ad-CBO's performance in optimizing static and dynamic objective functions. As a real-time application, we test the efficiency of Ad-CBO to find the optimal portfolio given stochastically evolving multi-asset prices in a financial market. The proposed Ad-CBO exhibits higher searching speed, lower tracking errors and regret bound than the CBO without stochastic diffusion
Paper Structure (9 sections, 7 theorems, 144 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 9 sections, 7 theorems, 144 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete CBO with Adaptive Average Drift and noise (Ad-CBO)
  4. Numerical Simulations
  5. Static case
  6. Dynamic case
  7. Regret bound for Ad-CBO in optimal portfolio problem
  8. Conclusion
  9. Upper bound of regret bound ($\beta=\infty$)

Key Result

Theorem 3.1

For a solution process $\{{\boldsymbol{x}}_n^i\}_{i=1}^N$ to discrete, we have

Figures (4)

  • Figure 1: Graphical representation of \ref{['Rastrigin']} with $d=2$
  • Figure 2: Determinant value $\Lambda(\sigma)$ when $h=0.1$ and $\lambda_0=1$.
  • Figure 3: Confidence interval for $L(\boldsymbol{x}_\infty)$ obtained by CBO with noise (blue) and Ad-CBO (red) according to $\lambda_1$ and $\sigma$, respectively. The solid lines with dots represent the expectations.
  • Figure 4: Wealth evolution: average of fifty simulations wealth based on CBO ($\sigma=0,1$), Ad-CBO ($\lambda_1=1$) and Adam-CBO ($\beta_1=0.9,\beta_2=0.99$).

Theorems & Definitions (16)

  • Theorem 3.1: Emergence of a global consensus
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Corollary 3.6
  • ...and 6 more