5  Convex optimization

5.1 Introduction

This is how we minimize \(\sigma\).

• Linear programming
  • George Dantzig (1914-2005)
• Quadratic programming
  • No-shorts efficient frontier
  • Karush-Kuhn-Tucker (KKT) conditions
  • Jagannathan, R. & Ma, T. (2003). Risk reduction in large portfolios: Why imposing the wrong constraints helps. ¹
  • Tam, A.S. (2021). Lagrangians and portfolio optimization.
• Boyd, S. & Vandenberghe, L. (2004). Convex Optimization. ²
  • Course website at Stanford: Convex Optimization I
  • Convex Optimization Short Course
  • Boyd Lectures on youtube: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
  • Affine combinations and convex sets
• Markowitz’s Critical Line Algorithm (CLA)
  • Markowitz, H.M. (1956). The optimization of a quadratic function subject to linear constraints. ³
  • Bailey, D.H. & López de Prado, M. (2013). An open-source implementation of the critical-line algorithm for portfolio optimization. ⁴
  • Markowitz, H.M., Starer, D., Fram, H., & Gerber, S. (2019). Avoiding the downside: A practical review of the Critical Line Algorithm for mean-semivariance portfolio optimization. ⁵
• Software:
  • CVXPY
  • robertmartin8/PyPortfolioOpt
  • scikit-portfolio
  • mdengler/cla

¹ Jagannathan & Ma (2003).

² Boyd & Vandenberghe (2004).

³ Markowitz (1956).

⁴ Bailey & López de Prado (2013).

⁵ Markowitz, Starer, Fram, & Gerber (2019).

5.2 No-shorts efficient frontier

Solve

\[ \vec{w}_{\ast} = \underset{w}{\mathrm{argmin}}\ \vec{w}^\intercal \, V \, \vec{w} \]

such that

\[ \vec{w} \cdot \vec{1} = 1 \]

\[ \vec{w} \cdot \vec{\mu} = r_{\ast} \]

and with further optional constraints

\[ A \, \vec{w} \geq \vec{b} \]

TODO: Discuss optimizing the no-shorts frontier.

5.3 Projection onto a simplex

• Condat, L. (2016). Fast projection onto the simplex and the \(l_1\) ball. ⁶

⁶ Condat (2016).

Bailey, D. H. & López de Prado, M. (2013). An open-source implementation of the critical-line algorithm for portfolio optimization. Algorithms, 6, 169–196. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2197616

Boyd, S. & Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press. https://web.stanford.edu/~boyd/cvxbook/

Condat, L. (2016). Fast projection onto the simplex and the \(l_1\) ball. Mathematical Programing Series A, 158, 575–585. https://link.springer.com/content/pdf/10.1007/s10107-015-0946-6.pdf

Jagannathan, R. & Ma, T. (2003). Risk reduction in large portfolios: Why imposing the wrong constraints helps. Journal of Finance, 58, 1651–1683. https://www.jstor.org/stable/3648224

Markowitz, H. M. (1956). The optimization of a quadratic function subject to linear constraints. Naval Research Logistics Quarterly, 3, 111–133.

Markowitz, H. M., Starer, D., Fram, H., & Gerber, S. (2019). Avoiding the downside: A practical review of the Critical Line Algorithm for mean-semivariance portfolio optimization. https://www.hudsonbaycapital.com/documents/FG/hudsonbay/research/599440_paper.pdf