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A Journey Through the 'Mathematical Underworld' of Portfolio Optimization

26 Pages Posted: 11 Feb 2013 Last revised: 5 Jul 2015

Marcos Lopez de Prado

Guggenheim Partners, LLC; Lawrence Berkeley National Laboratory; Harvard University - RCC

Date Written: February 16, 2013


* It has been estimated that the current size of the asset management industry is approximately US$58 trillion.

* Portfolio optimization is one of the problems most frequently encountered by financial practitioners. It appears in various forms in the context of Trading, Risk Management and Capital Allocation.

* The Critical Line Algorithm (CLA) is the only algorithm specifically designed for inequality-constrained portfolio optimization problems, which guarantees that the exact solution is found after a predefined number of iterations.

* Surprisingly, open-source implementations of CLA in a scientific language appear to be inexistent or unavailable.

* The lack of publicly available CLA software, commercially or open-source, means that trillions of dollars are likely to be suboptimally allocated as a result of practitioners using general-purpose quadratic optimizers.

* We believe that a large amount of financial firms and practitioners will benefit from our robust implementation of CLA in a scientific language.

The code is available in the author's website.

Keywords: portfolio selection, quadratic programming, portfolio optimization, constrained efficient frontier, turning point, Kuhn-Tucker conditions, risk aversion

JEL Classification: C02, G11, G14, D53

Suggested Citation

Lopez de Prado, Marcos, A Journey Through the 'Mathematical Underworld' of Portfolio Optimization (February 16, 2013). Available at SSRN: or

Marcos Lopez de Prado (Contact Author)

Guggenheim Partners, LLC ( email )

330 Madison Avenue
New York, NY 10017
United States


Lawrence Berkeley National Laboratory ( email )

1 Cyclotron Road
Berkeley, CA 94720
United States


Harvard University - RCC ( email )

26 Trowbridge Street
Cambridge, MA 02138
United States


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