Linear Versus Quadratic Portfolio Optimization and Risk Preferences
Summary
The document considers a portfolio problem that maximizes expected alpha subject to bounds on factor exposures, and asks how this linear formulation compares with Markowitz optimization. The responses clarify that the classical Markowitz mean-variance objective is quadratic in portfolio weights because it includes portfolio variance; a linear objective with linear constraints is a different problem and does not capture that risk penalty by itself.
One answer connects mean-variance optimization to assumptions about investor preferences and return distributions, and advises having a rationale for any constraints used in a linear model. Another points to earlier work on linear programming for mutual fund selection, while noting that such methods rely on particular structure and assumptions. The discussion gives conceptual guidance rather than a worked comparison or empirical results. It does not establish that LP is generally inferior: the appropriate formulation depends on the intended risk measure, constraints, and theoretical justification.
Key ideas
- Classical Markowitz optimization trades expected return against portfolio variance through a quadratic objective.
- Maximizing alpha subject to factor exposure bounds is a distinct linear optimization problem and does not inherently penalize variance.
- A linear formulation should be supported by a reason for its objective and constraints.
- Linear programming approaches to portfolio selection can rely on special assumptions or covariance structure.
- The document offers conceptual distinctions but no empirical comparison of the methods.
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# What's the disadvantage of using linear programming for portfolio optimization? # What's the disadvantage of using linear programming for portfolio optimization? I am a MFE student and we have project on the Markowitz portfolio optimization problem. i am wondering how much impact there will be, if I use a simpler linear optimizater instead of a quadratic one. Say, i have a target portfolio $x$, my alpha is $a$. I will try to maximize $xa$, and apply a factor exposure limit: $$l_0 < Ax < l_1$$ while $A$ is my factor exposure What's the biggest disadvantage of above approach, compared with the classic quadratic approach widely used in Markowitz portfolio optimization. Can anyone explain to me a bit? ## Answer by wsw (score 1) https://quant.stackexchange.com/a/19240 The Markowitz model for portfolio optimization (http://www.princeton.edu/~rvdb/542/lectures/lec17.pdf) is formulated as a quadratic programming (QP) problem, not an LP one. You cannot use an LP solver to solve a QP problem. ## Answer by phdstudent (score 0) https://quant.stackexchange.com/a/18812 The Markowitz setup assumes agents have mean-variance preferences (CARA utility when returns are normally distributed yields the same). So the standard markowitz optimization maximizes risk-return tradeoff, where risk is measured by variance and return by mean. It penalizes risk depending on the degree of risk aversion. If instead you use linear programming, that is ok, but you should have a strong theory behind on why those constraints matter. As I said before the markowitz setup is based on a strong theory. ## Answer by nbbo2 (score 0) https://quant.stackexchange.com/a/19251 After Markowitz published his famous paper, William F. Sharpe published "A Linear Programming Algorithm for Mutual Fund Portfolio Selection" (1967). I haven't re-read it 20 years, but AFAIR it relies on a special structure for the covariance matrix and some assumptions about the utility. Maybe reading this paper would tell you if you are onto something new or not. http://pubsonline.informs.org/doi/abs/10.1287/mnsc.13.7.499
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