Elementary symmetric polynomials and optimization

Themth elementary symmetric polynomial of degreen

is the sum of all terms containing a product of m variables. So, for example,

These polynomials came up in the previous post. The problem was choosing weights to minimize the variance of a weighted sum of random variables can be solved using elementary symmetric polynomials.

To state the optimization problem more generally, suppose you want to minimize

where the t_i and x_i are positive and thet_i sum to 1. You can use Lagrange multipliers to show that the solution is

Related posts

• Proof of optimization
• Symmetric funcions and U-statistics
• Minimize squared relative error
• Inverse optimization
• Differentially private stochastic gradient descent

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