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Optimization for Machine Learning with NumPy and SciPy

Solution: Constrained Optimization

Explore how to solve constrained optimization problems by formulating the dual problem and applying gradient-based methods. Understand the use of Lagrange multipliers and projection techniques to handle constraints effectively, enabling practical solutions for portfolio optimization and other machine learning challenges.

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Explanation

In the portfolio optimization problem,x1,x2,x3x_1,x_2,x_3 denote the fraction of each stock (STK1, STK2, STK3) in the portfolio. The goal is to maximize the diversity of the portfolio given by the EMP.

An EMP can be written as a standard minimization problem as follows:

where A∈Rm×dA \in \R^{m \times d} is the constraint matrix with the mm constraints and b∈Rmb \in \R^m is the constraint vector.

In the portfolio optimization problem x=[x1x2x3]∈R3x = ...