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Optimization for Machine Learning with NumPy and SciPy

Solution: Constrained Optimization

Explore constrained optimization by studying the portfolio optimization example and learning to derive and solve the dual problem. Understand the use of Lagrange multipliers, convex duality, and the projected gradient descent algorithm to address constraints efficiently. Gain practical skills to implement and stabilize optimization with NumPy, tackling numerical challenges commonly found in machine learning tasks.

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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 = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 ...