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LP Duality and Shadow Prices is a hard coding interview question covering linear programming, math, machine learning — the kind asked at FAANG (Facebook/Meta, Apple, Amazon, Netflix, Google) and AI-first companies like Anthropic, OpenAI, and Nvidia.
Given a linear program in standard form:
Primal: maximize c^T x subject to Ax <= b, x >= 0 Dual: minimize b^T y subject to A^T y >= c, y >= 0
Implement: 1. Formulate the dual from the primal 2. Verify complementary slackness: x_i * (A^T y - c)_i = 0 and y_j * (b - Ax)_j = 0 3. Compute shadow prices (dual optimal values) which represent the marginal value of relaxing each constraint by one unit
This is fundamental to understanding sensitivity analysis in LP and Lagrangian duality in ML (SVM dual, etc.).
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