A convex problem with a duality gap

ProofAdvanced⏱ 30 minAdvanced Optimizationduality gapslatercounterexample

Problem

Consider

\min_{x,y}\ e^{-x} \quad\text{subject to}\quad \frac{x^2}{y}\le0,\quad y>0 .

Show that the problem is convex, compute $p^\star$ and $d^\star$ , and identify which hypothesis of Slater's theorem fails.

Write a complete argument. Every inequality you use should be one you could state precisely.

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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