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### Interior-point methods: summary

- **Interior point methods (IPM)** are very reliable on small scale problems.
  - Example: SDP of dimension 100, SOCP with less than a thousand variables.
  - Most conic problems with a couple of hundred variables can be formulated and solved very quickly using preprocessors such as CVX.

- **IPM often efficient on larger problems** if KKT system has some structure (sparsity, blocks, etc).
  - Large scale linear programs with thousands of variables are routinely solved by free or commercial solvers using IPM (e.g., SDPT3, MOSEK, GLPK, CPLEX, etc.).
  - Much larger sparse LPs can also be solved efficiently using the same techniques.

- **Not workable for very large problems.**
  - For some problems, e.g., semidefinite programs, exploiting structure in IPM is hard.
  - First order methods (using the gradient only) seem to be the only option for extremely large problems.

*Source: A. d’Aspremont. Convex Optimization M2.*
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