MultiLevel Optimization

[Dana Wild]

Is it possible to do multilayer convex optimization in Yalmip? Say, convex-concave min-max optimization? 

If yes, can we do it by multiple application of solvesdp functions?

[Johan Löfberg]

YALMIP has various approaches to bilevel optimization with inner convex quadratic programs. There is a built-in naive solver branching directly on complementarities, or you can symbolically derive KKT conditions and optimize over these using either mixed-integer or nonlinear global solver, or you can do everything manually.

Some theory and manual implementations
http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Examples.BilevelProgramming

More theory, and the built-in solver
http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Tutorials.BilevelProgramming

Or derive the KKT and solve using an efficient MIQP solver (typically the best way in YALMIP)
http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Commands.Kkt

For the special case of minmax, you might want to look at robust optimization module instead. In some cases, the minmax problem can be explicitly posed as a robust counterpart, thus avoiding the whole bilevel stuff (which is inherently very hard)
http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Tutorials.RobustOptimization