How to get the explicit expression of the optimal solution of the quadratic multi-parametric programming problem?

[Shuai Lu]

Dear professor,

I am trying to getting the explicit expression of the optimal solution of the following convex quadratic parametric programming problem,

Theoretically, the optimal solution z(θ) should be a piecewise, convex, and quadratic function. Is Yalmip able to get the explicit expression of z(θ)? Will MPT toolbox work for this problem?

Besides, in my problem, θ is a scalar rather than a vector (i.e., m=1), which seems to make this problem easier.

I would appreciate it if you could give me some help or suggestions.

Best regards,

Shuai

[Johan Löfberg]

https://yalmip.github.io/tutorial/multiparametricprogramming/

[Shuai Lu]

Thank you very much, Professor.

Besides, if θ is a high-dimensional vector, for example, θ has 2000 elements, is it possible for MPT to obtain the solution in a reasonable time? Is this problem NP-hard?

On Friday, September 11, 2020 at 11:02:52 PM UTC+8 Johan Löfberg wrote:

https://yalmip.github.io/tutorial/multiparametricprogramming/

[Johan Löfberg]

2000 is just not gonna work.

MPT works well in 2D (as long it isn't numerically challengin), ok in 3d, maybe in 4d, and then you have to have luck. Complexity is exponential in dimension, and typically that complexity hits you hard already in low dimensions (in contrast to mixed-integer programming where you still can solve pretty large problems)

[Shuai Lu]

I got it. Thank you very much!

Best regards.