Quadratic optimization with quadratic constraints

Hi everyone,

I have an optimization problem with a quadratic objective function and quadratic constraint functions AND the problem is non-convex.

Is there any Matlab function which can do this? QUADPROG and FMINCON only allow linear constraints afaik. I also tried a solver by MOSEK (<http://mosek.com/>) but this only can deal with convex problems. Is there any tool/function for the non-convex case?

Thanks! René

Answer 1

[Steve Grikschat]

Hi Rene (sorry, can't get the accent right)

You can use FMINCON to solve this problem. However, there is no dedicated input for quadratic constraints. Instead, you must formulate them as nonlinear constraints.

Since your problem is non-convex, you should probably use the interior-point algorithm. http://www.mathworks.com/help/toolbox/optim/ug/brnoxzl.html#brnpd5f

Also, since your objective and constraint functions are quadratic, you can save a lot of time by providing outputs that compute the gradients (H*x + f) and a function that computes the Hessian (of the Lagrangian, in this case).

To do this, you'll need to create functions to compute your objective and constraints, as well as setting these options:

- Algorithm to 'interior-point' - GradObj to 'on' (user-computed 1st derivatives) - GradConstr to 'on' (same) - Hessian to 'user-supplied' (user-computed 2nd derivatives) - HessFcn to a handle to a function that computes the Hessian of the Lagrangian (see this page:http://www.mathworks.com/help/toolbox/optim/ug/fmincon.html#f186882)

Options Reference: http://www.mathworks.com/help/toolbox/optim/ug/f19175.html

+Steve

[Matt J]

Rene commented:

Sorry for responding that late, I had been busy with other stuff lately and just wanted to say "thank you" for your help, Steve!

Answer 2

[Matt J]

If you have single, L2-norm constraint, then this FEX submission should help,

http://www.mathworks.com/matlabcentral/fileexchange/53191-quadratic-minimization-with-2-norm-constraints

Answer 3

[Steve Grikschat]

As of R2020b, Optimization Toolbox now has a dedicated solver for second-order cone programming, which can be used to solve quadratic constrained problems.

https://www.mathworks.com/help//optim/ug/convert-quadratic-constraints-to-second-order-cone-constraints.html

https://www.mathworks.com/help//optim/ug/convert-qp-to-socp.html

Function reference:

https://www.mathworks.com/help//optim/ug/coneprog.html

coupled with a function to make a second-order cone constraint

https://www.mathworks.com/help//optim/ug/optim.coneprog.secondorderconeconstraint.html

For an example see

https://www.mathworks.com/help//optim/ug/cone-programming-mass-nonlinear-spring.html

[rui liu]

Hello, I have seen how to convert qp to socp(https://ww2.mathworks.cn/help/optim/ug/convert-qp-to-socp.html), however, I have some questions about the derivation process. So I want to ask if there are more documents about it, such as papers.

[Steve Grikschat]

Hi,

This equivalence seems to be stated quite often,but rarely with a thorough explanation. At least, none that I could find. (For example, this paper).

I searched and the closest I found was here.

[rui liu]

thank you very much!

[rui liu]

Hello, if there is some methods to accelerate the SOCP computation. I try to generate C code but it says "Function 'secondordercone' not supported for code generation. if there is other methods to make the solver process quick. Thank you!

[Steve Grikschat]

C code generation for coneprog is not supported in the current release, but is on our plans (see if you can access the R2024b pre-release notes which mentions this: https://www.mathworks.com/help/releases/R2024b/optim/release-notes.html)

In the meantime, an option that can have a notable effect on the solve time is the LinearSolver option

https://www.mathworks.com/help/optim/ug/coneprog.html#mw_3159abce-1444-4850-bb21-1b73934406cf

If your linear or cone constraints are dense (many nonzeros) or have dense columns, then tuning this option may be helpful.