Please help me convex problem

#1

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Can I use the CVX tool to slover this problem? Thank you!

(Peixi Liu) #2

The objective function is linear, but the left side of the second constraint is not concave, so this problem is not convex. You have to firstly transform this probem into a convex one equivalently.

(Mark L. Stone) #3

I’ll presume the relevant matrices are symmetric positive semidefinite, and that \tau also has a lower bound of zero (or greater) although I don’t see that explicitly stated in what is shown.

You can multiply the constraint on the 2nd line of constraints by \lambda_{max}. But the product term between \lambda_{min} and \tau is non-convex and will not be allowed by CVX. You could deal with that by not declaring \tau as a CVX variable, and instead, solve a separate CVX problem for each fixed value of \tau over a grid of its possible values. Then choose the solution with the best objective value from among these problems.

Actually, I believe that constraint is quasi-convex, so can be handled by using the bisection algorithm in section 4.2.5 “Quasiconvex optimization” of " Convex Optimization" by Boyd and Vandenberghe https://web.stanford.edu/~boyd/cvxbook/

1 Like
#4

I get. Thank you very much!

#5

I get. Thank you very much!