I want to solve the following optimization problem maximize f ( X ) = − log ⁡ d e t ( X + Y ) − a T ( X + Y ) − 1 a subject to X ⪰ W ,where the design variable X is symmetric positive semidefinite, W , Y are fixed symmetric positive semidefinite matrices, and a is a given vector. The inequality X ⪰ W means that X − W is symmetric positive semidefinite.I was wondering if there's hope to find an analytical solution to either the constrained or unconstrained problem. And if there is none, could I use convex optimization techniques to solve this numerically?

Question

I want to solve the following optimization problem  maximize   f  (  X  )  =  −  log  ⁡      d    e    t    (  X  +  Y  )  −      a    T    (  X  +  Y      )          −      1        a    subject to   X  ⪰  W  ,where the design variable   X is symmetric positive semidefinite,   W  ,  Y are fixed symmetric positive semidefinite matrices, and   a is a given vector. The inequality   X  ⪰  W means that   X  −  W is symmetric positive semidefinite.I was wondering if there&#039;s hope to find an analytical solution to either the constrained or unconstrained problem. And if there is none, could I use convex optimization techniques to solve this numerically?

Raiden Williamson · Accepted Answer

Make the substitution   Z  =  (  X  +  Y      )          −      1        ⇔  X  =      Z          −      1        −  Y, then the problem becomes convex:  maximize   log  ⁡      d    e    t    (  Z  )  −      a    T    Z  a    subject to   0  ≺  Z  ⪯  (  Y  +  W      )          −      1

I want to solve the following optimization problem <mtext>maximize </mtext> f (

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