d min lk (p X1, X2, ,xn) subject to P j=1 where n d n ||X₂||1 lk (p X1, X2, · ,xn) -ΣΣ ijlogp; Σlog Xil, Xi2, , Xid i=1

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d min lk (p X1, X2, ,xn) subject to P j=1 where n d n ||X₂||1 lk (p X1, X2, · ,xn) -ΣΣ ijlogp; Σlog Xil, Xi2, , Xid i=1 j=1 i=1 To derive p in theory, answer the following questions. Q1 Write down the Lagrangian function Lk (p, λ | X₁, X2, · ,xn) of this MLE problem, where λ is the dual variable. Find the dual function and dual problem with respective to X. Q2 Solve the dual problem from Q1 to find the optimal dual point \*. Q3 Write down the KKT conditions of this MLE problem and verify that P Σ=1&ij Σ₁=1 Σi=1 xij Hint: for Q3, the KKT conditions include the stationarity condition (derived from the Lagrangian function Lk in Q1) and the primal constraint. Pj = 1,