Consider the optimization problem min J(U₁, 4₂) UER2 with a) 1 - J(U₁, U₂) = \u² − 3u² + 2u² +5u₁-6u₂+C, CER. Determine

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Consider The Optimization Problem Min J U 4 Uer2 With A 1 J U U U 3u 2u 5u 6u C Cer Determine 1 (148.22 KiB) Viewed 20 times

Consider the optimization problem min J(U₁, 4₂) UER2 with a) 1 - J(U₁, U₂) = \u² − 3u² + 2u² +5u₁-6u₂+C, CER. Determine if the problem is convex on u € R². If it is not convex, introduce an inequality constraint h(U₁, U₂) ≤ 0 such that it becomes convex on u EUC R²! If you did not add a constraint or haven't been able to solve this task, use h(U₁, U₂) = -U₁ + 4.5 for the remaining tasks! Now consider the constrained problem min UER² d) e) f) s. t. J(U₁, U₂) h(U1, U2)<0 4₂ ≤ 1 with J(U₁, U₂) as before and h(u₁, U₂) from task a) (i.e. either the one you derived or the one given). b) State the Lagrangian and Karush-Kuhn-Tucker (KKT) conditions of first order for the given problem. c) Are the KKT conditions only necessary or also sufficient for the presented problem? Justify your answer! Find the optimal point (u, u) that solves the constrained problem. Explain how the constant c in J(U₁, U₂) changes the solution of the problem! Assume you want to solve the given problem numerically. Would you rather use the standard Newton method or the interior point method to do so? Ju- stify your answer and briefly describe, how your chosen method would solve the problem! Hint: You do not have to state all the equations involved, just the main idea of the method!