Consider the following constrained optimization problem: = min f(x1, x) x + x Subject to g(x,...

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Consider the following constrained optimization problem: = min f(x1, x₂) x² + x² Subject to g(x₁, x2) = 2x₁ + x₂ ≤ −5 OPT-1 (a) Write down the Wolfe dual of the problem (OPT-2) and eliminate (x, y) to obtain a problem in terms of the Lagrangian L as a function of only the (single) dual variable X. (b) By setting L'(2) = 0 show that λ = 2 is the optimal solution for the dual problem. Hence show that the optimal primal solution is (x, y) = (2, -1). (Hint: You must also check that L"(2) ≤0.). (c) As we know the solution occurs with equality in the constraint, write down the augmented Lagrangian Lo (x, A) for the problem (OPT-2) using an equality constraint 2x + y² = 3. Use the equations VL. (x,x) = 0 to verify that for arbitrary o> 0 the unconstrained stationary point (x₁(1, 0), ₂(1, 0)) of Lo(x, X) is (2, -1) when λ = 2. (d) By calculating V²L,((2, -1), 2) and show that this matrix is positive definite for a > 0. Consider the following constrained optimization problem: = min f(x1, x₂) x² + x² Subject to g(x₁, x2) = 2x₁ + x₂ ≤ −5 OPT-1 (a) Write down the Wolfe dual of the problem (OPT-2) and eliminate (x, y) to obtain a problem in terms of the Lagrangian L as a function of only the (single) dual variable X. (b) By setting L'(2) = 0 show that λ = 2 is the optimal solution for the dual problem. Hence show that the optimal primal solution is (x, y) = (2, -1). (Hint: You must also check that L"(2) ≤0.). (c) As we know the solution occurs with equality in the constraint, write down the augmented Lagrangian Lo (x, A) for the problem (OPT-2) using an equality constraint 2x + y² = 3. Use the equations VL. (x,x) = 0 to verify that for arbitrary o> 0 the unconstrained stationary point (x₁(1, 0), ₂(1, 0)) of Lo(x, X) is (2, -1) when λ = 2. (d) By calculating V²L,((2, -1), 2) and show that this matrix is positive definite for a > 0.

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a To write down the Wolfe dual of the problem OPT2 we need to convert the constrained minimization p... View the full answer