Chapter 4 – Quiz 2 – E600 Mathematics

1. True or false: conceptually, we can view constrained optimization with constraint set $C(\mathcal P)\subseteq dom(f)$ and objective $f$ as unconstrained optimization of the restricted function $f|_{C(\mathcal P)}$ .

False.

True.

2. Which statement about the Lagrangian function is not true? (Single choice)

Different local extremizers in the equality-constrained problem can be associated with different Lagrangian multipliers.

The first order condition of the equality-constrained problem is that the gradient of the Lagrangian function is equal to the zero vector.

Any interior local maximizer in the equality-constrained problem is a maximizer of the Lagrangian function.

Any interior local extremizer in the equality-constrained problem is a critical point of the Lagrangian function.

3. Select all true statements about the Lagrangian method. (Multiple choice)

Singularity points are defined as points where the constraint function $g$ is discontinuous.

We can summarize any collection of equality constraints by the requirement $g(x)=\mathbf 0$ using an appropriate function $g$ .

The implicit function theorem directly gives a sufficient condition to identify local extremizers.

Compared to unconstrained optimization, there are no additional border solution candidates for equality-constrained problems.

If there are singularity points, the constrained optimization problem can not have a solution.

The implicit function theorem is the central tool we use to generalize the methods of unconstrained optimization to equality-constrained problems.

4. What is the value of the objective function at the constrained global maximizers in $\max x_1\cdot x_2 \hspace{0.5cm}\text{subject to}\hspace{0.5cm} x_1^2 + x_2^2 = 8$ ? You can take for granted that a solution exists (because there are no border solution candidates, this means the solution(s) will be identified by the Lagrangian method). Write down your result as an integer, do not use spaces, letters, etc.

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