1.1Find a Stationary Point
Find the stationary point of $f(x,y)=x^2+2y^2-4x+8y$.
Solution
$\nabla f=\langle2x-4,4y+8\rangle=0$, so $(x,y)=(2,-2)$.
Optimization Practice
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Difficulty
Direct Practice
1.2Classify a Quadratic
Is $f(x,y)=3x^2+2xy+3y^2$ strictly convex?
Solution
The Hessian is $\begin{bmatrix}6&2\\2&6\end{bmatrix}$, with eigenvalues $4$ and $8$. It is positive definite, so the function is strictly convex.
Difficulty
Integrated Practice
2.1Write a KKT System
Write the KKT conditions for minimizing $f(x)$ subject to $g(x)\le0$.
Solution
$\nabla f(x)+\mu\nabla g(x)=0$, $g(x)\le0$, $\mu\ge0$, and $\mu g(x)=0$.
2.2Recognize a Convex Program
Why does a local minimum of a convex objective over a convex feasible set also solve the global problem?
Solution
Convexity prevents a feasible point from lying below a local minimum along any feasible line segment, so no feasible point has a smaller objective value.
Difficulty
Applied Problems
3.1Choose a Method
Choose a natural method for a smooth unconstrained problem whose Hessian is cheap to compute but may be indefinite.
Solution
Use safeguarded Newton steps with a line search or trust region, because an indefinite Hessian can give an ascent direction.