ORR kvizy

A scalar function of n real variables is guaranteed to have a minimum value provided

• the function is continuous.

• the function is smooth and convex and is analyzed on a convex set.

• the function is continuous and we analyze the function on a closed and bounded set.

A given real scalar function f(.) assumes a local minimum at x* if

• there exists some ε > 0 such that f(x) ≥ f(x*) for all x within a distance ε from x*.

• f(x) ≥ f(x*) for all x

• there exists some ε > 0 such that f(x) ≤ f(x*) for all x within a distance ε from x*.

Let's say that for some (unspecified) function, the Hessian matrix evaluated at the stationary point is:

Classify correctly the stationary point as one of the choices below

• minimum

• saddle point

• cannot decide based on the provided information. Some more analysis is needed

• maximum

The first-order necessary conditions of optimality for the optimization problem

min f(x)

subject to

g(x) = 0

are given by the following

• the optimization criterion f(x) is minimized or maximized

When checking if sufficient conditions of optimality for an equality-constrained optimization are satisfied, we need to compute the projected Hessian. We cannot just use the standard Hessian of the augmented cost function. Why?

• Projected Hessian extends the space in which we can search for a minimum.

• The standard Hessian imposes no restrictions on the vectors while in the constrained optimization the vectors are constrained.

• Projection of Hessian improves its numerical properties.

• 2

• 3

• 4

• 5

A local minimum of a convex function is also global. But is true that the convexity of the function is needed for a local minimum to be global?

• True

• False