Topological Optimization | Chapter 3

Consider the following optimization problem, in which a function δ has to be minimized.

• The solution does not exist because the problem is unbounded.

• The solution does not exist because no feasible points exist.

• The solution is x₁ = 10 and x₂ = 10/7.

Answer explanation

It is always possible to find a set of (x₁, x₂) that leads to a value of the function closer to zero, so no solution can exist. If a bound on x₁ and x₂ is set, a solution becomes possible.

For unconstrained optimization problems, local and global minima are located at stationary points.

• True

• False

Answer explanation

It is true that for an unconstrained problem local and global minima are located at stationary points.

For constrained problems, local minima might be located...

• At stationary points

• Outside the feasible set

• On the boundary of the feasible set

• At stationary points of the constraint equations

Answer explanation

For constrained problems, local minima are not even necessarily located at stationary points, as they may be located on the boundary of the feasible set.

With reference to the following function, which are its stationary points?

• All points highlighted with dashed vertical lines

• x₂, [x_3, x₄], x₅

• x₂, x_3, x₄, x₅, x₆

• x₂, x_3, x₄, x₅

• x₂, x₅

Answer explanation

The points x₂, [x₃, x₄] and x5 are stationary points. Note that ALL points between x₃ and x₄ are stationary points (it is a plateau!)

With reference to the following function, which are its local minima?

• x₁, x₅

• x₁, [x₃, x₄], x₅

• None of the listed answers is correct

• Only global minima exist for the current function

Draw a strictly convex function f : R → R.

Answer explanation

Which of the following functions are convex?

• Middle one

• Right one

• Left one

Answer explanation

Left function is strictly convex. Middle function is just convex. All strictly convex functions are also convex, therefore two right answers exist.

The generic optimization problem (P) is convex if the objective function and all __________ _________ are convex.

Answer explanation

The generic optimization problem (P) is convex if the objective function and all constraint functions g_i, i = 1,...,l, are convex.

When the feasible set is bounded and closed, a solution to the optimization problem always exists.

• This is true for any continuous objective function, not necessarily convex

• This is true for any objective function, not necessarily convex

• This is only true for convex objective functions

• This is never true

Answer explanation

When the feasible set is compact, i.e. bounded and closed, a solution always exists (this is true for any continuous objective function, not necessarily convex).

Why is Hessian matrix important?

• To check if a function is covex

• To check if a solution to the optimization problem exist

• To check if a point belongs to the feasible set

Answer explanation