Theory-Lesson 1 (KKT)

Question 1

What are the stages of the design process?

Answer 1

1) Conceptual design: Determites the principle of the solution
2)Preliminary design: Refining the conceptual designs and ranking them according to the design specifications.
3)Embodiment design: Design is developed with technical and economic criteria that lead to the final product.
4)Detail design: Completes the embodiment of products with final instructions about the components, the selection of materials, operation procedures and cost.

Question 2

Modelling guidelines

Answer 2

1)Simple but sufficient model.
2)Suitable molding of the problem.
3) Rigorous deduction phase.
4)Models should be validated before implemented.
5)Not criticized for failing.
6)GIGO.

Question 3

When is a function convex?

Answer 3

A subset X is convex if ax2+(1-a)x2 ε X, 0<=α<=1
For convex fuctions, local minimum is global minimum, optimal set is convex

Question 4

Which are the optimization methods?

Answer 4

The first category contains the optimality criteria (indirect method) and the second one the search methods (direct criteria)

Question 5

What is the exhaustive method?

Answer 5

It finds all the possible combinations between the DVs and the OFs
t=tsn.ofn,v^n.dv
where ts is the simulation time, n.of is the number of OFs,n.dv is the number of the design variables and n.v is the different values of each DV.
In the case where there are a lot of OFs and DVs this method is really unfeasible.

Question 6

What are the optimality conditions for an unconstrained problem?

Answer 6

If we assume that x* is the local minimum of f(x), then:
1)Necessary condition: Δf(x)=0 –>these points are called stationary points.
2)Sufficient condition:
Δ^2 f(x) to be positive definite.

Question 7

What’s a saddle point?

Answer 7

It’s a point for which Δf(x)=0 but is not an exremum.

Question 8

Which technique should we use for a constrained problem?

Answer 8

To check a point we think is a local minimum in a constrained optimization problem , we have to check it with respect to the Karush Kuhn Tacker optimality condition.
Firstly, we have to ckeck if the constraint we are dealing with is active or inactive. If we are dealing with an inactive constraint, we can use the method for the unconstrained problem (2 conditions).

For active constraints, the condition Δf(x*)=0 does not hold anymore because some constraints will block movement to this minimum.

We can’t get an improvement unless there is a vector d that its direction is both descending and towards the feasible region.

A vector d cannot exist if -Δf= Σ μi (Δ gi) => Σ μi (Δ gi)+Δf=0 for all active constraints iεI.

Inactive constraints can be included too but their μ=0 and so their μj*gj=0, so it doesn’t make any difference in the calculation.

Question 9

What are the necessary conditions for KKT optimality?

Answer 9

1) Δf(x) + Σ μι * Δgi(x)=0
2) gi(x)<=0 (feasibility)
3) μi * gi(x)=0 (complementary)
4) μi >=0 (non-negativity)

Question 10

KKT remarks

Answer 10

1) Necessity theorm helps identify poitns that are not optimal. A point si not optimal if it doesn’t satisfy the KKT optimality conditions.
2) Not all points that satisfy the KKT optimality conditions are optimal points.
3) If the problem is convex, the a local minimum is also a global minimum, and the KKT first order conditions are necessary as well as sufficient.