Optimal and Robust Control

1. Unconstrained minimization of quadratic function in the vector form

The quadratic function to be minimized has the following form

\(f(\mathrm x) = \frac{1}{2}\mathrm x^T\mathrm Q\mathrm x + \mathrm r^T\mathrm x\)
 
For the particular matrices Q and vectors r find the critical points of this optimization problem and examine if the function achieves a minimum or maximum or if the stationary point is just a saddle point.

a) Q = [1 1; 1 2]; r = [0; 1];

b) Q = [-1 1; 1 2];  r = [0; 1];

2. Unconstrained minimization of quadratic function in the scalar form

Find the minimum value of

\( f(x_1,x_2) = x_1^2 - x_1x_2 + x_2^2 + 3x_1 \)

Find the curvature (aka Hessian) matrix at the minimum. Sketch the contours, showing the gradient at several points.

3. Singularity of Hessian is inconclusive

The function \(f(x,y)=x^2+y^4\) has a minimum at the origin.

• Verify that the origin is a critical point.
• Show that the curvature matrix is singular at the origin.
• Prove that the critical point is indeed a minimum.

4. Optimum of a linear equality-constrained quadratic problem (QP)

Find an extremum (minimum or maximum) of the function

\(x_1+x_2^2+x_2x_3+2x_3^2\)

subject to

\(x_1+x_2+x_3=1\)

Solve the problem both without a specialized optimization software and with two distinct specialized optimization packages/solvers.

5. Optimum of a quadratic equality-constrained problem quadratic problem

Find an extremum (minimum or maximum) of the function

x 1 +x 2 2 +x 2 x 3 +2x 2 3  

subject to 1 2 (x 2 1 +x 2 2 +x 2 3 )= 1 .

Solve the problem both without a specialized optimization software and with two distinct specialized optimization packages/solvers.

6. Maximum of an inequality-constrained problem

Maximize

\(14x_1-x_1^2+6x_2-x_2^2+7\)

subject to

\(x_1+x_2\leq 2,\;x_1+2x_2\leq 3,\; x_1\geq 0,\; x_2\geq 0\)

As in the problem above, solve the problem both without a specialized optimization software and using two distinct specialized optimization packages/solvers.

7. Rectangle inside an ellipse

Find a rectangle of maximum perimeter that can be inscribed inside an ellipse. That is, maximize 4 (x+ y) subject to the constraint x 2 / a 2 +y 2 / b 2 =1 . Find a rectangle of maximum area 4 xy that can be inscribed inside an ellipse.

8. Meteor closes point of approach

A meteor is in a hyperbolic orbit described with respect to the earth at the origin by
x 2 a 2 −y 2 b 2 =1
Find its closest point of approach to a satellite that is in such an orbit that it has a constant position of ( x 1 ,y 1 ) . (Substitute numerical values for a and b if needed). Verify that the solution indeed yields a minimum.

9. Quadrangle with largest area

The lengths of all sides of a quadrangle are fixed, but the sides are linked at vertices freely, so the angles between them can vary, see the figure. Which configuration corresponds to the largest area of the quadrangle?

10. Regular point

Consider the following constraints in R²

x 1 ≥0,x 2 ≥0,x 2 −(x 1 −1) 2 ≤0 .

Show that the point x 1 =1,x 2 =0 is feasible but not regular.