Optimization algorithm QP vs MIQP

[Kristine]

Hello

I used gurobi to minimize a quadratic cost function subject to dynamic
constraint and range (continuous) constraints.
Problem 1 under can be solved by a QP and Problem 2 can be solved by
the MIQP.
However, i obtained the unexpected relation between the optimal cost
of the two problems that
J1 < J2 (while in fact from theory I expect J1>J2).
I can not understand this, therefore Im writing to ask if the
optimization algorithm assumes that the solution u(k) has to be
continuous when solving problem 1.

Where can I find information on how the (what) problem is solved?

PROBLEM 1
J1 = min_u sum_{k=0}^{k=N} [x(k)^2 + u(k)^2]
s.t.
x(k+1) = Ax(k) + Bu(k) + w(k)
m_min<=u(k)<=u_max


PROBLEM 2
J2 = min_u sum_{k=0}^{k=N} [x(k)^2 + u(k)^2]
s.t.
x(k+1) = Ax(k) + Bu(k) + w(k)
u(k) = u_min OR u_max

[Christopher Maes]

> I can not understand this, therefore Im writing to ask if the
> optimization algorithm assumes that the solution u(k) has to be
> continuous when solving problem 1.
>
> Where can I find information on how the (what)  problem is solved?

Hi Kristine,

Have you double checked that your formulation of both problems is correct?

I'm not sure I understand what you mean by a continuous solution. For
a QP, such as problem 1, variables in the solution may take on
fractional values. With a MIQP, such as problem 2, variables may be
constrained to take integer values.

The Wikipedia page on quadratic programming contains links to some
solution methods. It is a good place to start to learn how QPs are
solved:
http://en.wikipedia.org/wiki/Quadratic_programming

Thanks,
Chris

[Imre Polik]

Problem 2 has a smaller feasible set, so its objective value will be
worse. Since you are minimizing, this means J2>J1. The continuous
relaxation has a better objective value than the integer problem.
There's no contradiction, unless you mistyped the relation sign in
your posts.