The Feasibility Problem

Resources: Nocedal-Wright2006_Book_NumericalOptimization.pdf Chapter 15.4

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Inequality constraints on states may imply that for some $x_0$, there are no solutions to the MPC QP. We solve this by adding a slack variable $ϵ$.

Consider a Normal QP, we then modify it

$$\underset{z,ϵ}{\min }\frac{1}{2}{x}^{T}Gx+c^{T}x+{\zeta }^Tϵ,s.t.\{\begin{array}{ll}{x}_{t+1}=A{x}_t+B{u}_t,& t=0,\dots ,N-1\\ -ϵ+x^{low}\le x_t\le x^{high}+ϵ& t=1,\dots ,N\\ u^{low}\le u_t\le u^{high}& t=0,\dots ,N-1\end{array}$$

Where $\zeta$ is now a linear weight. We also implement an Exact Penalty, which means that if the normal problem is feasible, then we set $ϵ=0$ (no need for slack).

In some practical implementations you add a quadratic term to the objective function, ${ϵ}^{T}wϵ$.