User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Iterative Convex Overbounding

Iterative convex overbounding is a technique based on Young’s relation that is useful when solving an optimization problem with a BMI constraint.

The System
Consider the matrices $$ Q=Q^{T}\in \mathbb{R}^{n\times n}, B \in \mathbb{R}^{n\times m}, R \in \mathbb{R}^{m\times p}, D \in \mathbb{R}^{p\times q},S \in \mathbb{R}^{q\times r} and C \in \mathbb{R}^{r\times n}$$, where S and R are design variables in the BMI given by



\begin{align} \qquad Q + BRDSC + C^{T}S^{T}D^{T}R^{T}B^{T} < 0      \qquad    (1)\\ \end{align}                              $$

The LMI:Iterative Convex Overbounding
Suppose that S0 and R0 are known to satisfy (1). The BMI of (1) is implied by the LMI
 * $$\begin{bmatrix}

Q+\phi(R,S)+\phi^{\text{T}}(R,S) & B(R-R0)U &C^{\text{T}}(S-S0)^{\text{T}}V^{\text{T}}\\ \end{bmatrix} < 0 \qquad   (2) $$ where Φ(R,S) = B(RDS0+R0DS-R0DS0)C, W>0 is an arbitrary matrix, D=UV, and the matrices U and VT have full column rank. The LMI of (2) is equivalent to the BMI of (1) when R = R0 and S = S0, and is therefore non-conservative for values of R and S and are close to the previously known solutions R0 and S0.
 * &W^{\text{-1}}&0\\
 * &*&-W\\

Alternatively, the BMI of (1) is implied by the LMI
 * $$\begin{bmatrix}

Q+\phi(R,S)+\phi^{\text{T}}(R,S) &Z^{\text{T}}U^{\text{T}}(R-R0)^{\text{T}}B^{\text{T}}+V(S-S0)C\\
 * &Z\\

\end{bmatrix} < 0 \qquad   (3) $$ where Z > 0 is an arbitrary matrix, D = UV, and the matrices U and VT have full column rank. Again, the LMI of (4) is equivalent to the BMI of (2) when R = R0 and S = S0, and is therefore non-conservative for values of R and S and are close to the previously known solutions R0 and S0.

Conclusion:
A benefit of convex overbounding compared to a linearization approach, is that in addition to ensuring conservatism or error is reduced in the neighborhood of R = R0 and S = S0, the LMIs of (2) and (3) imply (1). Iterative convex overbounding is particularly useful when used to solve an optimization problem with BMI constraints.