# LMIs in Control/Matrix and LMI Properties and Tools/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 ${\displaystyle Q=Q^{T}\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},R\in \mathbb {R} ^{m\times p},D\in \mathbb {R} ^{p\times q},S\in \mathbb {R} ^{q\times r}andC\in \mathbb {R} ^{r\times n}}$ , where S and R are design variables in the BMI given by

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

## The LMI:Iterative Convex Overbounding

Suppose that S0 and R0 are known to satisfy (1). The BMI of (1) is implied by the LMI

${\displaystyle {\begin{bmatrix}Q+\phi (R,S)+\phi ^{\text{T}}(R,S)&B(R-R0)U&C^{\text{T}}(S-S0)^{\text{T}}V^{\text{T}}\\*&W^{\text{-1}}&0\\*&*&-W\\\end{bmatrix}}<0\qquad (2)}$

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.

Alternatively, the BMI of (1) is implied by the LMI

${\displaystyle {\begin{bmatrix}Q+\phi (R,S)+\phi ^{\text{T}}(R,S)&Z^{\text{T}}U^{\text{T}}(R-R0)^{\text{T}}B^{\text{T}}+V(S-S0)C\\*&Z\\\end{bmatrix}}<0\qquad (3)}$

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.