In optimiazation system, you have to weight the price for the different maturities in a way that reflect your confidence in each data point (influenced by liquidity). One way to do so is to weight, each price by its Black-Scholoes Vega (see Tankov (2003)). So when minimazing the squared differences of the sum your weighted option prices, you can use the following approximation (in term of call price).

$ \sum_{i=1}^{N} w_i (C^\theta(T_i,K_i)-C(T_i,K_i))^2=\sum_{i=1}^{N} \frac{ (C^\theta(T_i,K_i)-C(T_i,K_i))^2}{ Vega_i^2(T_i,K_i)} $

The advantage of this method is that the option prices scaled by its vega is approximatively equal to its implied volatility and implied volatility are more uniform across maturity and strike than option prices. As a proof, you can apply a taylor approximation.

$C^\theta\approx C+Vega_{BS}(\sigma^\theta-\sigma_{BS}) \Leftrightarrow \frac{C^\theta-C}{Vega}\approx (\sigma^\theta-\sigma)$

In optimiazation system, you have to weight the price for the different maturities in a way that reflect your confidence in each data point (influenced by liquidity). One way to do so is to weight, each price by its Black-Scholoes Vega (see Tankov (2003)). So when minimazing the squared differences of the sum your weighted option prices, you can use the following approximation (in term of call price).

$$ \sum_{i=1}^{N} w_i (C^\theta(T_i,K_i)-C(T_i,K_i))^2=\sum_{i=1}^{N} \frac{ (C^\theta(T_i,K_i)-C(T_i,K_i))^2}{ \text{Vega}_i^2(T_i,K_i)} $$

The advantage of this method is that the option prices scaled by its vega is approximately equal to its implied volatility and implied volatility are more uniform across maturity and strike than option prices. As a proof, you can apply a taylor approximation.

$$C^\theta\approx C+\text{Vega}_{BS}(\sigma^\theta-\sigma_{BS}) \Leftrightarrow \frac{C^\theta-C}{Vega}\approx (\sigma^\theta-\sigma)$$