The stress function ${\displaystyle \sigma }$  can be expanded as follows:

${\displaystyle \sigma (X)=\sum _{i<j\leq n}w_{ij}(d_{ij}(X)-\delta _{ij})^{2}=\sum _{i<j}w_{ij}\delta _{ij}^{2}+\sum _{i<j}w_{ij}d_{ij}^{2}(X)-2\sum _{i<j}w_{ij}\delta _{ij}d_{ij}(X)}$ 

Note that the first term is a constant ${\displaystyle C}$  and the second term is quadratic in ${\displaystyle X}$  (i.e. for the Hessian matrix ${\displaystyle V}$  the second term is equivalent to tr${\displaystyle X'VX}$ ) and therefore relatively easily solved. The third term is bounded by:

${\displaystyle \sum _{i<j}w_{ij}\delta _{ij}d_{ij}(X)=\,\operatorname {tr} \,X'B(X)X\geq \,\operatorname {tr} \,X'B(Z)Z}$ 

where ${\displaystyle B(Z)}$  has:

${\displaystyle b_{ij}=-{\frac {w_{ij}\delta _{ij}}{d_{ij}(Z)}}}$  for ${\displaystyle d_{ij}(Z)\neq 0,i\neq j}$ 

and ${\displaystyle b_{ij}=0}$  for ${\displaystyle d_{ij}(Z)=0,i\neq j}$ 

and ${\displaystyle b_{ii}=-\sum _{j=1,j\neq i}^{n}b_{ij}}$ .

Proof of this inequality is by the Cauchy-Schwarz inequality, see Borg^[3] (pp. 152–153).

Thus, we have a simple quadratic function ${\displaystyle \tau (X,Z)}$  that majorizes stress:

${\displaystyle \sigma (X)=C+\,\operatorname {tr} \,X'VX-2\,\operatorname {tr} \,X'B(X)X}$ 

${\displaystyle \leq C+\,\operatorname {tr} \,X'VX-2\,\operatorname {tr} \,X'B(Z)Z=\tau (X,Z)}$ 

The iterative minimization procedure is then:

• at the ${\displaystyle k^{th}}$  step we set ${\displaystyle Z\leftarrow X^{k-1}}$ 
• ${\displaystyle X^{k}\leftarrow \min _{X}\tau (X,Z)}$ 
• stop if ${\displaystyle \sigma (X^{k-1})-\sigma (X^{k})<\epsilon }$  otherwise repeat.

This algorithm has been shown to decrease stress monotonically (see de Leeuw^[2]).

Stress majorization and algorithms similar to SMACOF also have application in the field of graph drawing.^[4][5] That is, one can find a reasonably aesthetically appealing layout for a network or graph by minimizing a stress function over the positions of the nodes in the graph. In this case, the ${\displaystyle \delta _{ij}}$  are usually set to the graph-theoretic distances between nodes ${\displaystyle i}$  and ${\displaystyle j}$  and the weights ${\displaystyle w_{ij}}$  are taken to be ${\displaystyle \delta _{ij}^{-\alpha }}$ . Here, ${\displaystyle \alpha }$  is chosen as a trade-off between preserving long- or short-range ideal distances. Good results have been shown for ${\displaystyle \alpha =2}$ .^[6]