Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of ${\displaystyle n}$ ${\displaystyle m}$-dimensional data items, a configuration ${\displaystyle X}$ of ${\displaystyle n}$ points in ${\displaystyle r}$ ${\displaystyle (\ll m)}$-dimensional space is sought that minimizes the so-called stress function ${\displaystyle \sigma (X)}$. Usually ${\displaystyle r}$ is ${\displaystyle 2}$ or ${\displaystyle 3}$, i.e. the ${\displaystyle (n\times r)}$ matrix ${\displaystyle X}$ lists points in ${\displaystyle 2-}$ or ${\displaystyle 3-}$dimensional Euclidean space so that the result may be visualised (i.e. an MDS plot). The function ${\displaystyle \sigma }$ is a cost or loss function that measures the squared differences between ideal (${\displaystyle m}$-dimensional) distances and actual distances in r-dimensional space. It is defined as:

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

where ${\displaystyle w_{ij}\geq 0}$ is a weight for the measurement between a pair of points ${\displaystyle (i,j)}$, ${\displaystyle d_{ij}(X)}$ is the euclidean distance between ${\displaystyle i}$ and ${\displaystyle j}$ and ${\displaystyle \delta _{ij}}$ is the ideal distance between the points (their separation) in the ${\displaystyle m}$-dimensional data space. Note that ${\displaystyle w_{ij}}$ can be used to specify a degree of confidence in the similarity between points (e.g. 0 can be specified if there is no information for a particular pair).

A configuration ${\displaystyle X}$ which minimizes ${\displaystyle \sigma (X)}$ gives a plot in which points that are close together correspond to points that are also close together in the original ${\displaystyle m}$-dimensional data space.

There are many ways that ${\displaystyle \sigma (X)}$ could be minimized. For example, Kruskal recommended an iterative steepest descent approach. However, a significantly better (in terms of guarantees on, and rate of, convergence) method for minimizing stress was introduced by Jan de Leeuw. De Leeuw's iterative majorization method at each step minimizes a simple convex function which both bounds ${\displaystyle \sigma }$ from above and touches the surface of ${\displaystyle \sigma }$ at a point ${\displaystyle Z}$, called the supporting point. In convex analysis such a function is called a majorizing function. This iterative majorization process is also referred to as the SMACOF algorithm ("Scaling by MAjorizing a COmplicated Function").