Affine scaling

In

Several groups then independently came up with a variant of Karmarkar's algorithm. E. R. Barnes at IBM,^[2] a team led by R. J. Vanderbei at AT&T,^[3] and several others replaced the projective transformations that Karmarkar used by affine ones. After a few years, it was realized that the "new" affine scaling algorithms were in fact reinventions of the decades-old results of Dikin.^[1][4] Of the re-discoverers, only Barnes and Vanderbei et al. managed to produce an analysis of affine scaling's convergence properties. Karmarkar, who had also came with affine scaling in this timeframe, mistakenly believed that it converged as quickly as his own algorithm.^[5]: 346

Algorithm

Affine scaling works in two phases, the first of which finds a feasible point from which to start optimizing, while the second does the actual optimization while staying strictly inside the feasible region.

Both phases solve linear programs in equality form, viz.

minimize c ⋅ x

subject to Ax = b, x ≥ 0.

These problems are solved using an iterative method, which conceptually proceeds by plotting a trajectory of points strictly inside the feasible region of a problem, computing projected gradient descent steps in a re-scaled version of the problem, then scaling the step back to the original problem. The scaling ensures that the algorithm can continue to do large steps even when the point under consideration is close to the feasible region's boundary.^[5]: 337

Formally, the iterative method at the heart of affine scaling takes as inputs A, b, c, an initial guess x⁰ > 0 that is strictly feasible (i.e., Ax⁰ = b), a tolerance ε and a stepsize β. It then proceeds by iterating^[1]: 111

• Let D_k be the diagonal matrix with x^k on its diagonal.
• Compute a vector of dual variables:

  ${\displaystyle w^{k}=(AD_{k}^{2}A^{\operatorname {T} })^{-1}AD_{k}^{2}c.}$

• Compute a vector of reduced costs, which measure the slackness of inequality constraints in the dual:

  ${\displaystyle r^{k}=c-A^{\operatorname {T} }w^{k}.}$

• If ${\displaystyle r^{k}>0}$ and ${\displaystyle \mathbf {1} ^{\operatorname {T} }D_{k}r^{k}<\varepsilon }$, the current solution x^k is ε-optimal.
• If ${\displaystyle -D_{k}r^{k}\geq 0}$, the problem is unbounded.
• Update ${\displaystyle x^{k+1}=x^{k}-\beta {\frac {D_{k}^{2}r^{k}}{\|D_{k}r^{k}\|}}}$

Initialization

Phase I, the initialization, solves an auxiliary problem with an additional variable u and uses the result to derive an initial point for the original problem. Let x⁰ be an arbitrary, strictly positive point; it need not be feasible for the original problem. The infeasibility of x⁰ is measured by the vector

${\displaystyle v=b-Ax^{0}}$.

If v = 0, x⁰ is feasible. If it is not, phase I solves the auxiliary problem

minimize u

subject to Ax + uv = b, x ≥ 0, u ≥ 0.

This problem has the right form for solution by the above iterative algorithm,^[a] and

${\displaystyle {\begin{pmatrix}x^{0}\\1\end{pmatrix}}}$

is a feasible initial point for it. Solving the auxiliary problem gives

${\displaystyle {\begin{pmatrix}x^{*}\\u^{*}\end{pmatrix}}}$.

If u^* = 0, then x^* ≥0 is feasible in the original problem (though not necessarily strictly interior), while if u^* > 0, the original problem is infeasible.^[5]: 343

Analysis

While easy to state, affine scaling was found hard to analyze. Its convergence depends on the step size, β. For step sizes β ≤ 2/3, Vanderbei's variant of affine scaling has been proven to converge, while for β > 0.995, an example problem is known that converges to a suboptimal value.^[5]: 342 Other variants of the algorithm have been shown to exhibit chaotic behavior even on small problems when β > 2/3.^[6][7]

Notes

References