# Self-Regularity: a new paradigm for primal-dual interior-point algorithms

## By Gerald Moore

The fundamental topic of this book is the complexity analysis of primal-dual interior-point algorithms for linear programming problems.

Of course the classical algorithm for this problem is the famous Simplex method. This iterative method continually reduces the objective function by moving from vertex to vertex of the feasible set, and thus (theoretically) solves a linear programming problem exactly in a finite number of iterations. Although the Simplex method (almost) always converges quickly in practice, there are known examples where the number of iterations required depends exponentially on N, the size of the linear programming problem; e.g. the well-known Klee-Minty example. Thus an open problem was to create a new algorithm for solving linear programming problems, for which the number of iterations required was guaranteed to be only polynomial in N. The big disappointment, however, was that when such algorithms appeared, e.g. the Khachiyan algorithm in 1979, they performed much worse than the Simplex method in practice.

In 1984, however, the situation changed abruptly with the announcement of Karmarkar's algorithm. In contrast to the Simplex method, this new algorithm generated a sequence of iterates in the interior of the feasible set; hence the name interior-point method''. Most importantly, Karmarkar's algorithm was polynomial in N and he claimed that it could solve large linear programming problems much faster than the Simplex method. It was quickly recognised that there were strong links between this new idea and older rejected algorithms based on the idea of barrier functions''. For a number of years the area of linear optimisation was in a state of flux, but a consensus gradually developed that the new approach was best applied in a primal-dual framework'', i.e. both the primal and dual linear programming problems were solved simultaneously.

The present state of primal-dual interior-point algorithms is that they demand the solution of a highly-structured system of (slightly) nonlinear equations. This system is solved by introducing a non-negative homotopy parameter \mu and using Newton's method. Thus we require the solution for \mu=0, but it is easy to write down the solution for \mu=1. Hence \mu is gradually reduced from 1 to 0 with the solution for each value of \mu computed by Newton's method. One would like to state precisely how \mu should be decreased and precisely when the Newton iteration at each homotopy step should be terminated, so that not only does the algorithm perform well in practice but also the total amount of work required in terms of N can be proved to satisfy a realistic bound.

It is here that there has been a gap between theory and practice, whose elimination is one of the main purposes of this book. The homotopy methods that have performed best in practice are the so-called large update-neighbourhood'' methods, which reduce \mu to zero quickly and do not bother to solve the intermediate problems very accurately. The amount of work required for this type of method has only been proved to be O(N log N). On the other hand the so-called small update-neighbourhood'' methods, which reduce \mu to zero conservatively and are more careful in their solution of intermediate problems, do NOT perform so well in practice. The amount of work required for this type of method, however, has been proved to be O(N1/2 log N). Hence, on page 16, the authors ask the key question: Can we lower the complexity bounds of the large-update methods by improved analysis without solving any additional systems?'' Their answer is in the affirmative and uses the new idea of self-regular'' functions to control the accuracy of the Newton iterates. A self-regular function is a strictly convex twice continuously differentiable function defined on the positive real line, with global minimum at 1, and also satisfying important growth conditions. Using this idea, the authors prove that the amount of work required for the large update methods is close to O(N1/2 log N).

We have given the impression above that this book is mostly concerned with linear programming problems, but this is definitely not so. The larger part of the book applies self-regular functions to interior-point methods in three other areas of optimisation, namely

1. complementarity problems
2. semi-definite optimization
3. second-order conic optimization.
There are individual chapters which prove complexity results for each of these classes of problem.

In conclusion then, this is a theoretical book, proving important theoretical results for various interior-point algorithms. It is not suitable as an introduction to the area (although Chapter 1 is rather good in this respect) nor for those readers with mainly practical interests. Such people are better directed towards S.J. Wright's 1997 SIAM monograph. In general the book is easy to read, but I have still not recovered from the astonishing claim at the top of page 2.

There is no doubt that the major breakthroughs in the field of mathematical programming are always inaugurated in linear optimisation.''

Also with three authors available to proof-read each other, there are more typos then one would expect: e.g. the indentation in the pseudo-code on page 9 and the proximity function at the top of page 12.

Gerald Moore ( g.moore@imperial.ac.uk)
Department of Mathematics
Imperial College
London.

UK Nonlinear thanks Princeton University Press for providing a review copy of this book.

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