Interior point methods 25 years later

Abstract

Interior point methods for optimization have been around for more than 25 years now. Their presence has shaken up the field of optimization. Interior point methods for linear and (convex) quadratic programming display several features which make them particularly attractive for very large scale optimization. Among the most impressive of them are their lowdegree polynomial worst-case complexity and an unrivalled ability to deliver optimal solutions in an almost constant number of iterations which depends very little, if at all, on the problem dimension. Interior point methods are competitive when dealing with small problems of dimensions below one million constraints and variables and are beyond competition when applied to large problems of dimensions going into millions of constraints and variables.In this survey we will discuss several issues related to interior point methods including the proof of the worst-case complexity result, the reasons for their amazingly fast practical convergence and the features responsible for their ability to solve very large problems. The ever-growing sizes of optimization problems impose new requirements on optimization methods and software. In the final part of this paper we will therefore address a redesign of interior point methods to allow them to work in a matrix-free regime and to make them well-suited to solving even larger problems.

Solving large-scale optimization problems related to Bell’s Theorem

“…Indeed, an LP with n variables is solved to ε-optimality in O( √ n log( 1 ε )) iterations [19]. These methods are generally believed to be well-suited to the solution of very large scale optimization problems [20]. IPMs converge to the optimal solution in very few iterations: they usually need only O(log n) iterations to reach a solution of the problem with n variables.…”

“…The reader familiar with IPMs may skip this part. The reader interested in more detail on the theory and implementation of IPMs should consult the excellent book of Wright [19] and the recent survey by Gondzio [20], respectively. The matrix-free IPM itself is introduced by Gondzio [5].…”

A specialized primal-dual interior point method for the plastic truss layout optimization

“…At every member adding iteration described in Sect. 3.1, we generate the set K in (12) to identify and add the new members. In that case, the size of the problem grows and the new variables are appended to the problem…”

“…For more details and a survey of recent developments in the primal-dual interior point method, we refer the reader to [12] and the references therein.…”

“…However, a single iteration in these methods might be expensive when very largescale problems are solved. In such cases, an improvement in the efficiency might sometimes be delivered by replacing direct linear algebra techniques with well-suited iterative methods equipped with efficient preconditioners, see [6,7,12] and the references therein.…”