A Polynomial Arc-Search Interior-Point Algorithm for Linear Programming

Abstract

For interior-point algorithms in linear programming, it is well-known that the selection of the centering parameter is crucial for proving polynomility in theory and for efficiency in practice. However, the selection of the centering parameter is usually by heuristics and separate from the selection of the linesearch step size. The heuristics are quite different while developing practically efficient algorithms, such as MPC, and theoretically efficient algorithms, such as short-step path-following algorithm. This introduces a dilemma that some algorithms with the best-known polynomial bound are least efficient in practice, and some most efficient algorithms may not be polynomial. In this paper, we propose a systematic way to optimally select the centering parameter and linesearch step size at the same time, and we show that the algorithm based on this strategy has the best-known polynomial bound and may be very efficient in computation for real problems.

CurveLP-A MATLAB implementation of an infeasible interior-point algorithm for linear programming

“…Higher-order algorithms, however, had either a poorer polynomial bound than first-order algorithms [19] or did not even have a polynomial bound [5,20]. This dilemma was partially solved in [21] which proved that higher-order algorithms can achieve the best polynomial bound. An arc-search interior-point algorithm for linear programming was devised in [21].…”

“…This dilemma was partially solved in [21] which proved that higher-order algorithms can achieve the best polynomial bound. An arc-search interior-point algorithm for linear programming was devised in [21]. The algorithm utilized the first and second-order derivatives to construct an ellipse to approximate the central path.…”

“…Intuitively, searching along this ellipse should generate a larger step size than searching along any straight line. Indeed, it was shown in [21] that the arcsearch algorithm has the best polynomial bound and it may be very efficient in practical computation. This result was extended to prove a similar result for convex quadratic programming and the numerical test result was very promising [22].…”

“…The algorithms proposed in [21,22] assume that the starting point is feasible and the central path does exist. Available Netlib test problems are limited because most Netlib problems may not even have an interior-point as noted in [23].…”

“…To better demonstrate the claims in the previous papers, we propose an infeasible arc-search interior-point algorithm in this paper, which allows us to test a lot more Netlib problems. The proposed algorithm keeps a nice feature developed in [21,22], i.e., it searches optimizer along an arc (part of an ellipse). It also adopts some strategies used in MPC, such as using different step sizes for the vector of primal variables and the vector of slack variables.…”