Crash start of interior point methods

Abstract

The starting point used by an interior point algorithm for linear and convex quadratic programming may significantly influence the behaviour of the method. A widely used heuristic to construct such a point consists of dropping variable nonnegativity constraints and computing a solution which minimizes the Euclidean norm of the primal (or dual) point while satisfying the appropriate primal (or dual) equality constraints, followed by shifting the variables so that all their components are positive and bounded away from zero. In this Short Communication a new approach for finding a starting point is proposed. It relies on a few inexact Newton steps performed at the start of the solution process. A formal justification of the new heuristic is given and computational results are presented to demonstrate its advantages in practice. Computational experience with a large collection of small-and medium-size test problems reveals that the new starting point is superior to the old one and saves 20-40% of iterations needed by the primal-dual method. For larger and more difficult problems this translates into remarkable savings in the solution time.

External Constraint Handling for Solving Optimal Control Problems With Simultaneous Approaches and Interior Point Methods

“…To ensure reliable and efficient computation of the initialization algorithm, as well as the subsequent NLP iterations, several criteria [3] can be formulated regarding ideal initial points for IPMs. The ideal initial point should:…”

“…These constraints are then added in the OCP and the problem is repetitively solved until all original constraints are satisfied. However, a fundamental problem arises when implementing the same idea on interior point method (IPM) based solvers, since good performance hinges on the initial point to be feasible, or at least close to feasible [3].…”

Determining the best optimization method for large scale probabilistic supplier selection problem integrated with inventory management

“…There are many algorithm alternatives that we can use, some of them are active-set, interior point, SQP (sequential-quadratic-programming) and SQP-legacy [12]. For interior point method, it is an old one but several researchers were developed it to improve the performance, for example, by preconditioning the problem [13], improving on the starting point [14], and modified interior-point called infeasible interior-point algorithm to solve stochastic complementarity problems [15], besides some application researches were conducted to show the superiority of this method e.g. for atomic norm soft thresholding [16].…”

A New Class of Scaling Matrices for Scaled Trust Region Algorithms

“…Although specialized algorithms have been developed for some of these problems, the interior-point methods are still the main tool to tackle them. These methods require a feasible initial point [19]. It should be paid attention that proper data collection plays an important role in assessing the results obtained [39].…”