A Non-Archimedean Interior Point Method for Solving Lexicographic Multi-Objective Quadratic Programming Problems

TL;DR

This paper introduces a non-Archimedean interior point method that efficiently solves complex multi-objective quadratic programming problems, including infeasible and unbounded cases, with proven polynomial convergence.

Contribution

It presents the non-Archimedean IPM (NA-IPM), a novel algorithm that extends traditional methods to handle infeasibility, unboundedness, and lexicographic multi-objective problems more effectively.

Findings

Proves polynomial convergence of NA-IPM to a global optimum.

Demonstrates NA-IPM's ability to manage infeasibility and unboundedness transparently.

Validates effectiveness through four non-Archimedean programming test cases.

Abstract

This work presents a generalized implementation of the infeasible primal-dual Interior Point Method (IPM) achieved by the use of non-Archimedean values, i.e., infinite and infinitesimal numbers. The extended version, called here non-Archimedean IPM (NA-IPM), is proved to converge in polynomial time to a global optimum and to be able to manage infeasibility and unboundedness transparently, i.e., without considering them as corner cases: by means of a mild embedding (addition of two variables and one constraint) NA-IPM implicitly and transparently manages their possible presence. Moreover, the new algorithm is able to solve a wider variety of linear and quadratic optimization problems than its standard counterpart. Among them, the lexicographic multi-objective one deserves particular attention, since NA-IPM overcomes the issues that standard techniques (such as scalarization or preemptive approach) have. To support the theoretical properties of NA-IPM, the manuscript also shows four linear and quadratic non-Archimedean programming test cases where the effectiveness of the algorithm is verified. This also stresses that NA-IPM is not just a mere symbolic or theoretical algorithm but actually a concrete numerical tool, paving the way for its use in real-world problems in the near future.