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

Lorenzo Fiaschi, Marco Cococcioni

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.

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

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.

Paper Structure

This paper contains 19 sections, 13 theorems, 48 equations, 8 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Reviewing the standard Interior Point Method for Quadratic Programming
  3. Description of the standard IPM algorithm
  4. Alpha Theory
  5. Background in brief
  6. How to handle lexicographic optimization
  7. Non-Archimedean Interior Point Method
  8. Infeasibility and Unboundedness
  9. Convergence and complexity
  10. Numerical considerations and implementation issues
  11. Numerical Experiments
  12. Experiment 1: two-objective LP
  13. Experiment 2: unbounded problem
  14. Experiment 3: two-objective QP
  15. Experiment 4: three-objective QP
  16. ...and 4 more sections

Key Result

Proposition 1

Consider a lexicographic optimization problem whose objective functions $f_1,\,\ldots,\,f_n$ are real functions and the priority is induced by the natural order. Then, there exists an equivalent scalar program over the same domain, whose objective function is non-standard and has the following form: where $\frac{\beta_{i+1}}{\beta_i}\approx 0$, $i=1,\,\ldots,\,n$-1.

Figures (8)

  • Figure 1: The optimal segment for the primary objective is in red; $\xi_1$ is the optimal point a standard IPM would approach.
  • Figure 2: $\xi_2$ is the only optimal solution when both the objectives are considered.
  • Figure 3: Example of unbounded primal polyhedron.
  • Figure 4: Example of empty primal polyhedron.
  • Figure 5: The segment in green is the optimal region for the first objective and $\xi_1$ is its middle point. The starred one, $\xi_2$, is the global optimum instead.
  • ...and 3 more figures

Theorems & Definitions (35)

  • Definition 1
  • Definition 2: Monosemium
  • Definition 3: Leading monosemium
  • Definition 4: Leading monosemium function
  • Definition 5: Order of magnitude
  • Definition 6: Smallest order of magnitude
  • Definition 7: Infinitesimal number
  • Definition 8: Standard optimization problem
  • Definition 9: Non-Archimedean optimization problem
  • Proposition 1
  • ...and 25 more