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Almost Orthogonal Arrays: Theory and Search Three Ways

Luis Martínez, María Merino, Juan Manuel Montoya, Josué Tonelli-Cueto

TL;DR

This work tackles the challenge of constructing orthogonal arrays by relaxing their strict requirements to almost-orthogonal arrays (AOAs) defined via Tol$_t(A)$ and $U_{p,t}(A)$. It systematically connects AOAs to various discrepancy notions, surveys relaxations in experimental design and combinatorial settings, and builds AOAs using IP, local search, and algebraic methods. A key contribution is the extensive computational study showing competitive performance across parameter regimes, including optimal or near-optimal AOAs, with a public repository of results. The paper highlights the trade-offs between exact optimization, heuristics, and algebraic constructions, and demonstrates how symmetry can dramatically reduce search complexity while delivering high-quality AOAs with practical impact for experimental design and numerical integration.

Abstract

Orthogonal arrays play a fundamental role in many applications. However, constructing orthogonal arrays with the required parameters for an application usually is extremely difficult and, sometimes, even impossible. Hence there is an increasing need for a relaxation of orthogonal arrays to allow a wider flexibility. The latter has lead to various types of arrays under the name of ``nearly-orthogonal arrays'', and less often ``almost orthogonal arrays''. The aim of this paper is twofold. On the one hand, we review all the existing relaxations, comparing and discussing them in depth. On the other hand, we explore how to find almost orthogonal arrays three ways: using integer programming, local search meta-heuristics and algebraic methods. We compare all our search results with the ones existing in the literature, and we show that they are competitive, improving some of the existing arrays for many non-orthogonality measures. All our found almost orthogonal arrays are available at a public repository.

Almost Orthogonal Arrays: Theory and Search Three Ways

TL;DR

This work tackles the challenge of constructing orthogonal arrays by relaxing their strict requirements to almost-orthogonal arrays (AOAs) defined via Tol and . It systematically connects AOAs to various discrepancy notions, surveys relaxations in experimental design and combinatorial settings, and builds AOAs using IP, local search, and algebraic methods. A key contribution is the extensive computational study showing competitive performance across parameter regimes, including optimal or near-optimal AOAs, with a public repository of results. The paper highlights the trade-offs between exact optimization, heuristics, and algebraic constructions, and demonstrates how symmetry can dramatically reduce search complexity while delivering high-quality AOAs with practical impact for experimental design and numerical integration.

Abstract

Orthogonal arrays play a fundamental role in many applications. However, constructing orthogonal arrays with the required parameters for an application usually is extremely difficult and, sometimes, even impossible. Hence there is an increasing need for a relaxation of orthogonal arrays to allow a wider flexibility. The latter has lead to various types of arrays under the name of ``nearly-orthogonal arrays'', and less often ``almost orthogonal arrays''. The aim of this paper is twofold. On the one hand, we review all the existing relaxations, comparing and discussing them in depth. On the other hand, we explore how to find almost orthogonal arrays three ways: using integer programming, local search meta-heuristics and algebraic methods. We compare all our search results with the ones existing in the literature, and we show that they are competitive, improving some of the existing arrays for many non-orthogonality measures. All our found almost orthogonal arrays are available at a public repository.
Paper Structure (32 sections, 21 theorems, 192 equations, 2 figures, 5 tables, 1 algorithm)

This paper contains 32 sections, 21 theorems, 192 equations, 2 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview of almost-orthogonal arrays
  3. Orthogonal arrays
  4. Almost-orthogonal arrays, tolerance and unbalance
  5. Comparison with other relaxations of OAs
  6. Relaxations in the context of experimental designs
  7. Combinatorial relaxations
  8. Comparison with quadrature discrepancy measures
  9. Searching for AOAs of minimum tolerance and unbalance
  10. Trivial construction of AOAs out of OAs for strength two
  11. Symmetries for searching faster AOAs
  12. Symmetries of an array
  13. Symmetric Minimum Tolerance and Unbalance Problems
  14. Three families of symmetric arrays
  15. Bi-cyclic array
  16. ...and 17 more sections

Key Result

Theorem 2.3

Let $A$ be an $\mathrm{OA}(N,k,s,2)$, then $k\leq \frac{N-1}{s-1}.$

Figures (2)

  • Figure 1: Pareto Front for unbalance $\mathfrak u(25,7,2,1,\epsilon)$
  • Figure 2: Best normalized unbalances $\widehat{\mathop{\mathrm{U}}\nolimits}_{1,2}$ (left) and $\widehat{\mathop{\mathrm{U}}\nolimits}_{2,2}$ (right) by $\lambda=1$ (top) and $\lambda=2$ (bottom)

Theorems & Definitions (92)

  • Definition 2.1
  • Theorem 2.3: Rao's bound for $t=2$
  • Proposition 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Theorem 2.9
  • Remark 2.10
  • Corollary 2.11
  • ...and 82 more