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ADMM for 0/1 D-Opt and MESP relaxations

Gabriel Ponte, Marcia Fampa, Jon Lee, Luze Xu

TL;DR

This work introduces ADMM-based algorithms to efficiently solve convex relaxations of two NP-hard experimental-design problems: the $0/1$ D-optimality problem and the Maximum-Entropy Sampling problem. For D-Opt, it presents a natural bound solver with BVLS-style x-updates and closed-form Z-updates driven by eigen-decompositions, enabling fast, warm-startable upper bounds. For MESP, it develops ADMM solvers for the linx, factorization, and BQP bounds, deriving BVLS-like updates and closed-form spectral updates across the three relaxations; numerical results show strong performance for the factorization and BQP bounds, with linx being less competitive. Across extensive experiments, the ADMM relaxations often outperform general-purpose solvers in time and scale to larger instances, supporting their use as fast subproblem solvers inside branch-and-bound for experimental-design applications. The study highlights practical convergence behavior, the value of warm-starting, and avenues for adaptive parameter tuning and future NLP-bound challenges that could broaden applicability and efficiency in real-world B&B pipelines.

Abstract

The 0/1 D-optimality problem and the Maximum-Entropy Sampling problem are two well-known NP-hard discrete maximization problems in experimental design. Algorithms for exact optimization (of moderate-sized instances) are based on branch-and-bound. The best upper-bounding methods are based on convex relaxation. We present ADMM (Alternating Direction Method of Multipliers) algorithms for solving these relaxations and experimentally demonstrate their practical value.

ADMM for 0/1 D-Opt and MESP relaxations

TL;DR

This work introduces ADMM-based algorithms to efficiently solve convex relaxations of two NP-hard experimental-design problems: the D-optimality problem and the Maximum-Entropy Sampling problem. For D-Opt, it presents a natural bound solver with BVLS-style x-updates and closed-form Z-updates driven by eigen-decompositions, enabling fast, warm-startable upper bounds. For MESP, it develops ADMM solvers for the linx, factorization, and BQP bounds, deriving BVLS-like updates and closed-form spectral updates across the three relaxations; numerical results show strong performance for the factorization and BQP bounds, with linx being less competitive. Across extensive experiments, the ADMM relaxations often outperform general-purpose solvers in time and scale to larger instances, supporting their use as fast subproblem solvers inside branch-and-bound for experimental-design applications. The study highlights practical convergence behavior, the value of warm-starting, and avenues for adaptive parameter tuning and future NLP-bound challenges that could broaden applicability and efficiency in real-world B&B pipelines.

Abstract

The 0/1 D-optimality problem and the Maximum-Entropy Sampling problem are two well-known NP-hard discrete maximization problems in experimental design. Algorithms for exact optimization (of moderate-sized instances) are based on branch-and-bound. The best upper-bounding methods are based on convex relaxation. We present ADMM (Alternating Direction Method of Multipliers) algorithms for solving these relaxations and experimentally demonstrate their practical value.

Paper Structure

This paper contains 23 sections, 12 theorems, 77 equations, 9 figures, 9 tables.

Table of Contents

  1. Introduction
  2. 0/1 D-Optimality
  3. Update x
  4. Update Z
  5. MESP
  6. linx
  7. Update x.
  8. Update Z.
  9. DDFact
  10. Update x.
  11. Update Z.
  12. BQP
  13. Update W.
  14. Update E.
  15. Update Z.
  16. ...and 8 more sections

Key Result

Proposition 1

Given $Y^{t+1} \in \mathbb{S}^{m}$ and a positive scalar $\rho$. Let $\rho Y^{t+1} =: Q \Theta Q^ { \raisebox{\depth}{$\m@th\mathsf{T}$}}$ be the eigendecomposition, where $\Theta:= \mathop{\mathrm{{Diag}}}\nolimits(\theta_1,\dots,\theta_m)$ and $Q^ { \raisebox{\depth}{$\m@th\mathsf{T}$}} Q = Q

Figures (9)

  • Figure 1: Natural bound \ref{['prob']} for \ref{['dopt']}
  • Figure 2: Natural bound \ref{['prob']} for \ref{['dopt']}
  • Figure 3: \ref{['prob_ddfact']} bound for \ref{['MESP']}, varying $r:=\mathop{\mathrm{rank}}\nolimits(C)$ ($n=2000$, $s=140$)
  • Figure 4: \ref{['prob_ddfact']} bound for \ref{['MESP']}, varying $s$ ($n=2000$, $\mathop{\mathrm{rank}}\nolimits(C)=150$)
  • Figure 5: Behavior of the \ref{['bqp_original']} bound for \ref{['MESP']}, varying $s$ ($n=63$)
  • ...and 4 more figures

Theorems & Definitions (14)

  • Proposition 1
  • Corollary 1
  • Lemma 1: *[Lem. 13]Nikolov
  • Proposition 2: *[Prop. 2]Weijun
  • Lemma 2
  • Remark 1
  • Remark 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 4 more