Computing Experiment-Constrained D-Optimal Designs

TL;DR

This work addresses the generalized D-optimal design problem, allowing for nonlinear relationships in factor levels, and develops scalable algorithms suitable for cases where the number of candidate experiments grows exponentially with the factor dimension.

Abstract

In optimal experimental design, the objective is to select a limited set of experiments that maximizes information about unknown model parameters based on factor levels. This work addresses the generalized D-optimal design problem, allowing for nonlinear relationships in factor levels. We develop scalable algorithms suitable for cases where the number of candidate experiments grows exponentially with the factor dimension, focusing on both first- and second-order models under design constraints. Particularly, our approach integrates convex relaxation with pricing-based local search techniques, which can provide upper bounds and performance guarantees. Unlike traditional local search methods, such as the ``Fedorov exchange" and its variants, our method effectively accommodates arbitrary side constraints in the design space. Furthermore, it yields both a feasible solution and an upper bound on the optimal value derived from the convex relaxation. Numerical results highlight the efficiency and scalability of our algorithms, demonstrating superior performance compared to the state-of-the-art commercial software, JMP

Figures (5)

• Figure 1: A single run of the local search algorithm for solving the first-order model with the cardinality constraints for $d=18$. Heuristic means the local search found a locally improving move using the bit flip/swap heuristic, and IP means the algorithm failed to find an improving move with the heuristic and had to run Gurobi. In this example, the algorithm used the heuristic to find an improving move in every iteration, and the IP is only used in the last iteration to certify that the solution is a local optima.
• Figure 2: A single run of the local search algorithm for solving first-order model with knapsack constraints for $d=13$. Heuristic means the local search found a locally improving move using the bit flip/swap heuristic, and IP means the algorithm failed to find an improving move with the heuristic and had to run Gurobi to solve the IP.
• Figure 3: A single run of the local search algorithm for solving first-order model with knapsack constraints for $d=16$. The local search is initialized with a solution given by JMP, and the IP is needed in the first iteration to find an improving move. A single run of the local search algorithm. The Heuristic means the local search found a locally improving move using the bit flip/swap heuristic, and IP means the algorithm failed to find an improving move with the heuristic and had to run Gurobi to solve IP. The Gap is the difference between the value of the convex relaxation and the value of the solution.
• Figure 4: A single run of the column generation algorithm for the first-order model with knapsack constraints for $d=15$. The IP upper bound is the value of a dual feasible solution, and the Column Generation Value is the value of the column generation algorithm.
• Figure 5: A single run of the local search algorithm for solving second-order model with knapsack constraints for $d=16$. Heuristic means the local search found a locally improving move using the bit flip/swap heuristic and IP means the algorithm failed to find an improving move with the heuristic and had to run Gurobi. Using only the heuristic, the local search would have gotten stuck at iteration $60$ with a gap of $1.562$. Using the IP allows the algorithm to continue by improving the gap to $1.535$ at iteration $61$ and finally terminate at iteration $88$ with a gap of $1.406$.

Theorems & Definitions (22)

• Theorem 1
• Corollary 1
• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof