An Integer Programming Algorithm for Constructing Maximin Distance Designs From Good Lattice Point Sets

Q: What is the relationship between columns in LHDs Z b when N is even?

When N is even, Lemma 2 states that the relationship between columns in LHDs Z b is given by z(j)b = (N-1)^N - z(n+1-j)b for b = 0, 1, ..., N/2 - 1. This relationship allows for the evaluation of only the first half of the linear permutations, as the other LHDs have similar distance matrices. This is a significant finding as it reduces the number of LHDs that need to be evaluated, making the process more efficient and cost-effective.

Q: What are the strengths of Wang et al.'s method for N prime numbers?

Wang et al.'s method outperforms Xiao and Xu's (2017) construction for N prime numbers, using arrays of Costas (1984). It also outperforms Ba et al.'s (2015) SA algorithm for n = ph(N) and 7 <= N <= 30, in terms of L1-distance. When N is prime, it provides a formula to obtain the best N-run (N-1)-factor LHD, which is asymptotically optimal in terms of the maximin distance criterion. The average absolute correlation between two columns is smaller than 2/(N-2), making it more efficient for larger run sizes. However, it has limitations when N is not prime and the largest number of factors is less than N-1.

Q: What are the limitations of Van Dam et al.'s approach to constructing LHDs that maximize the L 1 - and L -distances?

The limitations of Van Dam et al.'s approach to constructing LHDs that maximize the L 1 - and L -distances include its computational demand and infeasibility to solve, which prevents it from being practically relevant. For example, to construct LHDs with 12 runs or more, the MILP problem C = [Z 0 , Z 1 , * * * , Z N -1 ] with Z b as in Section 3.2 becomes computationally demanding and often infeasible to solve. Additionally, their approach is limited to LHDs with two factors only, making it less applicable to larger-scale optimization problems.

Q: What is the problem formulation for constructing an N-run k-factor LHD that maximizes the L1-distance?

The problem formulation to construct an N-run k-factor LHD that maximizes the L1-distance involves defining binary variables y_u for each column in the candidate set C, and an integer decision variable t. The objective function is to maximize t, which maximizes the minimum L1-distance between two rows of the LHD. The constraints include ensuring that the final LHD has exactly k columns, and that the L1-distances between any two distinct rows of the LHD are larger than or equal to t. The constraints also ensure that t is a positive integer and the variables y_u are binary. The distance matrix A_1 defines the constraints within (4.5b). When N is even, the number of constraints within (4.5b) is (N/2)^2 + 1. The problem formulation is similar to that of Van Dam et al. (2009) but tailored to an attractive candidate set, with N^(ph(N))/2 binary decision variables. After solving the problem formulation, the output is the vector y, where the nonzero y_u values indicate the columns of C that are in the N-run k-factor LHD that maximizes the L1-distance.

Q: How does the IP algorithm compare to other benchmark algorithms in terms of L 1 -distance for different design problems?

The IP algorithm matches or improves upon the benchmark methods for most design problems, outperforming all benchmark designs in terms of the L 1 -distance for 16, 19, 20, 23, 25, 27, and 28 runs. For the other cases, the LHDs generated by the IP algorithm have the same L 1 -distance as the best benchmark designs, except for 9, 10, 15, 18, 24, and 29 runs. The 10-, 15-, 18-, and 24-run LHDs obtained by the genetic algorithm have a larger L 1 -distance than the IP algorithm's designs. The Gurobi solver certified that all LHDs constructed by the IP algorithm have the best possible L 1 -distance, except for 9, 10, 15, 18, 24, and 29 runs, where the upper bounds on the L 1 -distance are 119, 176, 168, 162, and 280, respectively. This indicates that better LHDs may be obtained by increasing the computing time of the solver. The IP algorithm can construct LHDs with more or fewer factors than ph(N ), making it a versatile tool for designing LHDs with different numbers of factors.

Q: How does the modified IP algorithm handle larger-sized LHDs?

The modified IP algorithm for constructing large designs faces complexity issues due to the cardinality-constrained optimization problem. However, it can still be computationally feasible for small and moderate-sized LHDs. To address larger-sized LHDs, the algorithm reduces the candidate set and introduces an extra step. This modification aims to improve performance and overcome the complexity challenges associated with the IP problem. The Gurobi solver has been used in previous experiments, demonstrating its ability to find good or optimal solutions for design problems with up to 30 runs and 29 factors within five minutes. The modifications made to the IP algorithm enhance its efficiency and effectiveness in constructing larger-sized LHDs.

Q: What is the reduced candidate set D?

The reduced candidate set D is the best N-run (N-1)-factor LHD constructed using the Nx(N-1) GLP set, Williams' transformation, and linear permutation. It results in the best L1-distance and allows for N-run LHDs with k <= N-1 factors, where N is a prime number. This set is used to generate LHDs with fewer than N runs using the leave-one-out method.

Question

1. What is the purpose of computer experiments and how do they relate to computer models?

2. What is the L q-distance in LHDs?

3. What is a good lattice point set?

4. What is the relationship between columns in LHDs Z b when N is even?

Accepted Answer

Computer experiments permit the study of complex systems that are simulated using computer models. A computer model uses algorithms and sets of mathematical equations to provide the best representation possible of the link between the input factors and responses of the system. However, many of these models are computationally expensive since they involve solving complicated partial differential equations numerically. Therefore, one of the main goals of computer experiments is to build an efficient, computationally-cheap surrogate model that approximates the computer model well. To achieve this, they demand cost-effective experimental designs that gather high-quality data from the computer model, using a limited number of runs.

Accepted Answer

The L q-distance between any two distinct rows in an LHD X is the minimum L q-distance between any two distinct rows. It is calculated by taking the element-wise q-th root of the distance matrix A q, where 1 n is an n x 1 vector of ones. The L q-distance of an LHD X, denoted as d q (X), is the minimum L q-distance between any two distinct rows of the design. An LHD that maximizes d q (X) is called a maximin L q-distance LHD. This distance is used to compare and prefer LHDs based on the maximin distance criterion. The larger the L q-distance, the more distinct the rows are in the LHD. The L q-distance can be calculated for different values of q, with q = 1 being the L 1-distance. Fully correlated vectors induce the same absolute element-wise distance vectors, as shown in Lemma 1. This distance matrix and the L q-distance are key components in generating LHDs and comparing their distinctiveness.

Accepted Answer

A good lattice point set (GLP) is an N x n set of elements x i,j = ih j (mod N) for i = 1, . . ., N and j = 1, . . ., n. It consists of positive integers smaller than and coprime to N, arranged in a specific order. Each column of the GLP set is a permutation of the elements in Z N. GLP sets are also known as LHDs (Linearly Decreasing Hash Functions). They can be constructed for any n <= ph(N), where ph(N) is the number of positive integers smaller than and coprime to N. GLP sets have applications in various fields, including cryptography and data structures. Zhou and Xu (2015) demonstrated that linear permutations of the columns of a GLP set can produce better LHDs in terms of the L1-distance. Additionally, the Williams' transformation can further improve the performance of GLP sets.

Accepted Answer

When N is even, Lemma 2 states that the relationship between columns in LHDs Z b is given by z(j)b = (N-1)^N - z(n+1-j)b for b = 0, 1, ..., N/2 - 1. This relationship allows for the evaluation of only the first half of the linear permutations, as the other LHDs have similar distance matrices. This is a significant finding as it reduces the number of LHDs that need to be evaluated, making the process more efficient and cost-effective.

Accepted Answer

Wang et al.'s method outperforms Xiao and Xu's (2017) construction for N prime numbers, using arrays of Costas (1984). It also outperforms Ba et al.'s (2015) SA algorithm for n = ph(N) and 7 <= N <= 30, in terms of L1-distance. When N is prime, it provides a formula to obtain the best N-run (N-1)-factor LHD, which is asymptotically optimal in terms of the maximin distance criterion. The average absolute correlation between two columns is smaller than 2/(N-2), making it more efficient for larger run sizes. However, it has limitations when N is not prime and the largest number of factors is less than N-1.

Accepted Answer

The limitations of Van Dam et al.'s approach to constructing LHDs that maximize the L 1 - and L -distances include its computational demand and infeasibility to solve, which prevents it from being practically relevant. For example, to construct LHDs with 12 runs or more, the MILP problem C = [Z 0 , Z 1 , * * * , Z N -1 ] with Z b as in Section 3.2 becomes computationally demanding and often infeasible to solve. Additionally, their approach is limited to LHDs with two factors only, making it less applicable to larger-scale optimization problems.

Accepted Answer

The problem formulation to construct an N-run k-factor LHD that maximizes the L1-distance involves defining binary variables y_u for each column in the candidate set C, and an integer decision variable t. The objective function is to maximize t, which maximizes the minimum L1-distance between two rows of the LHD. The constraints include ensuring that the final LHD has exactly k columns, and that the L1-distances between any two distinct rows of the LHD are larger than or equal to t. The constraints also ensure that t is a positive integer and the variables y_u are binary. The distance matrix A_1 defines the constraints within (4.5b). When N is even, the number of constraints within (4.5b) is (N/2)^2 + 1. The problem formulation is similar to that of Van Dam et al. (2009) but tailored to an attractive candidate set, with N^(ph(N))/2 binary decision variables. After solving the problem formulation, the output is the vector y, where the nonzero y_u values indicate the columns of C that are in the N-run k-factor LHD that maximizes the L1-distance.

Accepted Answer

The IP algorithm matches or improves upon the benchmark methods for most design problems, outperforming all benchmark designs in terms of the L 1 -distance for 16, 19, 20, 23, 25, 27, and 28 runs. For the other cases, the LHDs generated by the IP algorithm have the same L 1 -distance as the best benchmark designs, except for 9, 10, 15, 18, 24, and 29 runs. The 10-, 15-, 18-, and 24-run LHDs obtained by the genetic algorithm have a larger L 1 -distance than the IP algorithm's designs. The Gurobi solver certified that all LHDs constructed by the IP algorithm have the best possible L 1 -distance, except for 9, 10, 15, 18, 24, and 29 runs, where the upper bounds on the L 1 -distance are 119, 176, 168, 162, and 280, respectively. This indicates that better LHDs may be obtained by increasing the computing time of the solver. The IP algorithm can construct LHDs with more or fewer factors than ph(N ), making it a versatile tool for designing LHDs with different numbers of factors.

Accepted Answer

The modified IP algorithm for constructing large designs faces complexity issues due to the cardinality-constrained optimization problem. However, it can still be computationally feasible for small and moderate-sized LHDs. To address larger-sized LHDs, the algorithm reduces the candidate set and introduces an extra step. This modification aims to improve performance and overcome the complexity challenges associated with the IP problem. The Gurobi solver has been used in previous experiments, demonstrating its ability to find good or optimal solutions for design problems with up to 30 runs and 29 factors within five minutes. The modifications made to the IP algorithm enhance its efficiency and effectiveness in constructing larger-sized LHDs.

Accepted Answer

The reduced candidate set D is the best N-run (N-1)-factor LHD constructed using the Nx(N-1) GLP set, Williams' transformation, and linear permutation. It results in the best L1-distance and allows for N-run LHDs with k <= N-1 factors, where N is a prime number. This set is used to generate LHDs with fewer than N runs using the leave-one-out method.

Accepted Answer

The modified IP algorithm is compared to the SA algorithm of Ba et al. (2015) and the genetic algorithm of Liefvendahl and Stocki (2006) for constructing large LHDs. The computational setup of the algorithms is the same as before, but preliminary experiments revealed that the benchmark algorithms are computationally demanding for large numbers of runs or factors. To address this, an additional stopping rule was imposed, setting the maximum computing time to that of the Gurobi solver. The SA algorithm was executed 100 times, and the best LHD obtained within 300 seconds was reported. Similarly, 500 generations of the genetic algorithm were used, and the best LHD obtained within this time frame was recorded.

Accepted Answer

Prime number run sizes, such as 31, 47, 71, and 97, are used in design problems to assess the quality of LHDs obtained from subsets of columns of the N-run (N-1)-factor LHDs. These design problems allow for a comparison of LHDs obtained from different algorithms, such as modified IP (M-IP), SA, and genetic algorithms. The M-IP algorithm outperforms the others in 13 out of 20 design problems for large run sizes. For 31-run LHDs, all are optimal among LHDs obtained from subsets of columns of the initial 31-run 30-factor LHD. For 47, 71, and 97 runs, the relative gaps given by the solver ranged from 1.0% to 20.7%, indicating the effectiveness of the algorithms in optimizing LHDs with prime number run sizes.

Accepted Answer

The M-IP algorithm shows potential in generating attractive LHDs for large LHDs with general run sizes. It performs better than benchmark algorithms in terms of L1-distance for design problems with 8, 10, and 11 factors. The algorithm uses the leave-one-out method when the number of runs is not a prime. It starts with LHDs from Table 3 and produces better designs for 44 to 96 runs and 23 to 72 factors. The M-IP algorithm is particularly effective when the number of factors is at least eight, making it suitable for large LHDs with a significant number of runs relative to the number of factors.

Accepted Answer

The IP algorithm is computationally effective for small and moderate design problems. Numerical experiments showed that it outperforms benchmark algorithms in 47 out of 58 design problems. It is particularly effective for constructing LHDs with around 100 runs, supporting the idea that it will outperform benchmark algorithms for larger run sizes. However, for 29 runs and 28 factors, the IP algorithm did not generate a better LHD than Wang et al. (2018). This may be due to insufficient optimization time by the Gurobi solver. Additional computations revealed that, after four hours, the solver found an L 1 -distance of 275, which is larger than all benchmark designs in Table 2. Therefore, it is recommended to complete the optimization of the IP algorithm for better results.

An Integer Programming Algorithm for Constructing Maximin Distance Designs From Good Lattice Point Sets

Chat with Paper

AI Agents for this Paper

Most frequently asked questions

1. What is the purpose of computer experiments and how do they relate to computer models?

2. What is the L q-distance in LHDs?

3. What is a good lattice point set?

4. What is the relationship between columns in LHDs Z b when N is even?

5. What are the strengths of Wang et al.'s method for N prime numbers?

6. What are the limitations of Van Dam et al.'s approach to constructing LHDs that maximize the L 1 - and L -distances?

7. What is the problem formulation for constructing an N-run k-factor LHD that maximizes the L1-distance?

8. How does the IP algorithm compare to other benchmark algorithms in terms of L 1 -distance for different design problems?

9. How does the modified IP algorithm handle larger-sized LHDs?

10. What is the reduced candidate set D?

11. How does the modified IP algorithm compare to other algorithms?

12. How do prime number run sizes affect LHD quality?

13. How does the M-IP algorithm perform in generating LHDs for large LHDs with general run sizes?

14. What is the IP algorithm's effectiveness for small and moderate design problems?

Citations

Construction of orthogonal maximin distance designs

Tuning differential evolution algorithm for constructing uniform projection designs

References

Minimax and maximin distance designs

Exploratory designs for computational experiments

Experimental Designs Balanced for the Estimation of Residual Effects of Treatments

Choosing the Sample Size of a Computer Experiment: A Practical Guide

50 years of integer programming 1958-2008 : from the early years to the state-of-the-art

Related Papers (5)

Implicit Enumeration for the Pure Integer 0/1 Minimax Programming Problem

A Solution Approach for Two-Stage Stochastic Nonlinear Mixed Integer Programs

A Construction-Based Approach to Process Plant Layout Using Mixed-Integer Optimization

A Biobjective Perspective for Mixed-Integer Programming

On the relaxation of multi-level dynamic lot-sizing models