Optimal Exact Designs of Multiresponse Experiments under Linear and Sparsity Constraints

We propose a computational approach to constructing exact designs on finite design spaces that are optimal for multiresponse regression experiments under a combination of the standard linear and specific ‘sparsity’ constraints. The linear constraints address, for example, limits on multiple resource consumption and the problem of optimal design augmentation, while the sparsity constraints control the set of distinct trial conditions utilized by the design. The key idea is to construct an artificial optimal design problem that can be solved using any existing mathematical programming technique for univariate-response optimal designs under pure linear constraints. The solution to this artificial problem can then be directly converted into an optimal design for the primary multivariate-response setting with combined linear and sparsity constraints. We demonstrate the utility and flexibility of the approach through dose-response experiments with constraints on safety, efficacy, and cost, where cost also depends on the number of distinct doses used.

💡 Research Summary

The paper addresses the problem of constructing exact (integer‑valued) optimal designs for multi‑response regression experiments when the design must satisfy both linear resource constraints and sparsity constraints that limit the number of distinct experimental conditions (support points). Traditional optimal design theory largely focuses on approximate (continuous) designs and on linear constraints only; however, many practical studies—especially dose‑response clinical trials—require additional constraints such as a maximum number of dose levels, fixed overhead costs per dose, or safety/efficacy bounds.
The authors introduce a unified framework called Linear and Sparsity (LAS) constraints. Linear constraints are expressed in the familiar form ∑ a(x,k) w(x) ≤ b(k) for k = 1,…,K, covering resource limits, inclusion/exclusion requirements, and balance constraints. Sparsity constraints are modeled by introducing binary indicator variables s_w(x) = 1 if w(x) > 0 and 0 otherwise, and then adding terms ∑ c(x,k) s_w(x) to the linear inequalities. This yields a mixed integer linear formulation:
∑ a(x,k) w(x) + ∑ c(x,k) s_w(x) ≤ b(k), ∑ w(x) = N,
where N is the total number of trials. The formulation can capture a wide variety of practical restrictions, including (i) a bound on the total number of support points, (ii) support‑dependent cost structures, (iii) separation constraints that force support points to lie in distinct subsets, and (iv) lower/upper replication limits that apply only when a point is used.
The central technical contribution is a transformation that converts the original multi‑response optimal design problem into an auxiliary problem that fits the standard single‑response, linear‑constraint setting for which many mathematical programming (MP) solvers already exist. The key steps are:

1. Replace each multi‑response information matrix H(x) (generally full‑rank) by a rank‑one surrogate f(x)f(x)ᵀ, where f(x) can be obtained via a suitable factorization or by stacking the response vectors. This yields a surrogate information matrix M̃(w) = ∑ w(x) f(x)f(x)ᵀ that is compatible with single‑response optimality criteria.
2. Keep the original LAS constraints unchanged, but introduce the binary support indicators s_w(x) as additional decision variables.
3. Formulate the resulting problem as a mixed‑integer linear program (MILP), mixed‑integer second‑order cone program (MISOCP), or mixed‑integer semidefinite program (MISDP) depending on the chosen optimality criterion (e.g., D‑optimality leads to a determinant‑maximization problem that can be handled via SOC or SDP relaxations).
   Because the auxiliary problem’s size grows only linearly with the number of candidate points (each point contributes one integer replication variable and one binary support variable), existing solvers such as CPLEX, Gurobi, or MOSEK can solve it efficiently for moderate‑size design spaces (hundreds to a few thousand points). Once an optimal solution (ŵ, ŝ) is obtained, the binary indicators are simply discarded, and the integer replication vector ŵ directly provides an optimal exact design for the original multi‑response problem.
   The authors demonstrate the methodology on two realistic case studies. The first is a dose‑response clinical trial with 75 patients, constraints on the maximum number of toxic responses, a minimum efficacy requirement, and a cost structure where each distinct dose incurs a fixed overhead. The LAS formulation forces the design to use exactly five dose levels, each allocated between 10 and 20 patients. The resulting design attains a D‑optimality value equal to or better than that of the best approximate design while fully respecting all constraints. The second example illustrates a block design for accelerated lifetime testing, where sparsity constraints enforce that each stress level appears at most once, and linear constraints model total test time and budget. Again, the proposed approach yields feasible designs with superior optimality scores compared with heuristic alternatives.
   Beyond these examples, the paper emphasizes several broader implications: (i) the approach is compatible with a wide range of optimality criteria (D, A, c, I, G, MV, and their convex combinations); (ii) it leverages existing MP infrastructure, avoiding the need to develop specialized algorithms for each new constraint type; (iii) it provides exact integer solutions, eliminating the post‑processing rounding step required by approximate designs. Consequently, the proposed LAS‑to‑single‑response transformation offers a practical, theoretically sound, and computationally tractable pathway for researchers and practitioners who must design multi‑response experiments under complex, real‑world constraints.