A semidefinite programming approach to maximum von Neumann entropy experimental design for multiresponse models

Semidefinite programming (SDP) offers a powerful computational framework for approximate optimal experimental design by leveraging the geometry of the positive semidefinite cone. While SDP formulations are well established for classical criteria such as D–, A–, and E–optimality, their extension to the von Neumann (vN) entropy criterion remains largely unexplored. The vN-optimal design criterion is particularly suited for multiresponse models with correlated outputs, promoting balanced information allocation through entropy maximization of the Fisher Information Matrix (FIM). Drawing on the analogy between quantum information theory—where the vN entropy quantifies uncertainty of density operators—and statistical inference—where the FIM captures parameter uncertainty—we develop a tractable convex formulation for vN–optimal design. Leveraging recent advances that unify semidefinite and exponential cone programming, our approach enables efficient computation via interior-point methods. The proposed framework is illustrated through three applications: (i) dose–response design for joint efficacy–toxicity modeling, (ii) optimal sensor placement in multimodal systems, and (iii) parameter estimation in nonlinear kinetic models. An accompanying equivalence theorem is established to assess vN–optimality of the resulting designs.