NV-CuPc Interaction: Optimal Experimental Design

• NV-CuPc Interaction is a framework for designing experiments that precisely estimate parameters in diamond NV centers interacting with CuPc molecules.
• The methodology leverages D-optimality, convex relaxations, and second-order cone programming to efficiently construct optimal experimental designs under strict physical and statistical constraints.
• Extensions using mixed-integer SOCP, robust and Bayesian designs, and approximation algorithms enable practical adaptation to experimental uncertainty and complex design spaces.

The NV-CuPc interaction refers to the optimal experimental design problem where the aim is precise estimation of model parameters associated with nitrogen-vacancy (NV) centers in diamond that interact with copper phthalocyanine (CuPc) molecules. This concept is intrinsically linked to the statistical theory of $D$-optimal designs, convex relaxations, mixed-integer second-order cone programming (MISOCP), and robust optimization in experimental design. Modern approaches enable provable, computationally efficient construction of experimental designs that maximize parameter identifiability under physical and statistical constraints, even in the presence of complex interactions or uncertainty in system parameters.

1. $D$-Optimality Criterion and Information Matrices

The foundation of design for experiments on NV-CuPc interactions is the $D$-optimal criterion. The experiment is abstracted by $s$ candidate trials, each associated with an observation matrix $A_i \in \mathbb{R}^{m \times \ell_i}$. For a weight vector $w = (w_1, \dots, w_s)\geq0$ representing the proportion of effort allocated to each experimental run, the Fisher information matrix is

$$M(w) = \sum_{i=1}^s w_i A_i A_i^T \in \mathbb{S}_m^+.$$

The $D$-optimality criterion is then

$\Phi_D(w) = (\det M(w))^{1/m},$

which is strictly positive when $M(w)$ is nonsingular. Maximizing $\log\det M(w)$ is equivalent, due to the monotonicity of the logarithm and the homogeneous scaling of determinants, and this convex objective underpins most design procedures.

2. Second-Order Cone Programming Reformulation

It has been observed that all classical optimal design criteria based on determinants or traces of $M(w)$ (including $D_K, A_K, G, I$) are second-order cone (SOC) representable (Sagnol et al., 2013). For $D$-optimality, the key reformulation is the constraint: $$t \leq (\det M(w))^{1/m} \Longleftrightarrow t^m \leq \det(M(w)),$$ which can be encoded via SOCs using:

• The geometric mean representation over rotated cones,
• Cholesky-type reparameterizations to encode determinant maximization as geometric means of lower-triangular factors' diagonals.

Specifically, for a full-rank $K \in \mathbb{R}^{m\times k}$, one uses block Cholesky decompositions to represent $D_K$-optimality: $$\Phi_{D|K}(M(w)) = \max_{Q,L\succeq 0} (\det L)^{2/k}\quad\text{subject to }\sum_i A_i Q_i = K L,\,\|Q_{i,\bullet}^{(j)}\|\le 1.$$ This enables the full optimal design problem (with arbitrary linear constraints) to be cast as an SOCP.

3. Mixed-Integer SOCP for Exact Designs

Physical constraints in NV-CuPc experiments often require exact rather than fractional allocations: $n_i \in \mathbb{N}_0$, $\sum_i n_i = N$ runs. The exact $D$-optimal design problem is thereby formulated as a MISOCP: $$\begin{align*} &\max_{n_i \in \mathbb{Z}_+, Q_i, t_{ij}, J} \prod_{j=1}^m J_{jj}^{1/m} \ \text{subject to:} \qquad &\sum_{i=1}^s A_i Q_i = J \text{ (lower-triangular)}, \ &\|Q_{i,\bullet}^{(j)}\|^2 \leq t_{ij} n_i/N,\quad \sum_{i=1}^s t_{ij} \leq J_{jj},\; t_{ij} \geq 0, \ &\sum_{i=1}^s n_i = N,\quad n_i \in \mathbb{Z}_+. \end{align*}$$ Off-the-shelf MISOCP solvers (e.g., CPLEX, MOSEK) can process these formulations directly, returning provably optimal integer-valued designs under arbitrary linear constraints.

4. Branch-and-Cut Solution Strategy

The MISOCP is solved with branch-and-cut. Each node in the branch-and-bound tree corresponds to partial assignments of integer allocations $n_i$; at each node, a continuous SOCP relaxation is solved to optimality using interior-point methods, providing upper bounds. Branching is performed on fractional $n_i$; advanced solvers generate additional SOC or linear cutting planes to tighten relaxations. Termination occurs when the best integer feasible design's objective equals the most recent bound within numerical tolerances.

This approach provides a globally optimal design and a certificate of optimality, in contrast to standard heuristics (such as vertex-exchange or greedy addition), which systematically fail to attain the optimum in many scenarios.

5. Extensions to Robust, Bayesian, and Mixed-Factor Designs

Robustness to parameter uncertainty and mixed response models, relevant to NV-CuPc studies with uncertain system Hamiltonian or mixed outcome types, motivates further generalizations:

• EW $D$-optimality maximizes the determinant of the average Fisher information under a prior distribution $Q(d\theta)$ or bootstrap sample from pilot studies (Lin et al., 1 May 2025):

$$\xi^* = \arg\max_\xi |\bar M(\xi)|, \quad \bar M(\xi) = \sum_{i=1}^m w_i E[F(x_i, \Theta)].$$

Existence, support-size, and verification via the General Equivalence Theorem are ensured under regularity conditions.

• Bayesian $D$-optimality averages the log-determinant criterion over parameter priors, with local and global designs constructed via point-exchange algorithms and empirical averaging (Kang et al., 2023).
• These methods extend naturally to models with mixed continuous/discrete factors or qualitative/quantitative responses, with dedicated aggregation and rounding schemes.

6. Approximation Algorithms and Practical Regimes

Where exact optimization is computationally prohibitive (large $n$, moderate $k$, high $m$), randomized approximation algorithms achieve near-optimality in expectation (Singh et al., 2018):

• The “randomized $1/e$-approximation” samples $k$-subsets $S$ with probability $\propto \prod_{i\in S} x_i^*$ (where $x^*$ solves the relaxed convex program), guaranteeing

$$\mathbb{E}[\det(\sum_{i\in S} a_ia_i^T)] \geq (1/e)^m \det(\sum x_i^* a_i a_i^T).$$

• For $k \gg m$, the “asymptotic $(1-\epsilon)$-approximation” gives arbitrarily high expected efficiency, provided $k \geq 4m/\epsilon + 12\log(1/\epsilon)/\epsilon^2$.

In practice, these methods deliver 70–90% efficiency even in moderately overdetermined settings, and the Poisson rounding scheme further simplifies design with repetitions.

7. Design Spaces, Equilibrium Measures, and Cubature Connections

For experiments where the design space $S$ is a classical compact set (ball, box, simplex), results from pluripotential theory indicate that the equilibrium measure $\mu_S$ yields the exact solution (in terms of moments) to certain convex relaxations of the D-optimal design problem (Henrion et al., 6 Sep 2024):

• The unique optimal moments up to degree $2n$ coincide with those of $\mu_S$.
• Any atomic cubature reproducing these moments gives an approximate D-optimal design.
• In these domains, Chebyshev or Dirichlet cubature rules with positive weights recover the maximizing design, and sequences of such atomic designs converge (in the weak-star topology) to $\mu_S$ as $n\to\infty$.

This connection serves as both a theoretical benchmark and a practical guide for experiments such as NV-CuPc measurements constrained to such domains.

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In summary, the theory and algorithms for $D$-optimal experimental design provide a rigorous and computationally tractable pathway for optimizing experiments in NV-CuPc systems. Second-order cone programming and its mixed-integer extensions enable exact design with provable guarantees, while robust, Bayesian, and approximation methods ensure tractable solutions under model uncertainty or computational constraints. The link to equilibrium measures and cubature rules reveals deep geometric and analytic structure in the selection of optimal experiments on classical domains. For practical applications, standard solvers implementing these principles can decisively improve efficiency in physical and chemical parameter estimation compared to traditional heuristic approaches.