Iterative Power Algorithm for Global Optimization with Quantics Tensor Trains

Abstract

Optimization algorithms play a central role in chemistry since optimization is the computational keystone of most molecular and electronic structure calculations. Herein, we introduce the iterative power algorithm (IPA) for global optimization and a formal proof of convergence for both discrete and continuous global search problems, which is essential for applications in chemistry such as molecular geometry optimization. IPA implements the power iteration method in quantics tensor train (QTT) representations. Analogous to the imaginary time propagation method with infinite mass, IPA starts with an initial probability distribution and iteratively applies the recurrence relation ${\rho }_{k+1}(\mathbf{x})=U(\mathbf{x}){\rho }_k(\mathbf{x})/\Vert U{\rho }_k{\Vert }_{L^1}$, where $U(\mathbf{x})=e^{-V(\mathbf{x})}$ is defined in terms of the potential energy surface (PES) $V(\mathbf{x})$. Upon convergence, the probability distribution becomes a delta function, so the global minimum can be obtained as the position expectation value. QTT representations are generated by fast adaptive interpolation of multidimensional arrays to bypass the curse of dimensionality and the need to evaluate $V(\mathbf{x})$ for all possible $\mathbf{x}$. We illustrate the capabilities of IPA for global search optimization of two multidimensional PESs, including a differentiable model PES of a DNA chain and a discrete non-differentiable PES, $V(p)=\text{mod}(N,p)$, that resolves the prime factors of an integer $N$, with $p$ in the space of prime numbers folded as a $d$-dimensional $2_1\times 2_2\times \cdots \times 2_d$ tensor. We find that IPA resolves multiple degenerate global minima even when separated by large energy barriers in the highly rugged landscape of the potentials. Therefore, IPA should be of great interest for a wide range of other optimization problems ubiquitous in molecular and electronic structure calculations.

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