Large-scale portfolio optimization with variational neural annealing.

We employ a variational neural annealing approach for solving large-scale constrained portfolio optimization problems, mapped onto classical Ising-like Hamiltonians, and study its performance under finite annealing times. The method enables efficient exploration of high-dimensional feasible spaces and achieves competitive solutions on standard financial indices. To characterize the underlying optimization dynamics, we analyze the decay of the residual energy as a function of annealing steps and observe a power-law behavior for fixed problem size. Extending this analysis across different portfolio sizes, we examine whether the optimization dynamics are consistent with a dynamical finite-size scaling form and find that the residual energy data collapse onto a single scaling curve within the studied range. This scaling analysis yields an effective annealing-time exponent that quantifies how computational effort grows with problem size. Our results demonstrate that finite-size scaling concepts provide a useful framework for understanding and forecasting the performance of neural annealing-based optimization methods in realistic, constrained settings.