  Solving optimization problems is challenging for existing digital computers
and even for future quantum hardware. The practical importance of diverse
problems, from healthcare to financial optimization, has driven the emergence
of specialised hardware over the past decade. However, their support for
problems with only binary variables severely restricts the scope of practical
problems that can be efficiently embedded. We build analog iterative machine
(AIM), the first instance of an opto-electronic solver that natively implements
a wider class of quadratic unconstrained mixed optimization (QUMO) problems and
supports all-to-all connectivity of both continuous and binary variables.Beyond
synthetic 7-bit problems at small-scale, AIM solves the financial transaction
settlement problem entirely in analog domain with higher accuracy than quantum
hardware and at room temperature. With compute-in-memory operation and
spatial-division multiplexed representation of variables, the design of AIM
paves the path to chip-scale architecture with 100 times speed-up per
unit-power over the latest GPUs for solving problems with 10,000 variables. The
robustness of the AIM algorithm at such scale is further demonstrated by
comparing it with commercial production solvers across multiple benchmarks,
where for several problems we report new best solutions. By combining the
superior QUMO abstraction, sophisticated gradient descent methods inspired by
machine learning, and commodity hardware, AIM introduces a novel platform with
a step change in expressiveness, performance, and scalability, for optimization
in the post-Moores law era.
