Feedback and constraints in physical optimizers

Extremizing a quadratic form can be computationally straightforward or difficult depending on the feasible domain over which variables are optimized. For example, maximizing E = x(T)Vx for a real-symmetric matrix V with x constrained to a unit ball in R-N can be performed simply by finding the maximum (principal) eigenvector of V, but can become computationally intractable if the domain of x is limited to corners of the +/- 1 hypercube in R-N (i.e., x is constrained to be a binary vector). Many gain-loss physical systems, such as coherently coupled arrays of lasers or optical parametric oscillators, naturally solve minimum/maximum eigenvector problems (of a matrix of coupling coefficients) in their equilibration dynamics. In this paper we discuss recent case studies on the use of added nonlinear dynamics and real-time feedback to enforce constraints in such systems, making them potentially useful for solving difficult optimization problems. We consider examples in both classical and quantum regimes of operation.