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Optimization problems occurring in a wide variety of physical designproblems, including but not limited to optical engineering, quantum control,structural engineering, involve minimization of a simple cost function of thestate of the system (e.g. the optical fields, the quantum state) while beingconstrained by the physics of the system. The physics constraints often makessuch problems non-convex and thus only locally solvable, leaving open thequestion of finding the globally optimal design. In this paper, I considerdesign problems whose physics is described by bi-affine equality constraints,and show that under assumptions on the stability of these constraints and thephysical system being non-resonant, gradient descent globally solves a typicalphysical design problem. The key technical contributions of this paper are (i)outline a criteria that ensure the convergence of gradient descent to anapproximate global optima in the limit of large problem sizes, and (ii) userandom matrix theory to outline ensembles of physically motivated problemswhich, on an average, satisfy this convergence criteria.

Gradient descent globally solves average-case non-resonant physical design problems