We propose and study a dimensionality-reduced first order method for solving complex optimization problems of high-dimensional search space. We demonstrate that the proposed method is very efficient in design problems where the computational bottleneck is mostly due to the time-consuming nature of the forward problem in contrast to the complexity of the function behavior in the search space or other computational overheads. Many industrial problems are of this nature including design problems based on back testing or simulation of an evolutionary equation or a dynamic system in time, frequency or other (hybrid) domains, such as Electromagnetic, Quantum equations, Navier-Stokes PDEs, etc. The premise of efficiency improvement in the proposed framework is a better modelling and utilization of the complexity distribution among the components of an inverse design problem. As a particular case study, we list some of the existing optimization problems related to energy production, distribution and utilization at the industrial level. We briefly overview the different complexity components of these problems at a high level, and make suggestions as what industrial problems can be facilitated through the proposed framework.
A First Order Method for Solving Convex Bilevel Optimization Problems