Abstract: For ten years, the primary focus in continuous optimization has been on the development of first-order methods, due to problems arising from big data being too large to be solved by second-order algorithms, such as interior-point methods.
First-order methods have been devised for handling a rich array of objective functions, but almost invariably, the algorithms require the feasible region to be of particularly simple forms, such as a ball, a box or a simplex. The simple form is required by numerous algorithms, for example, so that orthogonal projections onto the feasible region can be done efficiently.
We present a framework whereby a general convex conic optimization problem is transformed into an equivalent convex optimization problem whose only constraints are linear equations and whose objective function is Lipschitz continuous. Virtually any subgradient method can be applied to solve the equivalent problem, as the only orthogonal projections are onto a subspace (the same subspace at every iteration).
We also outline various research pursuits that have been motivated by the new framework.