Convex Restriction: A Sufficient Convex Condition for Solvability and Feasibility of Nonlinear Equations

Convex restriction identifies the convex subset of a general nonconvex feasible set described by nonlinear equality and inequality constraints. Nonlinear optimization problems often rely on convex relaxations, which can be interpreted as convex outer approximations of the feasible set. Convex restrictions address the problem of constructing the inner approximation of the feasible set and is especially useful for establishing robustness of the solution under uncertainties. A general framework is developed for a system with sparse nonlinear structure, and an application to the electric power systems will be presented. The procedure results in analytical convex quadratic inequality constraints that provide nearly tight approximation of the actual feasible set for many of the IEEE test cases.