  A constrained optimization problem is primal infeasible if its constraints
cannot be satisfied, and dual infeasible if the constraints of its dual problem
cannot be satisfied. We propose a novel iterative method, named
proportional-integral projected gradient method (PIPG), for detecting primal
and dual infeasiblity in convex optimization with quadratic objective function
and conic constraints. The iterates of PIPG either asymptotically provide a
proof of primal or dual infeasibility, or asymptotically satisfy a set of
primal-dual optimality conditions. Unlike existing methods, PIPG does not
compute matrix inverse, which makes it better suited for large-scale and
real-time applications. We demonstrate the application of PIPG in quasiconvex
and mixed-integer optimization using examples in constrained optimal control.
