Approximated Perspective Relaxations: a Project&Lift Approach
The Perspective Reformulation (PR) of a Mixed-Integer NonLinear Program with semi-continuous variables is obtained by replacing each term in the (separable) objective function with its convex envelope. Solving the corresponding Perspective Relaxation requires appropriate techniques. Under some rather restrictive assumptions, the Projected PR can be defined where the integer variables are eliminated by projecting the solution set on the space of the continuous variables only. This approach produces a simple piecewise-convex problem with the same structure as the original one; however, this prevents the use of general-purpose solvers, in that some variables are then only implicitly represented in the formulation. We show how to construct an Approximated Projected PR whereby the projected formulation is "lifted" back to the original variable space, with the integer variables expressing one piece of the obtained piecewise-convex function; in some cases, this produces a reformulation of the original problem with exactly the same size and structure as the standard continuous relaxation but with a substantially improved bound. While the bound can be weaker than that of the PR, this approach can be applied in many more cases and allows direct use of off-the-shelf MIQP software; this is shown to be beneficial in different applications. In the process we also relax some of the other restrictive assumptions of the original development, such as the need for the objective function to be quadratic and the need for the left endpoint of the domain of the variables to be non-negative.