Improving the Approximated Projected Perspective Reformulation by dual information

Abstract

We propose an improvement of the Approximated Projected Perspective Reformulation (AP 2 R) of [1] for the case in which constraints linking the binary variables exist. The new approach requires to solve the Perspective Reformulation (PR) once, and then use the corresponding dual information to reformulate the problem prior to applying AP 2 R, thereby combining the root bound quality of the PR with the reduced relaxation computing time of AP 2 R. Computational results for the cardinalityconstrained Mean-Variance portfolio optimization problem show that the new approach is competitive with state-of-the-art ones.

Decompositions of Semidefinite Matrices and the Perspective Reformulation of Nonseparable Quadratic Programs

“…If the strong duality property holds for (10) (e.g., if Q 0), then solving (11) provides the best possible lower bound and the corresponding optimal solution (x, y). This could be used to compute the optimal δ as in (9), but in fact that is already provided by the dual variables of the diag(F ) ≥ 0 constraint. This is important as typically one does not want to solve (11) at all iterations of an enumerative approach to the original (1), for the large-scale SDP (11) is rather costly to solve.…”

Perspective Cuts for the ACOPF with Generators

“…Let F be the feasible subset of C n defined by Eq. (2)-(6) and (7). We call ACOPFG C the formulation min…”

“…The addition of PCs does not add further difficulties in the problem formulation except for the condition that they should be generated iteratively as their number is not finite. We can alternatively apply the AP2R technique [6,7], which works in two phases. The first phase is a projection where the optimal value of z g for the continuous relaxation of Eq.…”

Strengthening the sequential convex MINLP technique by perspective reformulations

“…The Approximated Projected PR [5] can still be used in this case; it yields an intermediate relaxation that provides a stronger bound than the original continuous relaxation, although possibly weaker than that of the (PR). The approach can also be improved by using dual information [6] so that the bound is the same, but only at the root node of the enumeration tree, while it becomes weaker than that of the true (PR) as branching proceeds. Furthermore, the advantage of the approach-that of producing a problem with basically the same shape as the original one-can also be a disadvantage with general nonlinear terms g ij , as it requires use of general (convex) nonlinear solvers for tackling continuous relaxations.…”