4 research outputs found
Intersection disjunctions for reverse convex sets
We present a framework to obtain valid inequalities for optimization problems
constrained by a reverse convex set, which is defined as the set of points in a
polyhedron that lie outside a given open convex set. We are particularly
interested in cases where the closure of the convex set is either
non-polyhedral, or is defined by too many inequalities to directly apply
disjunctive programming. Reverse convex sets arise in many models, including
bilevel optimization and polynomial optimization. Intersection cuts are a
well-known method for generating valid inequalities for a reverse convex set.
Intersection cuts are generated from a basic solution that lies within the
convex set. Our contribution is a framework for deriving valid inequalities for
the reverse convex set from basic solutions that lie outside the convex set. We
begin by proposing an extension to intersection cuts that defines a two-term
disjunction for a reverse convex set. Next, we generalize this analysis to a
multi-term disjunction by considering the convex set's recession directions.
These disjunctions can be used in a cut-generating linear program to obtain
disjunctive cuts for the reverse convex set.Comment: 24 page
Outer Approximation Algorithms for DC Programs and Beyond
We consider the well-known Canonical DC (CDC)
optimization problem, relying on an alternative equivalent formulation based on a polar characterization of the constraint, and
a novel generalization of this problem, which we name Single Reverse Polar problem (SRP). We study the theoretical properties of the new class of (SRP) problems, and contrast them with those of (CDC)problems.
We introduce of the concept of ``approximate oracle'' for the optimality conditions of (CDC) and (SRP), and make a thorough study of the impact of approximations in the optimality conditions onto
the quality of the approximate optimal solutions, that is the feasible solutions which satisfy them. Afterwards, we develop very general hierarchies of convergence conditions, similar but not identical for (CDC) and (SRP), starting from very abstract ones and moving towards more readily implementable ones. Six and three different sets of conditions are proposed for (CDC) and (SRP), respectively.
As a result, we propose very general algorithmic schemes, based on approximate oracles and the developed hierarchies, giving rise to many different implementable algorithms, which can be
proven to generate an approximate optimal value in a finite number of steps, where the error can be managed and controlled. Among them, six different implementable algorithms for (CDC) problems, four of which are new and can't be reduced to the original cutting plane algorithm for (CDC) and its modifications; the connections of our
results with the existing algorithms in the literature are outlined. Also, three cutting plane algorithms for solving (SRP) problems are proposed, which seem to be new and cannot be reduced to each other