476 research outputs found
Mixed-integer convex representability
Motivated by recent advances in solution methods for mixed-integer convex
optimization (MICP), we study the fundamental and open question of which sets
can be represented exactly as feasible regions of MICP problems. We establish
several results in this direction, including the first complete
characterization for the mixed-binary case and a simple necessary condition for
the general case. We use the latter to derive the first non-representability
results for various non-convex sets such as the set of rank-1 matrices and the
set of prime numbers. Finally, in correspondence with the seminal work on
mixed-integer linear representability by Jeroslow and Lowe, we study the
representability question under rationality assumptions. Under these
rationality assumptions, we establish that representable sets obey strong
regularity properties such as periodicity, and we provide a complete
characterization of representable subsets of the natural numbers and of
representable compact sets. Interestingly, in the case of subsets of natural
numbers, our results provide a clear separation between the mathematical
modeling power of mixed-integer linear and mixed-integer convex optimization.
In the case of compact sets, our results imply that using unbounded integer
variables is necessary only for modeling unbounded sets
Mixed-Integer Convex Representability
We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer assignments is finite. We develop a characterization for the more general case of unbounded integer variables together with a simple necessary condition for representability which we use to prove the first known negative results. Finally, we study representability of subsets of the natural numbers, developing insight towards a more complete understanding of what modeling power can be gained by using convex sets instead of polyhedral sets; the latter case has been completely characterized in the context of mixed-integer linear optimization.United States. National Science Foundation. (Grant CMMI-1351619
Extended Formulations in Mixed-integer Convex Programming
We present a unifying framework for generating extended formulations for the
polyhedral outer approximations used in algorithms for mixed-integer convex
programming (MICP). Extended formulations lead to fewer iterations of outer
approximation algorithms and generally faster solution times. First, we observe
that all MICP instances from the MINLPLIB2 benchmark library are conic
representable with standard symmetric and nonsymmetric cones. Conic
reformulations are shown to be effective extended formulations themselves
because they encode separability structure. For mixed-integer
conic-representable problems, we provide the first outer approximation
algorithm with finite-time convergence guarantees, opening a path for the use
of conic solvers for continuous relaxations. We then connect the popular
modeling framework of disciplined convex programming (DCP) to the existence of
extended formulations independent of conic representability. We present
evidence that our approach can yield significant gains in practice, with the
solution of a number of open instances from the MINLPLIB2 benchmark library.Comment: To be presented at IPCO 201
Mixed-integer bilevel representability
We study the representability of sets that admit extended formulations using
mixed-integer bilevel programs. We show that feasible regions modeled by
continuous bilevel constraints (with no integer variables), complementarity
constraints, and polyhedral reverse convex constraints are all finite unions of
polyhedra. Conversely, any finite union of polyhedra can be represented using
any one of these three paradigms. We then prove that the feasible region of
bilevel problems with integer constraints exclusively in the upper level is a
finite union of sets representable by mixed-integer programs and vice versa.
Further, we prove that, up to topological closures, we do not get additional
modeling power by allowing integer variables in the lower level as well. To
establish the last statement, we prove that the family of sets that are finite
unions of mixed-integer representable sets forms an algebra of sets (up to
topological closures)
Pure-state -representability in current-spin-density-functional theory
This paper is concerned with the pure-state -representability problem for
systems under a magnetic field. Necessary and sufficient conditions are given
for a spin-density matrix to be representable by a Slater
determinant. We also provide sufficient conditions on the paramagnetic current
for the pair to be Slater-representable in the
case where the number of electrons is greater than 12. The case is
left open
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