A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer gomory cuts for 0-1 programming

 We establish a precise correspondence between lift-and-project cuts for mixed 0-1 programs, simple disjunctive cuts (intersection cuts) and mixed-integer Gomory cuts. The correspondence maps members of one family onto members of the others. It also maps bases of the higher-dimensional cut generating linear program (CGLP) into bases of the linear programming relaxation. It provides new bounds on the number of facets of the elementary closure, and on the rank, of the standard linear programming relaxation of the mixed 0-1 polyhedron with respect to the above families of cutting planes. Based on the above correspondence, we develop an algorithm that solves (CGLP) without explicitly constructing it, by mimicking the pivoting steps of the higher dimensional (CGLP) simplex tableau by certain pivoting steps in the lower dimensional (LP) simplex tableau. In particular, we show how to calculate the reduced costs of the big tableau from the entries of the small tableau and based on this, how to identify a pivot in the small tableau that corresponds to one or several improving pivots in the big tableau. The overall effect is a much improved lift-and-project cut generating procedure, which can also be interpreted as an algorithm for a systematic improvement of mixed integer Gomory cuts from the small tableau. Received: October 5, 2000 / Accepted: March 19, 2002 Published online: September 5, 2002 Research was supported by the National Science Foundation through grant #DMI-9802773 and by the Office of Naval Research through contract N00014-97-1-0196.     

Al-promoted increase of surface area and adsorption capacity in ordered mesoporous silica materials with a cubic structure

The structure of ordered mesoporous materials presenting a cubic structure undergoes a strong modification after adding aluminium alkoxides to the synthesis gel. As a result, an outstanding increase in the surface area and pore volume is observed, together with changes in the mesopore ordering. Hydrated aluminium species influence the chemical environment of the micelles during synthesis, which seems to induce the accumulation of stacking faults in the mesopore framework, and give rise to closer structural packing. The adsorption and release of ibuprofen as a test molecule correlates well with the textural changes observed.     

Lexicography and degeneracy: can a pure cutting plane algorithm work?

We discuss an implementation of the lexicographic version of Gomory’s fractional cutting plane method for ILP problems and of two heuristics mimicking the latter. In computational testing on a battery of MIPLIB problems we compare the performance of these variants with that of the standard Gomory algorithm, both in the single-cut and in the multi-cut (rounds of cuts) version, and show that they provide a radical improvement over the standard procedure. In particular, we report the exact solution of ILP instances from MIPLIB such as stein15, stein27, and bm23, for which the standard Gomory cutting plane algorithm is not able to close more than a tiny fraction of the integrality gap. We also offer an explanation for this surprising phenomenon.     

Generalized intersection cuts and a new cut generating paradigm

Intersection cuts are generated from a polyhedral cone and a convex set S whose interior contains no feasible integer point. We generalize these cuts by replacing the cone with a more general polyhedron C. The resulting generalized intersection cuts dominate the original ones. This leads to a new cutting plane paradigm under which one generates and stores the intersection points of the extreme rays of C with the boundary of S rather than the cuts themselves. These intersection points can then be used to generate in a non-recursive fashion cuts that would require several recursive applications of some standard cut generating routine. A procedure is also given for strengthening the coefficients of the integer-constrained variables of a generalized intersection cut. The new cutting plane paradigm yields a new characterization of the closure of intersection cuts and their strengthened variants. This characterization is minimal in the sense that every one of the inequalities it uses defines a facet of the closure.     

A hard integer program made easy by lexicography

A small but notoriously hard integer program formulated by Donald Knuth fifty years ago is solved by three versions of a lexicographic algorithm using Gomory cuts. The lexicographic cutting plane algorithms are faster than CPLEX on this problem by a factor of at least 10.     

Bees algorithm for generalized assignment problem

Bees algorithm (BA) is a new member of meta-heuristics. BA tries to model natural behavior of honey bees in food foraging. Honey bees use several mechanisms like waggle dance to optimally locate food sources and to search new ones. This makes them a good candidate for developing new algorithms for solving optimization problems. In this paper a brief review of BA is first given, afterwards development of a BA for solving generalized assignment problems (GAP) with an ejection chain neighborhood mechanism is presented. GAP is a NP-hard problem. Many meta-heuristic algorithms were proposed for its solution. So far BA is generally applied to continuous optimization. In order to investigate the performance of BA on a complex integer optimization problem, an attempt is made in this paper. An extensive computational study is carried out and the results are compared with several algorithms from the literature.     

Integer programming and convex analysis: Intersection cuts from outer polars

Recently a new, geometrically motivated approach was proposed [1] for integer programming, based on generating intersection cuts from some convex setS whose interior contains the linear programming optimum but no feasible integer point. Larger sets tend to produce stronger cuts, and in this paper convex analysis is used to construct sets as large as possible within the above requirements. Information is generated from all problem constraints within a unit cubeK containing The key concept is that of outer polars, viewed as maximal convex extensions of the ballB circumscribingK, relative to the problem constraints. The outer polarF ^* of the feasible setF overB is shown to be a convex set whose boundary contains all feasible vertices ofK, and whose interior contains no feasible integer point. The existence of a dual correspondence betweenF andF ^*, and the fact that polarity is inclusion-reversing, leads to a dualization of operations onF. As one possible procedure based on this approach, we construct a generalized intersection cut, that can be strengthened whenever some vertex ofF is cut off. This makes it possible to fruitfully combine intersection cuts with implicit enumeration or branch-and-bound. While valid for arbitrary integer programs, the theory developed here is relevant primarily to (pure or mixed-integer) 0–1 problems. Other topics discussed include: generalized polars, intersection cuts from related problems, connections with asymptotic theory.This paper was presented at the 7th Mathematical Programming Symposium, 1970, The Hague, The Netherlands.The research underlying this paper was partially supported by the National Science Foundation under grant GP-31699 and by the Office of Naval Research under contract N00014-67-A-0314-0007 NR 047-048.     

Nonlinear 0–1 programming: II. Dominance relations and algorithms

A nonlinear 0–1 program can be restated as a multilinear 0–1 program, which in turn is known to be equivalent to a linear 0–1 program with generalized covering (g.c.) inequalities. In a companion paper [6] we have defined a family of linear inequalities that contains more compact (smaller cardinality) linearizations of a multilinear 0–1 program than the one based on the g.c. inequalities. In this paper we analyze the dominance relations between inequalities of the above family. In particular, we give a criterion that can be checked in linear time, for deciding whether a g.c. inequality can be strengthened by extending the cover from which it was derived. We then describe a class of algorithms based on these results and discuss our computational experience. We conclude that the g.c. inequalities can be strengthened most of the time to an extent that increases with problem density. In particular, the algorithm using the strengthening procedure outperforms the one using only g.c. inequalities whenever the number of nonlinear terms per constraint exceeds about 12–15, and the difference in their performance grows with the number of such terms. Research supported by the National Science Foundation under grant ECS 7902506 and by the Office of Naval Research under contract N00014-75-C-0621 NR 047-048.     

The galerkin method and feedback control of linear distributed parameter systems

In the development of feedback control theory for distributed parameter systems (DPS), i.e., systems described by partial differential equations, it is important to maintain the finite dimensionality of the controller even though the DPS is infinite dimensional. Since this dimension is directly related to the available on-line computer capacity, it must be finite (and not very large) in order to make the controller implementable from an engineering standpoint. In previous work, it has been our intention to investigate what can be accomplished by finite-dimensional control of infinite-dimensional systems; in particular, we have concentrated on controller design and closed-loop stability. The starting point for all of this is some means for producing a finite-dimensional approximation—a reduced-order model—of the actual DPS. When the “modes” of the DPS are known, the natural candidate for model reduction is projection onto the modal subspace spanned by a finite number of critical modes. Unfortunately, in real engineering systems, these modes are never known exactly and some other reasonable approximation must be used. In this paper, the model reduction is based on the well-known Galerkin procedure. We generate the Galerkin reduced-order model and develop a finite-dimensional controller from it; then we analyze the stability of this controller in closed loop with the actual DPS. Our results indicate conditions under which model reduction based on consistent Galerkin approximations will lead to stable finite-dimensional control.