## Section: New Results

### Generic solvers

Participants : Sophie Michel, Andrew Miller, Céline Saubatte, François Vanderbeck, Benoit Vignac.

#### Branching in branch-and-price: a generic scheme for the non-binary case

We have pursued our work on developing a branching scheme that is compatible with the column generation procedure and that implies no strutural modifications to the pricing problem. We have carry on a comprehensive experimental study that demonstrates the efficiency of what could have been seen otherwise as a theoretic scheme. Moreover, although the scheme was initially presented for a binary integer program, we have shown (in theory and in our computational experiments) how it extends to a general mixed integer program [22]

#### Primal heuristics based on column generation

In the past decade, significant progress has been achieved in developing generic primal heuristics that made their way into commercial mixed integer programming (MIP) solver. Extensions to the context of a column generation solution approach are not straightforward. The Dantzig-Wolfe decomposition principle can indeed be exploited in greedy, local search, rounding or truncated exact methods. The price coordination mechanism can bring a global view that may be lacking in some “myopic” approaches based on a compact formulation. However, the dynamic generation of variables requires specific adaptation of heuristic paradigms.

Based on our application specific experience with these tehcniques [122] , [130] , [162] , [163] , and on a review of generic classes of column generation based primal heuristics, in [24] we focus on a so-called “diving” method in which we introduce diversification based on Limited Discrepancy Search. While being a general purpose approach, the implementation of our heuristic illustrates the technicalities specific to column generation. The method is numerically tested on variants of the cutting stock and vehicle routing problems.

#### PMaP: A **P**arallel **Ma**cro
**P**artitioning framework for solving mixed integer
programs

In [133] we propose a new framework (the Parallel Macro Partitioning (PMaP) framework) that partitions the feasible domain of an MIP by using concepts derived from recently developed primal heuristics. The ideas from these heuristics, which include LP-and-fix, RINS, and local branching, among others, enable us both to define quickly many subproblems whose feasibility region is much smaller than that of the original problem. They also allow us to generate complementary cuts in order to ensure that the solver does not search the same region on many different processors. The result is that PMaP is able to quickly partition the feasible region at a high level into a large set of subproblem MIPs that can be solved simultaneously in parallel on separate processors. Initial computational resources suggest that PMaP has significant promise as a framework capable of bringing many processors to bear effectively on difficult problems. In particular, they suggest that the overall approach of high-level partitioning is more promising as a means of enumeration than classical branch-and-bound in terms of being able to use effectively hundreds or even thousands of processors in order to solve difficult problems. Given the increasing prevalence that parallel computing architectures are likely to plan in the future, it seems at least possible that our approach, when it reaches maturity, may come to be seen as a breakthrough achievement.

Partially as a result of the development of this software, Mahdi Namazifar spent a summer internship at IBM's T.J. Watson Research Center in 2008 working on these and other ideas. We expect to integrate PMaP into COIN-OR (maintained by an IBM team at Watson) in the near future.

#### Other mixed integer programming heuristics

Most MIP heuristics that have been developed perform best on
problems with binary variables. In [132] we propose methods
for problems with *general* (i.e., not necessary binary)
integer variables. Called *randomized rounding* heuristics,
these methods do much more than simply rounding a single LP
fractional solution. They attempt to find feasible solutions by
randomly walking within a specially-constructed polyhedron, and then
performing rounding operations from the points traversed. The
polyhedron in question is expressed as the convex hull of some of
the “interesting" extreme points of the LP relaxation of the
original problem, where the extreme points chosen to be of interest
have a high probability of being in the region of the LP relaxation
pointed to by the objective function. Preliminary computational
results for this heuristic approach suggest that it may be the most
effective primal heuristic known for certain classes of general MIP
problems.

#### Mixed integer non-linear programming

While our research on MINLP problems is just beginning, there is already evidence of potential for significant success. For example, we have been considering how to exploit the information generated by solvers as they make branching decisions (in particular, which variable to branch on) during the branch-and-bound process. This has allowed us to define a structured family of disjunctive cuts whose generation requires no additional effort beyond that necessary to perform classical strong branching. Even for MIPs, these ideas seem capable of singificantly reducing the size of branch-and-bound trees; moreover, the ideas themselves are directly applicable to MINLPs, and we are currently investigating how best to apply them [110] .

Another factor in defining effective branch-and-solvers for MINLPs
is defining solvers that are capable of *re* -optimizing NLP
problems efficiently. Classical interior point methods have
well-known difficulties in performing warm re-starts; for this and
other methods, *active set* methods seem to have more potential
for efficient re-optimization. In [43] ,
we have defined a new active set method for linear programming and
generalized it for quadratic programming problems. This algorithm
has a number of desirable properties for these problems (including
re-optimization capabilities), and we are hopeful to generalize this
method further to other families of NLP problems.

Multilinear functions appear in many global optimization problems, including blending and electricity transmission, among many others. In [30] , the authors study convex envelopes for a product of variables that have lower and upper bounds, and which is itself bounded. Since global branch-and-bound solvers for such problems use polyhedral relaxations of such sets to compute bounds, having tight relaxations can improve performance. For two variables, the well-known McCormick inequalities define the convex hull for an unbounded product. For a bounded product, we define valid linear inequalities that support as many points of the convex hull as possible. Though uncountably infinite in number, these inequalities can be separated for exactly in polynomial time. We have extended extensions of such results to products of more than two variables, considering both convex hull descriptions and separation.

For global optimization problems with sums of multiple product terms, a common technique for creating relaxations for these problems is to decompose each product term into bilinear terms (i.e., products of two variables) and then use the McCormick envelope for each term separately. The authors of [31] are investigating an approach which generates a (stronger) relaxation directly from the multilinear terms. Computational results demonstrate that significant advantages can result from using this approach.