Refine
Document Type
- ZIB-Report (2)
- Master's Thesis (1)
Language
- English (3)
Has Fulltext
- yes (3)
Keywords
- mixed-integer programming (3) (remove)
Institute
- Mathematical Optimization (3) (remove)
Modern solving software for mixed-integer programming (MIP)
incorporates numerous algorithmic components whose behavior is
controlled by user parameter choices,
and whose usefulness dramatically varies depending on the progress of the solving process.
In this thesis, our aim is to
construct a phase-based solver that dynamically reacts
on phase transitions with an appropriate change of its component behavior.
Therefore, we decompose the branch-and-bound solving process into three distinct phases:
The first phase objective is to find a feasible solution. During the second phase,
a sequence of incumbent solutions gets constructed
until the incumbent is eventually optimal. Proving
optimality is the central objective of the remaining third phase.
Based on the MIP-solver SCIP we construct a phase-based solver to make use of the phase concept in two steps:
First, we identify promising components for every solving phase individually and show that their
combination is beneficial on a test bed of practical MIP instances.
We then present and evaluate three heuristic criteria to make use of the phase-based solver
in practice, where it is infeasible to distinguish between the last two phases
before the termination of the solving process.
The selection of a good branching variable is crucial for small search trees in Mixed Integer Programming. Most modern solvers employ a strategy guided by history information, mainly the variable pseudo-costs, which are used to estimate the objective gain. At the beginning
of the search, such information is usually collected via an expensive look-ahead strategy called strong-branching until variables are considered reliable.
The reliability notion is thereby mostly based on fixed-number thresholds, which may lead to ineffective branching decisions on problems with highly varying objective gains.
We suggest two new notions of reliability motivated by mathematical statistics that take into account the sample variance of the past observations on each variable individually. The first method prioritizes additional strong-branching look-aheads on variables whose pseudo-costs
show a large variance by measuring the relative error of a pseudo-cost confidence interval. The second method performs a two-sample Student-t test for filtering branching candidates with a high probability to be better than the best history candidate.
Both methods were implemented in the MIP-solver SCIP and computational results on standard MIP test sets are presented.
Modern MIP solvers employ dozens of auxiliary algorithmic components to support the branch-and-bound search in finding and improving primal solutions and in strengthening the dual bound.
Typically, all components are tuned to minimize the average running time to prove optimality. In this article, we take a different look at the run of a MIP solver. We argue that the solution process consists of three different phases, namely achieving feasibility, improving the incumbent solution, and proving optimality. We first show that the entire solving process can be improved by adapting the search strategy with respect to the phase-specific aims using different control tunings. Afterwards, we provide criteria to predict the transition between the individual phases and evaluate the performance impact of altering the algorithmic behavior of the MIP solver SCIP at the predicted phase transition points.