838 research outputs found
Searching for optimal integer solutions to set partitioning problems using column generation
We describe a new approach to produce integer feasible columns to a set partitioning problem directly in
solving the linear programming (LP) relaxation using column generation. Traditionally, column generation
is aimed to solve the LP relaxation as quick as possible without any concern of the integer properties of
the columns formed. In our approach we aim to generate the columns forming the optimal integer solution
while simultaneously solving the LP relaxation. By this we can remove column generation in the branch
and bound search. The basis is a subgradient technique applied to a Lagrangian dual formulation of the set
partitioning problem extended with an additional surrogate constraint. This extra constraint is not relaxed
and is used to better control the subgradient evaluations. The column generation is then directed, via the
multipliers, to construct columns that form feasible integer solutions. Computational experiments show that
we can generate the optimal integer columns in a large set of well known test problems as compared to both
standard and stabilized column generation and simultaneously keep the number of columns smaller than
standard column generation
Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics
The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session
On generalized surrogate duality in mixed-integer nonlinear programming
The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global -optimality with spatial branch and bound is a tight, computationally
tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solvers
can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we
exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and
challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the
algorithm’s ability to generate strong dual bounds through extensive computational experiments
Rigorous design of distillation columns using surrogate models based on Kriging interpolation
The economic design of a distillation column or distillation sequences is a challenging problem that has been addressed by superstructure approaches. However, these methods have not been widely used because they lead to mixed-integer nonlinear programs that are hard to solve, and require complex initialization procedures. In this article, we propose to address this challenging problem by substituting the distillation columns by Kriging-based surrogate models generated via state of the art distillation models. We study different columns with increasing difficulty, and show that it is possible to get accurate Kriging-based surrogate models. The optimization strategy ensures that convergence to a local optimum is guaranteed for numerical noise-free models. For distillation columns (slightly noisy systems), Karush–Kuhn–Tucker optimality conditions cannot be tested directly on the actual model, but still we can guarantee a local minimum in a trust region of the surrogate model that contains the actual local minimum.The authors gratefully acknowledge the financial support of the Ministry of Economy and Competitiveness of Spain, under the project CTQ2012-37039-C02-02
On the implementation of a global optimization method for mixed-variable problems
We describe the optimization algorithm implemented in the open-source
derivative-free solver RBFOpt. The algorithm is based on the radial basis
function method of Gutmann and the metric stochastic response surface method of
Regis and Shoemaker. We propose several modifications aimed at generalizing and
improving these two algorithms: (i) the use of an extended space to represent
categorical variables in unary encoding; (ii) a refinement phase to locally
improve a candidate solution; (iii) interpolation models without the
unisolvence condition, to both help deal with categorical variables, and
initiate the optimization before a uniquely determined model is possible; (iv)
a master-worker framework to allow asynchronous objective function evaluations
in parallel. Numerical experiments show the effectiveness of these ideas
- …