A generic method for formulating pulp and paper optimization problems is presented. Two ongoing projects in the framework of the DOTS project illustrate the method: optimization of sizing quality at a specialty paper mill, and optimization of the water and broke systems at a coated paper mill. Explicit and implicit formulations are compared, and different usages of external simulators in conjunction with optimization are discussed. The problems are solved using MATLAB/TOMLAB. Some results from different optimization algorithms are also presented.
Powerful response surface methods based on kriging and radial basis function (RBF) interpolation have been developed for expensive, i.e. computationally costly, global nonconvex optimization. We have implemented some of these methods in the solvers rbfSolve and EGO in the TOMLAB Optimization Environment (http://www.tomopt.com/tomlab/). In this paper we study algorithms based on RBF interpolation. The practical performance of the RBF algorithm is sensitive to the initial experimental design, and to the static choice of target values. A new adaptive radial basis interpolation (ARBF) algorithm, suitable for parallel implementation, is presented. The algorithm is described in detail and its efficiency is analyzed on the standard test problem set of Dixon-Szego. Results show that it outperforms the published results of rbfSolve and several other solvers.
A mixed-integer constrained extension of the radial basis function (RBF) interpolation algorithm for computationally costly global non-convex optimization is presented. Implementation in TOM-LAB (http://tomlab.biz) solver rbfSolve is discussed. The algorithm relies on mixed-integer nonlinear (MINLP) sub solvers in TOMLAB, e.g. OQNLP, MINLPBB or the constrained DIRECT solvers (glcDirect or glcSolve). Depending on the initial experimental design, the basic RBF algorithm sometimes fails and make no progress. A new method how to detect when there is a problem is presented. We discuss the causes and present a new faster and more robust Adaptive RBF (ARBF) algorithm. Test results for unconstrained problems are discussed.
The optimization environment TOMLAB, http://tomopt.com, has seen a tremendous growth during the last years. Most state-of-the-art optimization software has been hooked up, e.g. KNITRO, SNOPT and CONOPT for large-scale nonlinear programming, and CPLEX and Xpress-MP for large-scale mixed-integer programming. Unique tools for global black-box mixed-integer nonconvex problems have been developed. Originally developed for MATLAB, now TOMLAB is available for LabView as TOMVIEW and .NET as TOMNET. TOMLAB is interfaced with the modelling language AMPL and the DIFFPACK package for advanced PDE solutions. This talk gives an overview over the latest developments.
Improvements of the adaptive radial basis function algo-rithm (ARBF) for computationally costly optimization are presented. A new target value search algorithm and modifications to improve robustness and speed are discussed. The algoritm is implemented in solver ARBFMIP in the TOM-LAB Optimization Environment (http://tomopt.com/). Solvers in TOMLAB are used to solve global and local subproblems. Results and comparisons with other solvers are presented for global optimization test problems. Performance on costly real-life applications are reported.
The container loading problem aims at optimal packing of boxes of different dimensions into available containers with respect to some objective function. This problem arises in areas like distribution and logistics. Since the scale of distribution and amount of distributed goods grows rapidly, so does the need for more energy effective and environmentally friendly distribution methods.This paper considers a single container loading problem motivated by a real life application. The formulation includes an objective function for maximizing utilization of a container and additional constraints such as a predefined order and priorities between boxes. A short evaluation of the algorithm is presented as well as some directions for future development.
When dealing with costly objective functions in optimization, one good alternative is to use a surrogate model approach. A common feature for all such methods is the need of an initial set of points, or "experimental design", in order to start the algorithm. Since the behavior of the algorithms often depends heavily on this set, the question is how to choose a good experimental design. We investigate this by solving a number of problems using different designs, and compare the outcome with respect to function evaluations and a root mean square error test of the true function versus the surrogate model produced. Each combination of problem and design is solved by 3 different solvers available in the TOMLAB optimization environment. Results indicate two designs as superior.
Response surface methods based on kriging and radial basis function (RBF) interpolationhave been successfully applied to solve expensive, i.e. computationally costly,global black-box nonconvex optimization problems.In this paper we describe extensions of these methods to handle linear, nonlinear, and integer constraints. In particular, algorithms for standard RBF and the new adaptive RBF (ARBF) aredescribed. Note, however, while the objective function may be expensive, we assume that any nonlinear constraints are either inexpensive or are incorporated into the objective function via penalty terms. Test results are presented on standard test problems, both nonconvexproblems with linear and nonlinear constraints, and mixed-integernonlinear problems (MINLP). Solvers in the TOMLAB OptimizationEnvironment (http://tomopt.com/tomlab/) have been compared,specifically the three deterministic derivative-free solversrbfSolve, ARBFMIP and EGO with three derivative-based mixed-integernonlinear solvers, OQNLP, MINLPBB and MISQP, as well as the GENOsolver implementing a stochastic genetic algorithm. Results showthat the deterministic derivative-free methods compare well with thederivative-based ones, but the stochastic genetic algorithm solver isseveral orders of magnitude too slow for practical use.When the objective function for the test problems is costly to evaluate, the performance of the ARBF algorithm proves to be superior.
This paper describes an Integer Programming model for generating stable loading patterns for the Pallet Loading Problem under several stability criteria. The results obtained during evaluation show great improvement in the number of stable patterns in comparison with results reported earlier. Moreover, most of the solved cases also ensure optimality in terms of utilization of a pallet.
Response surface methods show promising results for global optimization of costly non convex objective functions, i.e. the problem of finding the global minimum when there are several local minima and each function value takes considerable CPU time to compute. Such problems often arise in industrial and financial applications, where a function value could be a result of a time-consuming computer simulation or optimization. Derivatives are most often hard to obtain. The problem is here extended with linear and nonlinear constraints, and the nonlinear constraints can be costly or not. A new algorithm that handles the constraints, based on radial basis functions (RBF), and that preserves the convergence proof of the original RBF algorithm is presented. The algorithm takes advantage of the optimization algorithms in the Tomlab optimization environment (www.tomlab.biz). Numerical results are presented for standard test problems.
Response surface methods based on kriging and radial basis function (RBF) interpolation have been successfully applied to solve expensive, i.e. com-putationally costly, global black-box nonconvex optimization problems. We describe extensions of these methods to handle linear, nonlinear and integer constraints. In particular standard RBF and new adaptive RBF (ARBF) algorithms are discussed. Test results are presented on standard test problems, both nonconvex problems with linear and nonlinear constraints, and mixed-integer nonlinear problems. Solvers in the TOMLAB Optimization Environment (http://tomopt.com/tomlab/) are compared; the three deterministic derivative-free solvers rbfSolve, ARBFMIP and EGO with three derivative-based mixed-integer nonlinear solvers, OQNLP, MINLPBB and MISQP as well as GENO implementing a stochastic genetic algorithm. Assuming that the objective function is costly to evaluate the performance of the ARBF algorithm proves to be superior.