90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
- 90-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
- 90-01 Instructional exposition (textbooks, tutorial papers, etc.)
- 90-02 Research exposition (monographs, survey articles)
- 90-03 Historical (must also be assigned at least one classification number from Section 01)
- 90-04 Explicit machine computation and programs (not the theory of computation or programming)
- 90-06 Proceedings, conferences, collections, etc.
- 90-08 Computational methods
90Bxx Operations research and management science
90Cxx Mathematical programming [See also 49Mxx, 65Kxx]
Computational Bounds for Elevator Control Policies by Large Scale Linear Programming
- We computationally assess policies for the elevator control problem by a new column-generation approach for the linear programming method for discounted infinite-horizon Markov decision problems. By analyzing the optimality of given actions in given states, we were able to provably improve the well-known nearest-neighbor policy. Moreover, with the method we could identify an optimal parking policy. This approach can be used to detect and resolve weaknesses in particular policies for Markov decision problems.
New Trust Region SQP Methods for Continuous and Integer Optimization
- In this thesis new algorithms are presented that address nonlinear optimization problems. The algorithms belong to the class of sequential quadratic programming (SQP) methods. Two problem formulations that arise frequently in real-world applications are considered. Both have in common that functions are nonlinear and the formulations contain equality and inequality constraints. For the one class of problems the domain of all variables is R. These problems are called nonlinear programming (NLP) problems. Many applications also require that some of the featured variables are restricted to the domain Z. Problems with additional integer variables are called mixed-integer nonlinear programs (MINLP) and are also considered here.
This work is motivated by the advancement of an algorithm for solving MINLPs that was first discussed by Exler and Schittkowski (2007). The algorithm adapts concepts of SQP methods to mixed-integer nonlinear optimization. The new approach replaces the continuous quadratic problems by mixed-integer quadratic problems. The aim is to profit from the fast local convergence properties of SQP methods at least with respect to the continuous variables when integer variables remain fixed. Two new versions of the underlying algorithm of Exler and Schittkowski are presented.
It is well-known that SQP methods might not converge for any arbitrary starting point. To obtain global convergence, techniques of trust region methods are employed by the new algorithms. The first version of an algorithm for MINLPs presented in this thesis employs the Linf-penalty function as merit function. Applying this penalty function might lead to a slow convergence. The so-called Maratos effect requires the reduction of the step length so that fast convergence is lost. Hence, safeguards have to be added. The presented algorithm calculates additional second order correction (SOC) steps. Calculating SOC steps is a frequently used approach to obtain fast local convergence. There also exist other techniques. The SOC steps require additional function evaluations. Frequently, function values of mixed-integer problems arising in the field of engineering are evaluated by running time-consuming simulation tools, where a single function evaluation can take minutes or even hours. Thus, the goal of the development of an efficient method has to be that the number of needed function evaluations is as small as possible.
For that reason the investigation of methods that obtain fast local convergence without calculating SOC steps is the key aspect of this thesis. As a fundamental theory is available for NLPs, whereas MINLPs lack in most of these concepts, the main part of this thesis presents and analyzes a new trust region SQP algorithm addressing NLPs. The algorithm proposed here avoids the calculation of SOC steps by using an augmented Lagrangian function as merit function. In trust region methods a differentiable merit function, such as an augmented Lagrangian function, was employed in the past for equality constrained problems. Methods that also treat inequality constraints, transform these constraints into equality constraints. The new algorithm does not reformulate the underlying problem.
The proposed algorithm for NLPs is described in detail. The global and local convergence properties of the new algorithm are investigated. Under suitable assumptions it is shown that for any arbitrary starting point the sequence of generated iterates contains at least one accumulation point that is a Karush-Kuhn-Tucker point of the underlying NLP. Under certain conditions fast local convergence is proved, as full SQP steps will be accepted close to a solution. Thus, no additional SOC steps are required.
Due to the insight that is gained by the development of the algorithm for NLPs, an additional version of the algorithm for MINLPs can be stated. The second algorithm also enhances the algorithm of Exler and Schittkowski (2007), but does not calculate SOC steps anymore and the extra function evaluations are avoided.
All presented algorithms are implemented in FORTRAN and completely documented. The code is evaluated on a set of almost 500 test problems. Numerical results show the good performance of the new algorithms. The numerical tests of the algorithm for NLPs indicate that the theoretical convergence results hold in practice. Moreover, the efficiency of the second algorithm for MINLPs that does not calculate SOC steps has improved compared to the first version with SOC steps
An exact column-generation approach for the lot-type design problem
- We consider a fashion discounter distributing its many branches with
integral multiples from a set of available lot-types. For the problem of
approximating the branch and size dependent demand using those lots
we propose a tailored exact column generation approach assisted by fast
algorithms for intrinsic subproblems, which turns out to be very efficient
on our real-world instances.
The Integrated Size and Price Optimization problem
- We present the Integrated Size and Price Optimization Problem (ISPO)
for a fashion discounter with many branches. Based on a two-stage
stochastic programming model with recourse, we develop an exact
algorithm and a production-compliant heuristic that produces small
optimality gaps. In a field study we show that a distribution of
supply over branches and sizes based on ISPO solutions is
significantly better than a one-stage optimization of the
distribution ignoring the possibility of optimal pricing.