3,290 research outputs found
Qualitative Characteristics and Quantitative Measures of Solution's Reliability in Discrete Optimization: Traditional Analytical Approaches, Innovative Computational Methods and Applicability
The purpose of this thesis is twofold. The first and major part is devoted to
sensitivity analysis of various discrete optimization problems while the second
part addresses methods applied for calculating measures of solution stability
and solving multicriteria discrete optimization problems.
Despite numerous approaches to stability analysis of discrete optimization
problems two major directions can be single out: quantitative and qualitative.
Qualitative sensitivity analysis is conducted for multicriteria discrete optimization
problems with minisum, minimax and minimin partial criteria. The main
results obtained here are necessary and sufficient conditions for different stability
types of optimal solutions (or a set of optimal solutions) of the considered
problems.
Within the framework of quantitative direction various measures of solution
stability are investigated. A formula for a quantitative characteristic called
stability radius is obtained for the generalized equilibrium situation invariant
to changes of game parameters in the case of the H¨older metric. Quality of the
problem solution can also be described in terms of robustness analysis. In this
work the concepts of accuracy and robustness tolerances are presented for a
strategic game with a finite number of players where initial coefficients (costs)
of linear payoff functions are subject to perturbations.
Investigation of stability radius also aims to devise methods for its calculation.
A new metaheuristic approach is derived for calculation of stability
radius of an optimal solution to the shortest path problem. The main advantage
of the developed method is that it can be potentially applicable for
calculating stability radii of NP-hard problems.
The last chapter of the thesis focuses on deriving innovative methods based
on interactive optimization approach for solving multicriteria combinatorial
optimization problems. The key idea of the proposed approach is to utilize
a parameterized achievement scalarizing function for solution calculation and
to direct interactive procedure by changing weighting coefficients of this function.
In order to illustrate the introduced ideas a decision making process is
simulated for three objective median location problem.
The concepts, models, and ideas collected and analyzed in this thesis create
a good and relevant grounds for developing more complicated and integrated
models of postoptimal analysis and solving the most computationally challenging
problems related to it.Siirretty Doriast
Mathematical programs with complementarity constraints: convergence properties of a smoothing method
In this paper, optimization problems with complementarity constraints are considered. Characterizations for local minimizers of of Orders 1 and 2 are presented. We analyze a parametric smoothing approach for solving these programs in which is replaced by a perturbed problem depending on a (small) parameter . We are interested in the convergence behavior of the feasible set and the convergence of the solutions of for In particular, it is shown that, under generic assumptions, the solutions are unique and converge to a solution of with a rate . Moreover, the convergence for the Hausdorff distance , between the feasible sets of and is of order
Stochastic on-time arrival problem in transit networks
This article considers the stochastic on-time arrival problem in transit
networks where both the travel time and the waiting time for transit services
are stochastic. A specific challenge of this problem is the combinatorial
solution space due to the unknown ordering of transit line arrivals. We propose
a network structure appropriate to the online decision-making of a passenger,
including boarding, waiting and transferring. In this framework, we design a
dynamic programming algorithm that is pseudo-polynomial in the number of
transit stations and travel time budget, and exponential in the number of
transit lines at a station, which is a small number in practice. To reduce the
search space, we propose a definition of transit line dominance, and techniques
to identify dominance, which decrease the computation time by up to 90% in
numerical experiments. Extensive numerical experiments are conducted on both a
synthetic network and the Chicago transit network.Comment: 29 pages; 12 figures. This manuscript version is made available under
the CC-BY-NC-ND 4.0 license
https://creativecommons.org/licenses/by-nc-nd/4.0
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