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The Modified Quasi-newton Method For Nonconvex Unconstrained Optimization Problems

Posted on:2015-04-04Degree:MasterType:Thesis
Country:ChinaCandidate:X J QuFull Text:PDF
GTID:2180330431955991Subject:Applied Mathematics
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Quasi-Newton method has become one of the most effective methods for solvingunconstrained optimization problems because of its fast convergence and favorablenumerical effect. However some illustrations show that during the process of solvingnon-convex minimization problems the quasi-Newton method does not guaranteeglobal convergence. Accordingly, the improvement on quasi-Newton method, whichaimed to ensure its global convergence while being applied to solve non-convexminimal problems, has been becoming an important issue. In this paper, by the use ofdifferent information of the objective function and on the basis of the modification ofthe quasi-Newton equation, we proposed three types of modifications ofquasi-Newton method and predicted their global convergence while solvingnon-convex minimal problems with these methods.In the first chapter, we introduce some basic knowledge on unconstrainedoptimization,the structure of descent algorithm and some regular linear search. Andthen we also introduce some theories on quasi-Newton method, the research progresson this work. Finally, we introduce the focus and innovation of this paper.In the second and third chapters, based on the second-order Taylor’s expansion ofobjective function and the first-order Taylor’s expansion of gradient function, wepropose a generalized quasi-Newton equations with an argument in [0,1], andtogether with the a modified BFGS-type algorithm and a modified-type DFPalgorithm which is based on the generalized quasi-Newton equation. After that, weanalyze the global convergence of the proposed algorithms and report some numericalexperiments.In the fourth chapter, we analyze the fourth-order Taylor’s expansion by the useof three or four order tensor, and then we propose a tensor-based quasi-Newtonequation and deduce a tensor-based quasi-Newton BFGS-type update formula onquasi-Newton matrices as well as the cautious version of this formulation. At last, wepropose a corresponding algorithm by employing this cautious version. Weestablished the global convergence of the proposed algorithm when the objectivefunction is twice continuously differentiable. Numerical results indicate that thealgorithm has excellent performance.
Keywords/Search Tags:Non-convex unconstrained optimization, modified quasi-Newtonmethod, tensor-based quasi-Newton method, global convergence
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