Chen Disser (1121212)
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SOLVING NONLINEAR CONSTRAINED OPTIMIZATION PROBLEMSTHROUGH CONSTRAINT PARTITIONINGBYYIXIN CHENB.S., University of Science and Technology of China, 1999M.S., University of Illinois at Urbana-Champaign, 2001c 2005 by Yixin Chen. All rights reserved.DISSERTATIONSubmitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Computer Sciencein the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2005Urbana, Illinoiscontinuous, and mixed spaces.
It facilitates constraint partitioning by providing a set ofnecessary conditions, one for each subproblem, to characterize the local optima. It furtherAbstractreduces the complexity by defining a much smaller search space in each subproblem forbacktracking. Since resolving the global constraints only incurs a small amount of overhead,our approach leads to a significant reduction of complexity.Our partition-and-resolve approach has achieved substantial improvements over existingIn this dissertation, we propose a general approach that can significantly reduce the complexity in solving discrete, continuous, and mixed constrained nonlinear optimization (NLP)problems.
A key observation we have made is that most application-based NLPs have structured arrangements of constraints. For example, constraints in AI planning are often localized into coherent groups based on their corresponding subgoals. In engineering designproblems, such as the design of a power plant, most constraints exhibit a spatial structurebased on the layout of the physical components. In optimal control applications, constraintsare localized by stages or time.methods in solving AI planning and mathematical programming problems. In this dissertation, we present SGPlan, a planner based on constraint partitioning that has significantlyimproved the solution time and quality on many application domains when compared toother leading planners.
We also describe our implementation that has successfully incorporated ESP with ASPEN, a planning system for spacecraft exploration developed by NASA.The ESP planner performs 100 to 1000 times faster than the original ASPEN on NASA’sbenchmarks and can generate solutions of much higher quality.Constraint partitioning has led to a major breakthrough in solving mathematical pro-We have developed techniques to exploit these constraint structures by partitioning theconstraints into subproblems related by global constraints. Constraint partitioning leads tomuch relaxed subproblems that are significantly easier to solve.
However, there exist globalconstraints relating multiple subproblems that must be resolved. Previous methods cannotexploit such structures using constraint partitioning because they cannot resolve inconsistentgramming problems in operations research and engineering applications. In this dissertation,we have applied our method to solve some large-scale continuous and mixed-integer NLPsin standard benchmarks. We have solved some large-scale problems that were not solvableby other leading optimization packages and have improved the solution quality on manyproblems.global constraints efficiently.We have developed a mathematical theory that provides strong necessary and sufficientanalytical conditions for limiting the subspace to be searched when resolving the globalconstraints. We have proposed the theory of extended saddle points (ESP) to supportconstraint partitioning when solving NLPs.
Based on a novel penalty formulation, ESPoffers a necessary and sufficient condition for constrained local optima of NLPs in discrete,iiiivAcknowledgmentsI would like to thank my advisor, Professor Benjamin W. Wah, for his guidance during thecourse of my graduate study.
He taught me how to find important subjects, define suitableresearch problems, solve problems, and present the results in an informative and interestingway. The problem solving skills I learned from him will be invaluable for my career as ascientist. He also taught me to be confident, persistent, and hard-working in performingresearch.I would like to thank Professors Jiawei Han, Grigore Rosu, and Martin Wong for servingTo my wife, Juan, and my parentson my Ph.D. committee and for providing many useful comments and suggestions.
I wouldlike to thank Professor C. V. Ramamoorthy for his encouragement and support during mygraduate study.I would also like to thank Dr. Tao Wang, Dr. Xiao Su, Dr. Zhe Wu, Dr. Dong Lin, Dr.Minglun Qian, Dr. Honghai Zhang, Hang Yu, Dong Xin, Chih-Wei Hsu, Batu Sat, MarkRichard, and all other members in our research group for providing insightful comments onthe work and for providing a friendly environment for me to work in.I wish to thank my wife, Juan, and my parents for their love and support.Finally, I would like to acknowledge the support of the National Science FoundationGrant IIS 03-12084 and National Aeronautics and Space Administration Grant NCC 2-481.vviTable of Contents2.3.1Discrete-time discrete-state methods .
. . . . . . . . . . . . . . . . .392.3.2Discrete-time mixed-state methods . . . . . . . . . . . . . . . . . . .402.3.3Continuous-time mixed-state methods . . . . . . . . . . . . . . . . .402.3.4Remarks on existing planning methods . . . . . . . . . . . . . . . . .41Chapter 33.1List of Tables . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiChapter 1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.1Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2Observations on Constraint Structure . . . . . . .
. . . . . . . . . . . . . . .31.2.1Example I: The TRIMLON Benchmark . . . . . . . . . . . . . . . . .31.2.2Example II: The AIRPORT Planning Problem . . . . . . . . . . . . .71.3Research Goal and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .111.4Contributions and Significance of Research . . . . . . . . . . . . . . . .
. . .131.5Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15Chapter 22.1Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Existing Penalty Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .182.1.1Global optimal penalty methods . . . . . . . . . . . . . . . .
. . . . .202.1.2Local optimal penalty methods . . . . . . . . . . . . . . . . . . . . .25Existing Partitioning Methods for Mathematical Programming . . . . . . . .302.2.1Subspace partitioning methods . . . . . . . . . . . . . . . . . . . . .312.2.2Separable programming methods . . . . . . . .
. . . . . . . . . . . .352.2.3Remarks on existing partitioning methods . . . . . . . . . . . . . . .38Existing Planning Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . .383.23.33.4Necessary and Sufficient Extended Saddle-Point Condition . . . . . . . . . .443.1.1ESPC for continuous optimization . . . . . . . . . .
. . . . . . . . . .453.1.2ESPC for discrete optimization . . . . . . . . . . . . . . . . . . . . .493.1.3ESPC for mixed optimization . . . . . . . . . . . . . . . . . . . . . .523.1.4Search procedures for finding extended saddle points . . . . . . . . .56ESPC under Constraint Partitioning . . . .
. . . . . . . . . . . . . . . . . .593.2.1Basic definitions for partitioned MINLPs . . . . . . . . . . . . . . . .593.2.2Necessary and sufficient ESPC for partitioned subproblems . . . . . .603.2.3The partition-and-resolve procedure . . . . . . . . . . . . . . . . . . .63Asymptotic Convergence of Stochastic Partitioned Search . . . . . . . .
. . .643.3.1Constraint Partitioned Simulated Annealing (CPSA) Algorithm . . .663.3.2Asymptotic convergence of CPSA . . . . . . . . . . . . . . . . . . . .71Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82Chapter 44.12.24.22.3viiTheory of Extended Saddle Points . . . .
. . . . . . . . . . . . . 44Applications on Planning . . . . . . . . . . . . . . . . . . . . . . . 84Applications on PDDL2.2 Planning . . . . . . . . . . . . . . . . . . . . . . .844.1.1Locality of mutual-exclusion constraints in temporal planning . .
. .854.1.2System architecture of SGPlang . . . . . . . . . . . . . . . . . . . . .954.1.3Updating the penalties of violated global constraints . . . . . . . . . 1034.1.4Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Applications on ASPEN Planning . . . . . . . . . .
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