Curtin University Homepage
  • Library
  • Help
    • Admin

    espace - Curtin’s institutional repository

    JavaScript is disabled for your browser. Some features of this site may not work without it.
    View Item 
    • espace Home
    • espace
    • Curtin Theses
    • View Item
    • espace Home
    • espace
    • Curtin Theses
    • View Item

    Global optimization for nonconvex optimization problems

    191768_Ruan2013.pdf (1.679Mb)
    Access Status
    Open access
    Authors
    Ruan, Ning
    Date
    2012
    Supervisor
    Prof. Kok Lay Teo
    Prof. David Gao
    Type
    Thesis
    Award
    PhD
    
    Metadata
    Show full item record
    School
    Department of Mathematics and Statistics
    URI
    http://hdl.handle.net/20.500.11937/1936
    Collection
    • Curtin Theses
    Abstract

    Duality is one of the most successful ideas in modern science [46] [91]. It is essential in natural phenomena, particularly, in physics and mathematics [39] [94] [96]. In this thesis, we consider the canonical duality theory for several classes of optimization problems.The first problem that we consider is a general sum of fourth-order polynomial minimization problem. This problem arises extensively in engineering and science, including database analysis, computational biology, sensor network communications, nonconvex mechanics, and ecology. We first show that this global optimization problem is actually equivalent to a discretized minimal potential variational problem in large deformation mechanics. Therefore, a general analytical solution is proposed by using the canonical duality theory.The second problem that we consider is a nonconvex quadratic-exponential optimization problem. By using the canonical duality theory, the nonconvex primal problem in n-dimensional space can be converted into a one-dimensional canonical dual problem, which is either a concave maximization or a convex minimization problem with zero duality gap. Several examples are solved so as to illustrate the applicability of the theory developed.The third problem that we consider is quadratic minimization problems subjected to either box or integer constraints. Results show that these nonconvex problems can be converted into concave maximization dual problems over convex feasible spaces without duality gap and the Boolean integer programming problem is actually equivalent to a critical point problem in continuous space. These dual problems can be solved under certain conditions. Both existence and uniqueness of the canonical dual solutions are presented. A canonical duality algorithm is presented and applications are illustrated.The fourth problem that we consider is a quadratic discrete value selection problem subjected to inequality constraints. The problem is first transformed into a quadratic 0-1 integer programming problem. The dual problem is thus constructed by using the canonical duality theory. Under appropriate conditions, this dual problem is a maximization problem of a concave function over a convex continuous space. Theoretical results show that the canonical duality theory can either provide a global optimization solution, or an optimal lower bound approximation to this NP-hard problem. Numerical simulation studies, including some relatively large scale problems, are carried out so as to demonstrate the effectiveness and efficiency of the canonical duality method. An open problem for understanding NP-hard problems is proposed.The fifth problem that we consider is a mixed-integer quadratic minimization problem with fixed cost terms. We show that this well-known NP-hard problem in R2n can be transformed into a continuous concave maximization dual problem over a convex feasible subset of Rn with zero duality gap. We also discuss connections between the proposed canonical duality theory approach and the classical Lagrangian duality approach. The resulting canonical dual problem can be solved under certain conditions, by traditional convex programming methods. Conditions for the existence and uniqueness of global optimal solutions are presented. An application to a decoupled mixed-integer problem is used to illustrate the derivation of analytic solutions for globally minimizing the objective function. Numerical examples for both decoupled and general mixed-integral problems are presented, and an open problem is proposed for future study.The sixth problem that we consider is a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation, the nonconvex primal problem can be converted into a canonical dual problem (i.e., either a concave maximization problem with zero duality gap). Illustrative applications to quadratic minimization with multiple quadratic constraints, box/integer constraints, and general nonconvex polynomial constraints are discussed, along with insightful connections to classical Lagrangian duality. Conditions for ensuring the existence and uniqueness of global optimal solutions are presented. Several numerical examples are solved.The seventh problem that we consider is a general nonlinear algebraic system. By using the least square method, the nonlinear system of m quadratic equations in n-dimensional space is first formulated as a nonconvex optimization problem. We then prove that, by using the canonical duality theory, this nonconvex problem is equivalent to a concave maximization problem in Rm, which can be solved by well-developed convex optimization techniques. Both existence and uniqueness of global optimal solutions are discussed, and several illustrative examples are presented.The eighth problem that we consider is a general sensor network localization problem. It is shown that by the canonical duality theory, this nonconvex minimization problem is equivalent to a concave maximization problem over a convex set in a symmetrical matrix space, and hence can be solved by combining a perturbation technique with existing optimization techniques. Applications are illustrated and results show that the proposed method is potentially a powerful one for large-scale sensor network localization problems.

    Related items

    Showing items related by title, author, creator and subject.

    • Iterative algorithms for envelope-constrained filter design.
      Tseng, Chien H. (1999)
      The design of envelope-constrained (EC) filters is considered for the time-domain synthesis of filters for signal processing problems. The objective is to achieve minimal noise enhancement where the shape of the filter ...
    • Optimal control problems with constraints on the state and control and their applications
      Li, Bin (2011)
      In this thesis, we consider several types of optimal control problems with constraints on the state and control variables. These problems have many engineering applications. Our aim is to develop efficient numerical methods ...
    • Joint pricing and production planning of multiple products
      Mardaneh, Elham (2010)
      Many industries are beginning to use innovative pricing techniques to improve inventory control, capacity utilisation, and ultimately the profit of the firm. In manufacturing, the coordination of pricing and production ...
    Advanced search

    Browse

    Communities & CollectionsIssue DateAuthorTitleSubjectDocument TypeThis CollectionIssue DateAuthorTitleSubjectDocument Type

    My Account

    Admin

    Statistics

    Most Popular ItemsStatistics by CountryMost Popular Authors

    Follow Curtin

    • 
    • 
    • 
    • 
    • 

    CRICOS Provider Code: 00301JABN: 99 143 842 569TEQSA: PRV12158

    Copyright | Disclaimer | Privacy statement | Accessibility

    Curtin would like to pay respect to the Aboriginal and Torres Strait Islander members of our community by acknowledging the traditional owners of the land on which the Perth campus is located, the Whadjuk people of the Nyungar Nation; and on our Kalgoorlie campus, the Wongutha people of the North-Eastern Goldfields.