Canonical Duality Theory

Author : Yi Chen
Publisher :
ISBN 13 :
Total Pages : 250 pages
Book Rating : 4.:/5 (974 download)

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Download or read book Canonical Duality Theory for Global Optimization Problems and Applications written by Yi Chen and published by . This book was released on 2015 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The canonical duality theory is studied, through a discussion on a general global optimization problem and applications on fundamentally important problems. This general problem is a formulation of the minimization problem with inequality constraints, where the objective function and constraints are any convex or nonconvex functions satisfying certain decomposition conditions. It covers convex problems, mixed integer programming problems and many other nonlinear programming problems. The three main parts of the canonical duality theory are canonical dual transformation, complementary-dual principle and triality theory. The complementarydual principle is further developed, which conventionally states that each critical point of the canonical dual problem is corresponding to a KKT point of the primal problem with their sharing the same function value. The new result emphasizes that there exists a one-to-one correspondence between KKT points of the dual problem and of the primal problem and each pair of the corresponding KKT points share the same function value, which implies that there is truly no duality gap between the canonical dual problem and the primal problem. The triality theory reveals insightful information about global and local solutions. It is shown that as long as the global optimality condition holds true, the primal problem is equivalent to a convex problem in the dual space, which can be solved efficiently by existing convex methods; even if the condition does not hold, the convex problem still provides a lower bound that is at least as good as that by the Lagrangian relaxation method. It is also shown that through examining the canonical dual problem, the hidden convexity of the primal problem is easily observable. The canonical duality theory is then applied to dealing with three fundamentally important problems. The first one is the spherically constrained quadratic problem, also referred to as the trust region subproblem. The canonical dual problem is onedimensional and it is proved that the primal problem, no matter with convex or nonconvex objective function, is equivalent to a convex problem in the dual space. Moreover, conditions are found which comprise the boundary that separates instances into 2hard case3 and 2easy case3. A canonical primal-dual algorithm is developed, which is able to efficiently solve the problem, including the 2hard case3, and can be used as a unified method for similar problems. The second one is the binary quadratic problem, a fundamental problem in discrete optimization. The discussion is focused on lower bounds and analytically solvable cases, which are obtained by analyzing the canonical dual problem with perturbation techniques. The third one i is a general nonconvex problem with log-sum-exp functions and quartic polynomials. It arises widely in engineering science and it can be used to approximate nonsmooth optimization problems. The work shows that problems can still be efficiently solved, via the canonical duality approach, even if they are nonconvex and nonsmooth." -- Abstract.