On topology optimization and canonical duality method

Topology optimization for general materials is correctly formulated as a bi-level knapsack problem, which is considered to be NP-hard in global optimization and computer science. By using canonical duality theory (CDT) developed by the author, the linear knapsack problem can be solved analytically to obtain global optimal solution at each design iteration. Both uniqueness, existence, and NP-hardness are discussed. The novel CDT method for general topology optimization is refined and tested by both 2-D and 3-D benchmark problems. Numerical results show that without using filter and any other artificial technique, the CDT method can produce exactly 0-1 optimal density distribution with almost no checkerboard pattern. Its performance and novelty are compared with the popular SIMP and BESO approaches. Additionally, some mathematical and conceptual mistakes in literature are explicitly addressed. A brief review on the canonical duality theory for modeling multi-scale complex systems and for solving general nonconvex/discrete problems are given in Appendix. This paper demonstrates a simple truth: elegant designs come from correct model and theory. © 201

Canonical Duality Theory for Global Optimization problems and applications

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 complementary-dual 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 “hard case” and “easy case”. A canonical primal-dual algorithm is developed, which is able to efficiently solve the problem, including the “hard case”, 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 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.Doctor of Philosoph