Canonical Dual Algorithms for Global Optimization with Applications

Canonical duality theory provides a unified framework which can transform a nonconvex primal minimization problem to a canonical dual maximization problem over a convex domain without duality gap. But the global optimality is guaranteed by a certain positive definite condition and such condition is not always satisfied. The goal of this thesis aims to explore possible techniques that can be used to solve global optimization problems based on the canonical duality theory. Firstly, an algorithmic framework for canonical duality theory is established, which shows that the canonical dual algorithms can be developed in four aspects under the positive definite condition explicitly or implicitly, namely, (i) minimizing the primal problem, (ii) maximizing the canonical dual problem, (iii) solving a nonlinear equation caused by total complementary function, and (iv) solving a nonlinear equation caused by canonical dual function. Secondly, we show that if there exists a critical point of the canonical dual problem in the positive definite domain, by solving an equivalent semidefinite programming (SDP) problem, the corresponding global solution to the primal problem can be obtained easily via off-the-shelf software packages. A specific canonical dual algorithm is given for each problem, including sum of fourth-order polynomials minimization, nonconvex quadratically constrained quadratic program (QCQP), and boolean quadratic program (BQP). Thirdly, we propose a canonical primal-dual algorithm framework based on the total complementary function. Convergence analysis is discussed from the perspective of variational inequalities (VIs) and contraction methods. Specific canonical primal-dual algorithms for sum of fourth-order polynomials minimization is given as well. And a real-world application to the sensor network localization problem is illustrated. Next, a canonical sequential reduction approach is proposed to recover the approximate or global solution for the BQP problem. By fixing some previously known components, the original problem can be reduced sequentially to a lower dimension one. This approach is successfully applied to the well-known maxcut problem. Finally, we discuss the canonical dual approach applied to continuous time constrained optimal control. And it shows that the optimal control law for the n-dimensional constrained linear quadratic regulator can be achieved precisely via one-dimensional canonical dual variable, and for the optimal control problem with concave cost functional, an approximate solution can be obtained by introducing a linear perturbation term.Ph

Mean field approximation for solving QUBO problems

The Quadratic Unconstrained Binary Optimization (QUBO) problem is NP-hard. Some exact methods like the Branch-and-Bound algorithm are suitable for small problems. Some approximations like stochastic simulated annealing for discrete variables or mean-field annealing for continuous variables exist for larger ones, and quantum computers based on the quantum adiabatic annealing principle have also been developed. Here we show that the mean-field approximation of the quantum adiabatic annealing leads to equations similar to those of thermal mean-field annealing. However, a new type of sigmoid function replaces the thermal one. The new mean-field quantum adiabatic annealing can replicate the best-known cut values on some of the popular benchmark Maximum Cut problems

Discrete state transition algorithm for unconstrained integer optimization problems

A recently new intelligent optimization algorithm called discrete state transition algorithm is considered in this study, for solving unconstrained integer optimization problems. Firstly, some key elements for discrete state transition algorithm are summarized to guide its well development. Several intelligent operators are designed for local exploitation and global exploration. Then, a dynamic adjustment strategy "risk and restoration in probability" is proposed to capture global solutions with high probability. Finally, numerical experiments are carried out to test the performance of the proposed algorithm compared with other heuristics, and they show that the similar intelligent operators can be applied to ranging from traveling salesman problem, boolean integer programming, to discrete value selection problem, which indicates the adaptability and flexibility of the proposed intelligent elements. (C) 2015 Elsevier B.V. All rights reserved