Canonical Dual Problem Research Articles

On SDP method for solving canonical dual problem in post buckling of large deformed elastic beam

This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

MIMO Detection for High Order QAM by Canonical Dual Approach

We develop a canonical dual approach for solving the MIMO problem. First, a special linear transformation is introduced to reformulate the original problem into a{−1,1}constrained quadratic programming problem. Then, we derive a canonical dual problem which is piecewise continuous problem with no duality gap. Under certain conditions, the canonical problem becomes a concave maximization dual problem over a convex feasible domain. By getting the stationary point of the canonical dual problem, we can find either an optimal or approximate solution of the original problem. A gradient decent algorithm is proposed to solve the MIMO problem and simulation results are provided to demonstrate the effectiveness of the method.

Canonical Duality for Box Constrained Nonconvex and Nonsmooth Optimization Problems

This paper presents an application of the canonical duality theory for box constrained nonconvex and nonsmooth optimization problems. By use of the canonical dual transformation method, which is developed recently, these very difficult constrained optimization problems inRncan be converted into the canonical dual problems, which can be solved by deterministic methods. The global and local extrema can be identified by the triality theory. Some examples are listed to illustrate the applications of the theory presented in the paper.

Global solutions to fractional programming problem with ratio of nonconvex functions

This paper presents a canonical dual approach for minimizing a sum of quadratic function and a ratio of nonconvex functions in Rn. By introducing a parameter, the problem is first equivalently reformed as a nonconvex polynomial minimization with elliptic constraint. It is proved that under certain conditions, the canonical dual is a concave maximization problem in R2 that exhibits no duality gap. Therefore, the global optimal solution of the primal problem can be obtained by solving the canonical dual problem.

Global solutions to a class of CEC benchmark constrained optimization problems

This paper aims to solve a class of CEC benchmark constrained optimization problems that have been widely studied by nature-inspired optimization algorithms. Based on canonical duality theory, these challenging problems can be reformulated as a unified canonical dual problem over a convex set, which can be solved deterministically to obtain global optimal solutions in polynomial time. Applications are illustrated by some well-known CEC benchmark problems, and comparisons with other methods have demonstrated the effectiveness of the proposed approach.

Global optimal solutions to general sensor network localization problem

Sensor network localization problem is to determine the position of the sensor nodes in a network given pairwise distance measurements. Such problem can be formulated as a quartic polynomial minimization via the least squares method. This paper presents a canonical duality theory for solving this challenging problem. It is shown that the nonconvex minimization problem can be reformulated as a concave maximization dual problem over a convex set in a symmetrical matrix space, and hence can be solved efficiently by combining a general (linear or quadratic) perturbation technique with existing optimization techniques. Applications are illustrated by solving some relatively large-scale problems. Our results show that the general sensor network localization problem is not NP-hard unless its canonical dual problem has no solution in its positive definite domain. Fundamental ideas for solving general NP-hard problems are discussed.

Canonical dual solutions to nonconvex radial basis neural network optimization problem

Radial Basis Functions Neural Networks (RBFNNs) are tools widely used in regression problems. One of their principal drawbacks is that the formulation corresponding to the training with the supervision of both the centers and the weights is a highly non-convex optimization problem, which leads to some fundamental difficulties for the traditional optimization theory and methods. This paper presents a generalized canonical duality theory for solving this challenging problem. We demonstrate that by using sequential canonical dual transformations, the nonconvex optimization problem of the RBFNN can be reformulated as a canonical dual problem (without duality gap). Both global optimal solution and local extrema can be classified. Several applications to one of the most used Radial Basis Functions, the Gaussian function, are illustrated. Our results show that even for a one-dimensional case, the global minimizer of the nonconvex problem may not be the best solution to the RBFNNs, and the canonical dual theory is a promising tool for solving general neural networks training problems.

Canonical primal–dual algorithm for solving fourth-order polynomial minimization problems

This paper focuses on implementation of a general canonical primal–dual algorithm for solving a class of fourth-order polynomial minimization problems. A critical issue in the canonical duality theory has been addressed, i.e., in the case that the canonical dual problem has no interior critical point in its feasible space Sa+, a quadratic perturbation method is introduced to recover the global solution through a primal–dual iterative approach, and a gradient-based method is further used to refine the solution. A series of test problems, including the benchmark polynomials and several instances of the sensor network localization problems, have been used to testify the effectiveness of the proposed algorithm.

Resource Allocation and Adaptive Modulation in Uplink SC-FDMA Systems

In this paper, we study joint power and sub-channel allocation, and adaptive modulation in Single Carrier Frequency Division Multiple Access (SC-FDMA) which is adopted as the multiple access scheme for the uplink in the 3GPP-LTE standard. A sum-utility maximization problem is considered. Unlike OFDMA, in addition to the restriction of allocating a sub-channel to one user at most, the multiple sub-channels allocated to a user in SC-FDMA should be consecutive as well. This renders the resource allocation problem prohibitively difficult and the standard optimization tools (e.g., Lagrange dual approach widely used for OFDMA, etc.) can not help towards its optimal solution. We propose a novel optimization framework for the solution of this problem which is inspired from the recently developed canonical duality theory. We first formulate the optimization problem as binary-integer programming problem, and then transform this binary-integer programming problems into a continuous space canonical dual problem that is a concave maximization problem. Based on the solution of the continuous space dual problem, we derive joint power and sub-channel allocation algorithm whose computational complexity is polynomial. We provide conditions under which the proposed algorithms are optimal. We also propose an adaptive modulation scheme which selects an appropriate modulation strategy for each user. We compare the proposed algorithm with the existing algorithms in the literature to assess their performance. The results show a tremendous performance gain.

A joint resource optimization and adaptive modulation framework for uplink single‐carrier frequency‐division multiple access systems

SummaryIn this paper, we study joint resource allocation and adaptive modulation in single‐carrier frequency‐division multiple access systems, which is adopted as the multiple access scheme for the uplink in the 3GPP Long Term Evolution standard. We formulate an adaptive modulation and sum‐cost minimization (JAMSCmin) problem. Unlike orthogonal frequency‐division multiple access, in addition to the restriction of allocating a subchannel to one user at most, the multiple subchannels allocated to a user in single‐carrier frequency‐division multiple access systems should be consecutive as well. This renders the resource allocation problem prohibitively difficult and the standard optimization tools (e.g., Lagrange dual approach widely used for orthogonal frequency‐division multiple access, etc.) cannot help towards its optimal solution. We propose a novel optimization framework for the solution of this problem that is inspired from the recently developed canonical duality theory. We first formulate the optimization problem as binary‐integer programming (BIP) problem and then transform this BIP problem into continuous space canonical dual problem that is the concave maximization problem. Based on the solution of the canonical dual problem, we derive joint resource allocation and adaptive modulation algorithm, which has polynomial time complexity. We provide conditions under which the proposed algorithm is optimal. We compare the proposed algorithm with the existing algorithms in the literature. The results show a tremendous performance gain. Copyright © 2013 John Wiley & Sons, Ltd.

Extended canonical duality and conic programming for solving 0-1 quadratic programming problems

An extended canonical dual approach for solving 0-1 quadraticprogramming problems is introduced. We derive the relationshipbetween the optimal solutions to the extended canonical dual problemand the original problem and prove that there exists no duality gapin-between. The extended canonical dual approach leads to asufficient condition for global optimality, which is more generalthan known results of this kind. To solve the extended canonicaldual problem, we construct corresponding conic programming problemsand study their relationship to the extended canonical dual problem.Using this relationship, we design an algorithm for solving theextended canonical dual problem. Our work extends the known solvablesub-class of 0-1 quadratic programming problems.

Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality

This paper presents a canonical duality theory for solving a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation developed by the first author, the nonconvex primal problem can be converted into a canonical dual problem with zero duality gap. A general analytical solution form is obtained. Both global and local extrema of the nonconvex problem can be identified by the triality theory associated with the canonical duality theory. 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. Criteria for the existence and uniqueness of optimal solutions are presented. Several numerical examples are provided.

Canonical duality for solving nonconvex and nonsmooth optimization problem

This paper presents an application of the canonical duality theory for solving a class of nonconvex and nonsmooth optimization problems. It is shown that by use of the canonical dual transformation, these difficult optimization problems in Rn can be converted into a one-dimensional canonical dual problems, which can be solved to obtain all extremal points. Both global and local extremality conditions can be identified by the triality theory. Applications are illustrated.

Minimal distance between two non-convex surfaces

This article presents a global optimal solution to a quadratic minimization problem subjected to non-convex constraints. By using the canonical dual transformation developed by the first author, a canonical dual problem (with zero duality gap) is formulated which can be solved easily to obtain all critical points. The global minimizer is uniquely identified by the canonical duality theory. Applications are illustrated. The method and theory developed in this article can be generalized for solving general global optimization problems subjected to non-convex constraints.

Optimal solutions to a class of nonconvex minimization problems with linear inequality constraints

This paper presents a canonical duality theory and optimal solutions to a class of global optimization problems subjected to linear inequality constraints. By using the canonical dual transformation developed recently, a canonical dual problem is formulated, which is perfectly dual to the primal problem. The global minmizer can be identified by the triality theory. Results show that if the global extrema of the original problem are located on the boundary of the primal feasible space, the dual solution should be interior point of the dual feasible set. Several examples are illustrated to show how this theory works.

Solutions and optimality criteria for nonconvex quadratic-exponential minimization problem

This paper presents a set of complete solutions and optimality conditions for a nonconvex quadratic-exponential optimization problem. By using the canonical duality theory developed by the first author, the nonconvex primal problem in n-dimensional space can be converted into an one-dimensional canonical dual problem with zero duality gap, which can be solved easily to obtain all dual solutions. Each dual solution leads to a primal solution. Both global and local extremality conditions of these primal solutions can be identified by the triality theory associated with the canonical duality theory. Several examples are illustrated.

Canonical dual approach to solving 0-1 quadratic programming problems

By using the canonical dual transformation developed recently, we derive a pair of canonical dual problems for 0-1 quadratic programming problems in both minimization and maximization form. Regardless convexity, when the canonical duals are solvable, no duality gap exists between the primal and corresponding dual problems. Both global and local optimality conditions are given. An algorithm is presented for finding global minimizers, even when the primal objective function is not convex. Examples are included to illustrate this new approach.

Solutions and optimality criteria to box constrained nonconvex minimization problems

This paper presents a canonical duality theory for solvingnonconvex polynomial programming problems subjected to boxconstraints. It is proved that under certain conditions, the constrained nonconvex problems can be converted to the so-called canonical (perfect) dual problems, which can be solved by deterministic methods.Both global and local extrema of the primal problems can be identified by a triality theory proposed by the author. Applications to nonconvex integer programming and Boolean least squares problems are discussed. Examples are illustrated. A conjecture on NP-hard problems is proposed.

Complete Solutions and Extremality Criteria to Polynomial Optimization Problems

This paper presents a set of complete solutions to a class of polynomial optimization problems. By using the so-called sequential canonical dual transformation developed in the author's recent book [Gao, D.Y. (2000), Duality Principles in Nonconvex Systems: Theory, Method and Applications, Kluwer Academic Publishers, Dordrecht/Boston/London, xviii + 454 pp], the nonconvex polynomials in $$\mathbb{R}^n$$ can be converted into an one-dimensional canonical dual optimization problem, which can be solved completely. Therefore, a set of complete solutions to the original problem is obtained. Both global minimizer and local extrema of certain special polynomials can be indentified by Gao-Strang's gap function and triality theory. For general nonconvex polynomial minimization problems, a sufficient condition is proposed to identify global minimizer. Applications are illustrated by several examples.

Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints

This paper presents a duality theory for solving concave minimization problem and nonconvex quadratic programming problem subjected to nonlinear inequality constraints. By use of the canonical dual transformation developed recently, two canonical dual problems are formulated, respectively. These two dual problems are perfectly dual to the primal problems with zero duality gap. It is proved that the sufficient conditions for global minimizers and local extrema (both minima and maxima) are controlled by the triality theory discovered recently [5]. This triality theory can be used to develop certain useful primal-dual methods for solving difficult nonconvex minimization problems. Results shown that the difficult quadratic minimization problem with quadratic constraint can be converted into a one-dimensional dual problem, which can be solved completely to obtain all KKT points and global minimizer.