On quadratically constrained quadratic optimization problems and canonical duality theory

The following article has been included in a multiple retraction: Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems; Ning Ruan, David Y Gao. http://mms.sagepub.com/content/early/2015/07/08/1081286515591087.abstract In 2015 SAGE were made aware of concerns regarding the Special Issue of Mathematics & Mechanics of Solids on Advances in Canonical Duality Theory, guest-edited by Professor David Gao. At the request of the Guest Editor, the Special Issue has been retracted, due to conflict of interest regarding Professor Gao’s role as Guest Editor and co-author on a number of submitted papers. In addition the peer review process was less rigorous than the journal requires. The Guest Editor takes full responsibility for the retraction. The following articles that were due to appear in the Special Issue have therefore been retracted: Canonical Duality-Triality: Bridge between Nonconvex Analysis/Mechanics and Global Optimization in complex systems; David Y Gao, Ning Ruan, and Vittorio Latorre. http://mms.sagepub.com/content/early/2015/02/24/1081286514566533.abstract Canonical Dual Approach for Contact Mechanics Problems with Friction; Vittorio Latorre, Simone Sagratella, David Y Gao. http://mms.sagepub.com/content/early/2015/01/20/1081286514566534.abstract Canonical Duality Theory for Solving Non-Monotone Variational Inequality Problems; Guoshan Liu, David Y Gao, Shouyang Wang. http://mms.sagepub.com/content/early/2015/02/04/1081286514566535.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part I; Shu-Cherng Fang, David Y Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wen-Xun Xing. http://mms.sagepub.com/content/early/2015/02/24/1081286514566704.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part II; Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. http://mms.sagepub.com/content/early/2015/02/09/1081286514566723.abstract Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant-Kirchhoff Material; David Y Gao and E. Hajilarov. http://mms.sagepub.com/content/early/2015/07/06/1081286515591084.abstract Triality Theory and Complete Post-buckling Solutions of Large Deformed Beam by Canonical Dual Finite Element Method; Kun Cai, David Y Gao, Qinghua Qin. http://mms.sagepub.com/content/early/2015/06/28/1081286515591085.abstract Global Solutions to Spherically Constrained Quadratic Minimization via Canonical Duality Theory; Yi Chen, David Y Gao. http://mms.sagepub.com/content/early/2015/04/08/1081286515577122.abstract Unified Canonical Duality Methodology for Global Optimization; Vittorio Latorre, David Y Gao and N. Ruan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591305.abstract A Framework of Canonical Dual Algorithms for Global Optimization; Xiaojun Zhou, David Y Gao, Chunhua Yang. http://mms.sagepub.com/content/early/2015/07/22/1081286515592190.abstract Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems; Ning Ruan, David Y Gao. http://mms.sagepub.com/content/early/2015/07/08/1081286515591087.abstract Global Optimization Solutions to a Class of Non-convex Quadratic Minimization Problems with Quadratic Constraints; Yubo Yuan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591086.abstract On Minimal Distance between Two Non-Convex Surfaces; Daniel Morales-Silva, David Y Gao. http://mms.sagepub.com/content/early/2015/07/27/1081286515592949.abstract The Editor-in-Chief and SAGE strive to uphold the very highest standards of publication ethics and are committed to supporting the high standards of integrity of Mathematics & Mechanics of Solids. Authors, reviewers, editors and interested readers are encouraged to consult SAGE’s ethics statements and the Committee on Publication Ethics (COPE) website for guidelines on publication ethics.

D.Y. Gao together with some of his collaborators applied his Canonical duality theory (CDT) for solving a class of constrained optimization problems. Unfortunately, in several papers on this subject there are unclear statements, not convincing proofs, or even false results. It is our aim in this work to study rigorously this class of constrained optimization problems in finite dimensional spaces and to point out several false results published in the last ten years.

This chapter presents a comprehensive review and some new developments on canonical duality theory for nonconvex systems. Based on a tricanonical form for quadratic minimization problems, an insightful relation between canonical dual transformations and nonlinear (or extended) Lagrange multiplier methods is presented. Connections between complementary variational principles in nonconvex mechanics and Lagrange duality in global optimization are also revealed within the framework of the canonical duality theory. Based on this framework, traditional saddle Lagrange duality and the so-called biduality theory, discovered in convex Hamiltonian systems and d.c. programming, are presented in a unified way; together, they serve as a foundation for the triality theory in nonconvex systems. Applications are illustrated by a class of nonconvex problems in continuum mechanics and global optimization. It is shown that by the use of the canonical dual transformation, these nonconvex constrained primal problems can be converted into certain simple canonical dual problems, which can be solved to obtain all extremal points. Optimality conditions (both local and global) for these extrema can be identified by the triality theory. Some new results on general nonconvex programming with nonlinear constraints are also presented as applications of this canonical duality theory. This review brings some fundamentally new insights into nonconvex mechanics, global optimization, and computational science.

Canonical duality theory (CDT) is presented by its creator DY Gao as a theory which can be used for solving a large class of challenging real-world problems. It is the aim of this paper to study rigorously constrained optimization problems in finite dimensional spaces using the method suggested by CDT and to discuss several results published in the last ten years.

A novel canonical duality theory (CDT) is presented for solving general bilevel mixed integer nonlinear optimization governed by linear and quadratic knapsack problems. It shows that the challenging knapsack problems can be solved analytically in term of their canonical dual solutions. The existence and uniqueness of these analytical solutions are proved. NP-hardness of the knapsack problems is discussed. A powerful CDT algorithm combined with an alternative iteration and a volume reduction method is proposed for solving the NP-hard bilevel knapsack problems. Application is illustrated by benchmark problems in optimal topology design. The performance and novelty of the proposed method are compared with the popular commercial codes.

This paper presents a detailed study on analytical solutions to a general nonlinear boundary-value problem in finite deformation theory. Based on canonical duality theory and the associated pure complementary energy principle in nonlinear elasticity proposed by Gao in (Mech Res Commun 26:31–37, 1999, [6], Wiley Encyclopedia of Electrical and Electronics Engineering, 1999, [7], Meccanica 34:169–198, 1999, [8]), we show that the general nonlinear partial differential equation for deformation is actually equivalent to an algebraic (tensor) equation in stress space. For St Venant–Kirchhoff materials, this coupled cubic algebraic equation can be solved principally to obtain all possible solutions. Our results show that for any given external source field such that the statically admissible first Piola–Kirchhoff stress field has nonzero eigenvalues, the problem has a unique global minimal solution, which is corresponding to a positive-definite second Piola–Kirchhoff stress \(\mathbf{T}\), and at most eight local solutions corresponding to negative-definite \(\mathbf{T}\). Additionally, the problem could have 15 unstable solutions corresponding to indefinite \(\mathbf{T}\). This paper demonstrates that the canonical duality theory and the pure complementary energy principle play fundamental roles in nonconvex analysis and finite deformation theory.

This paper presents a canonical dual approach for solving a class of non-monotone variational inequality problems. It shows that by using the canonical dual transformation, these challenging problems can be reformulated as a canonical dual problem, which is equivalent to the primal problems in the sense that they have the same set of KKT points. Existence theorem for global optimal solutions is obtained. Based on the canonical duality theory, this dual problem can be solved via well-developed convex programming methods. Applications are illustrated with several examples.

A canonical d.c. (difference of canonical and convex functions) programming problem is proposed, which can be used to model general global optimization problems in complex systems. It shows that by using canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open problems are presented.

Recently, energy efficiency in wireless networks has become an important objective. Aside from the growing proliferation of smartphones and other high-end devices in conventional human-to-human (H2H) communication, the introduction of machine-to-machine (M2M) communication or machine-type communication into cellular networks is another contributing factor. In this paper, we investigate quality-of-service (QoS)-driven energy-efficient design for the uplink of long term evolution (LTE) networks in M2M/H2H co-existence scenarios. We formulate the resource allocation problem as a maximization of effective capacity-based bits-per-joule capacity under statistical QoS provisioning. The specific constraints of single carrier frequency division multiple access (uplink air interface in LTE networks) pertaining to power and resource block allocation not only complicate the resource allocation problem, but also render the standard Lagrangian duality techniques inapplicable. We overcome the analytical and computational intractability by first transforming the original problem into a mixed integer programming (MIP) problem and then formulating its dual problem using the canonical duality theory. The proposed energy-efficient design is compared with the spectral efficient design along with round robin (RR) and best channel quality indicator (BCQI) algorithms. Numerical results, which are obtained using the invasive weed optimization (IWO) algorithm, show that the proposed energy-efficient uplink design not only outperforms other algorithms in terms of energy efficiency while satisfying the QoS requirements, but also performs closer to the optimal design.

The bounds of feasible space on constrained nonconvex quadratic programming

This paper presents a robust canonical duality–triality theory for solving nonconvex programming problems under data uncertainty. This theory includes a robust canonical saddle-point theorem and robust canonical optimality conditions, which can be used to identify both robust global and local extrema of the primal problem. Two numerical examples are presented to illustrate that the robust Triality theory is particularly powerful for solving nonconvex optimization problems with data uncertainty.

Deterministic branch-and-bound algorithms for continuous global optimization often visit a large number of boxes in the neighborhood of a global minimizer, resulting in the so-called cluster problem (Du and Kearfott in J Glob Optim 5(3):253---265, 1994). This article extends previous analyses of the cluster problem in unconstrained global optimization (Du and Kearfott 1994; Wechsung et al. in J Glob Optim 58(3):429---438, 2014) to the constrained setting based on a recently-developed notion of convergence order for convex relaxation-based lower bounding schemes. It is shown that clustering can occur both on nearly-optimal and nearly-feasible regions in the vicinity of a global minimizer. In contrast to the case of unconstrained optimization, where at least second-order convergent schemes of relaxations are required to mitigate the cluster problem when the minimizer sits at a point of differentiability of the objective function, it is shown that first-order convergent lower bounding schemes for constrained problems may mitigate the cluster problem under certain conditions. Additionally, conditions under which second-order convergent lower bounding schemes are sufficient to mitigate the cluster problem around a global minimizer are developed. Conditions on the convergence order prefactor that are sufficient to altogether eliminate the cluster problem are also provided. This analysis reduces to previous analyses of the cluster problem for unconstrained optimization under suitable assumptions.

In this paper, we give some results which constitute an application of directional convexificators recently introduced by Dempe and Pilecka [Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming. J Global Optim. 2015;61:769–788]. After establishing mean value conditions in terms of directional convexificators, we formulate variational inequalities of Stampacchia and Minty type in terms of directional convexificators and use these variational inequalities as a tool to find out necessary and sufficient conditions for a point to be an optimal solution of an inherent optimization problem. An example illustrating our findings is also given.

Motivated by a work of Dempe and Pilecka [Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming. J Global Optim. 2015;61:769–788], where the authors introduced the notion of directional convexificators, we investigate a set valued optimization problem . Using a scalarization technique by mean of the Hiriart-Urruty signed distance function, together with directional convexificators, we give necessary optimality conditions of . Sufficient optimality conditions are obtained under a new generalized convexity. To illustrate the obtained results, an example is given.

We consider thermodynamic behaviour of thermal machines founded on kinetic rather than static origins. Their models, which are formulated for finite time transitions, simplify to models of classical thermodynamics in the limiting case of an infinite duration. An extended exergy is derived as a finite-time extension of the classical thermodynamic work delivered from a system of a body and its environment. With this quantity enhanced bounds can be determined for active continuous and cascade processes, in which there is an indirect energy exchange between two sybsystems through the working fluid of an engine, a refrigerator or a heat pump. These bounds refer to systems with finite exchange area or with a finite contact time. An economic framework of this theory is outlined. For both continuous and discrete processes, nonlinear thermodynamic models are derived from a combination of the energy balance and transfer equations. These models serve as constraints in the problem of work optimization. Variational and optimal control approaches are developed which are analogous to those found in analytical mechanics. Variational calculus is used along with some aspect of the canonical transformation theory to maximize work and discuss the role of a finite process intensity and of a finite duration. The optimality of a definite irreversible process for a finite-time transition of a controlled fluid is pointed out as well as a connection between the process duration, optimal dissipation and the optimal process intensity measured in terms of a hamiltonian, a dissipative quantity. It is shown that limits of the classical availability theory should be replaced by stronger limits which are obtained for finite time processes, and which are closer to reality. A hysteretic property of the generalized exergy describes a decrease of the maximum work received from an engine system and an increase of work added to a heat pump system, the features which are particularly important in high-rate regions of thermodynamic processes. For an infinite sequence of infinitesimal thermal machines, an optimal temperature strategy is obtained in the form similar to that known in the theory of simulated annealing.