Robust canonical duality theory for solving nonconvex programming problems under data uncertainty

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.

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 At the request of the Guest Editor, the Special Issue has been retracted, due to conflict of interest regarding Professor Gao’s role as Guest Editor and co-author on a number of submitted papers.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 The following articles that were due to appear in the Special Issue have therefore been retracted: Canonical Duality-Triality: Bridge between Nonconvex Analysis/Mechanics and Global Optimization in complex systems; David Y Gao, Ning Ruan, and Vittorio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.

In this paper, we design a Branch and Bound algorithm based on interval arithmetic to address nonconvex robust optimization problems. This algorithm provides the exact global solution of such difficult problems arising in many real life applications. A code was developed in MatLab and was used to solve some robust nonconvex problems with few variables. This first numerical study shows the interest of this approach providing the global solution of such difficult robust nonconvex optimization problems.

Understanding the impact of climate change and its variability on regional hydrology is predicated, to a large extent, on availability of long records of high quality climate data. Although recent advances in satellite technology have improved our observation capabilities, past records come from traditional in situ measurements that have large associated uncertainties. Further, the uncertainties in historical observations vary signicantly over space and in time. The information on uncertainty that is available with latest datasets is vastly ignored in the hydrologic literature. Since any relationship linking hydrologic and climatic variables is underpinned by historical data, ignoring this uncertainty information can be misleading. In this study, a methodology is developed to incorporate the observation uncertainty in climate data for hydrologic prediction by combining Bayesian learning theory and the concepts of robust optimization. A Bayesian variant of principal component analysis is used to propagate uncertainties in observations to derive a new representation of data with reduced dimensionality. Robust optimization is then used to obtain the parameters of the prediction model, a linear support vector machine in this case, by accounting for the uncertainty in data representation. The developed methodology is applied to make long-range prediction of Indian summer monsoon rainfall (ISMR) by capturing dynamic relationships between ISMR and sea surface temperature. The advantages of incorporating observation uncertainties for hydrologic predictions are discussed in the light of results obtained.

Nonlinear and nonconvex optimization problems are vital and fundamental problems in science and engineering fields. In this article, a novel finite-time circadian rhythms learning network (called FT-CRLN) is proposed for solving nonlinear and nonconvex optimization problems with periodic noises. Different from the traditional recurrent neural networks, the proposed FT-CRLN can suppress the periodic noise notably and achieve excellent convergence performance in solving nonlinear and nonconvex problems. The theoretical analysis and rigorous mathematical proof verify the superior convergence, high accuracy, and strong robustness of the proposed FT-CRLN. The simulation results demonstrate the effectiveness and robustness of the proposed FT-CRLN in solving nonlinear and nonconvex problems compared with other state-of-art neural networks.

A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. We hope that an insight into the inner workings of these methods will allow the reader to appreciate the unique marriage of task structure and generative models that allow these heuristic techniques to (provably) succeed. The monograph will lead the reader through several widely used nonconvex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.

Generalized gradient projection neural network models are proposed to solve nonsmooth convex and nonconvex nonlinear programming problems over a closed convex subset of ℝ n . By using Clarke’s generalized gradient, the neural network modeled by a differential inclusion is developed, and its dynamical behavior and optimization capabilities both for convex and nonconvex problems are rigorously analyzed in the framework of nonsmooth analysis and the differential inclusion theory. First for nonconvex optimization problems, the quasiconvergence results similar to those achieved by previous work are proved. For convex optimization problems, the global convergence results are proved, i.e. any trajectory of the neural network converges to an asymptotically stable equilibrium point, which is an optimal solution of the primal problem whenever it exists. This result remains valid even if the initial condition is chosen outside the feasible set. In addition, an asymptotic control result involving a Tykhonov-like regularization shows that any trajectory of the revised neural network can be forced to converge toward a particular optimal solution of the primal problem. Finally, two simulation optimization algorithms for solving the optimization and control problems are designed respectively, and three typical simulation experiments are illustrated to present the accuracy and efficiency of theoretical convergence results of this paper.

Adaptive scaling damped BFGS method without gradient Lipschitz continuity

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.

Robust optimization has come out to be a potent approach to study mathematical problems with data uncertainty. We use robust optimization to study a nonsmooth nonconvex mathematical program over cones with data uncertainty containing generalized convex functions. We study sufficient optimality conditions for the problem. Then we construct its robust dual problem and provide appropriate duality theorems which show the relation between uncertainty problems and their corresponding robust dual problems.

Optimization problems in real life often have problems with data that cannot be known precisely; constraints on the data are commonly referred as errors. This kind of data is called uncertainty. This uncertainty problem can be solved using Robust Optimization (RO). RO is growing rapidly with the participation of various kinds of research, especially the supply chain (distribution of food or goods between regions). It can be seen that RO is very active in providing support and contribution in various aspects of life by providing optimal results for an objective function and dealing with existing limitations and data uncertainty. This article discusses the background of the problem and the purpose of creating an article, provides an overview of bibliometric map analysis methods and discusses literature and studies. Critical review from OR database articles for supply chain problems are used as a reference, so at the end, it can be determined what novelty is an opportunity for further research.

Trajectory optimization for asteroid landing with two-phase free final time

A dynamic convexized method for nonconvex mixed integer nonlinear programming

Data uncertainty is a common feature in most of the real-life optimization problems. Despite that, the usual approach in mathematical optimization is to assume that all the input data are known deterministically and equal to some nominal values. Nevertheless, the optimal solution of the nominal problem can reveal itself suboptimal or even infeasible. An area where data uncertainty is a natural concern is portfolio optimization. As a matter of fact, in portfolio selection every optimization model deals with the estimate of the portfolio rate of return, and of either a risk or a safety measure. In the literature several techniques that are immune to data uncertainty have been proposed. These techniques are called robust. In this article we investigate two well-known robust techniques when applied to a specific portfolio selection problem, and compare the portfolios selected by the respective robust counterparts. Both the approaches allow the modeler to adjust the level of conservatism of the solution. We carried out extensive computational results based on real-life data from London Stock Exchange Market under different market behaviors.

Data envelopment analysis with uncertain data: An application for Iranian electricity distribution companies