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Finding sparse solutions to a system of equations and/or inequalities is an important topic in many application areas such as signal processing, statistical regression and nonparametric modeling. Various continuous relaxation models have been proposed and widely studied to deal with the discrete nature of the underlying problem. In this paper, we propose a quadratically constrained ℓ q (0 < q < 1) minimization model for finding sparse solutions to a quadratic system. We prove that solving the proposed model is strongly NP-hard. To tackle the computation difficulty, a first order necessary condition for local minimizers is derived. Various properties of the proposed model are studied for designing an active-set-based descent algorithm to find candidate solutions satisfying the proposed condition. In addition to providing a theoretical convergence proof, we conduct extensive computational experiments using synthetic and real-life data to validate the effectiveness of the proposed algorithm and to show the superior capability in finding sparse solutions of the proposed model compared with other known models in the literature. We also extend our results to a quadratically constrained ℓ q (0 < q < 1) minimization model with multiple convex quadratic constraints for further potential applications. Summary of Contribution: In this paper, we propose and study a quadratically constrained ℓ q minimization (0 < q < 1) model for finding sparse solutions to a quadratic system which has wide applications in sparse signal recovery, image processing and machine learning. The proposed quadratically constrained ℓ q minimization model extends the linearly constrained ℓ q and unconstrained ℓ 2 - ℓ q models. We study various properties of the proposed model in aim of designing an efficient algorithm. Especially, we propose an unrelaxed KKT condition for local/global minimizers. Followed by the properties studied, an active-set based descent algorithm is then proposed with its convergence proof being given. Extensive numerical experiments with synthetic and real-life Sparco datasets are conducted to show that the proposed algorithm works very effectively and efficiently. Its sparse recovery capability is superior to that of other known models in the literature. [ABSTRACT FROM AUTHOR]

This paper considers decisions of bidders with bounded rationality in different thinking levels for a first-price sealed-bid reverse auction. To characterize the nonequilibrium strategic thinking, we construct a mathematical model that incorporates the "level-k decision rule" to iteratively derive closed-form solutions. Then the effects of bidders' heterogeneous beliefs of thinking levels on their bid prices and expected payoffs are investigated. We find that under some assumptions, bidders will exhibit oscillating behavior in their bid prices and expected profits in terms of thinking levels. When the thinking level tends to infinity, the bid prices and expected profits converge to those in the conventional analysis with perfectly rational bidders. An interesting finding from theoretical analysis verified by numerical experiments is that any bidder with a thinking level above two should bid according to level-2 bidder's strategy to achieve the highest expected profit, and in this case the buyer will pay no more than level-2 bidder's bid price. [ABSTRACT FROM AUTHOR]

Clean energy is an important source to mitigate the pressure of resources shortage and environmental effects. While multiattribute reverse auction is frequently adopted for the procurement efficiency of clean energy devices, the associated winner determination problem is explicitly studied for a utility company/buyer to select the most appropriate supplier in the clean energy market. Considering the buyer's loss-averse behavior with incomplete information and suppliers' conflicting attributes in different types evaluated by multiple experts, we incorporate an I U L O W A f (induced uncertain linguistic ordered weighted averaging) operator and a revised TODIM (an acronym in Portuguese of Interactive and Multi-attribute Decision Making) measure that is consistent with prospect theory into the "benefits, opportunities, costs and risks" (BOCR) framework to develop a novel BOCR-uRTODIM solution method for multi-attribute decision making. Numerical experiments illustrate the effectiveness and robustness of the proposed method by comparing it with some known methods. An interesting result indicates that the risk aversion degree of the buyer increases as the variance of suppliers' attributes expands. The BOCR-uRTODIM could be a useful tool for utility companies to avoid losses and for associated suppliers to improve their attributes for a win in the clean energy industry. • Reverse auction for the clear energy device procurement is studied. • Utility companies' loss-averse behavior and incomplete information are considered. • Prospect theory is integrated into analytic hierarchy process for decision making. • A novel, effective, and robust method is proposed for determining the winner. • Managerial insights are drawn for utility companies in the clean energy industry. [ABSTRACT FROM AUTHOR]

In the paper, we study the inventory centralization problem with risk-averse newsvendors using cooperative game theory. Pareto-optimality and collective rationality conditions are adopted to characterize the stable profit allocations among coalition members. Pareto-optimal profit allocation rules are derived for exponential and power utilities while the necessary and sufficient conditions are shown for stable profit allocations. We also show that the game model has a nonempty core when the exponential and power utilities are adopted to describe the risk-averse behavior. [ABSTRACT FROM AUTHOR]

Suppliers may experience emotional/behavioral consequences of anticipated regrets that consist of winner and loser regrets in first- and second-price sealed-bid reverse auctions. Constructing mathematical models that incorporate regret theory to derive closed-form solutions of regret-anticipated suppliers’ bid decisions, this paper theoretically examines the effects of anticipated regrets on suppliers’ bid prices, buyer’s expected procurement cost and auction format decision. Comparing with the no regret scenario, we find that winner regret has adverse effects on the buyer’s expected procurement cost in first-price sealed-bid reverse auctions with regret-anticipated suppliers. To mitigate the adverse effects, we propose using the reserve price strategy for the buyer with theoretical analysis and numerical supports. An interesting analysis reveals that as the number of suppliers increases, the optimal reserve price increases or decreases depending on the degree of winner regret is lower or higher than that of loser regret. Also, the classical revenue equivalence theorem no longer holds when the degree of winner regret differs from that of loser regret. [ABSTRACT FROM AUTHOR]

In this paper, we significantly extend the applicability of state-of-the-art ELDP (equations for linearizing discrete product terms) method by providing a new linearization to handle more complicated non-linear terms involving both of discrete and bounded continuous variables. A general class of “representable programming problems” is formally proposed for a much wider range of engineering applications. Moreover, by exploiting the logarithmic feature embedded in the discrete structure, we present an enhanced linear reformulation model which requires half an order fewer equations than the original ELDP. Computational experiments on various engineering design problems support the superior computational efficiency of the proposed linearization reformulation in solving engineering optimization problems with discrete and bounded continuous variables. [ABSTRACT FROM AUTHOR]

In this paper, we propose a branch-and-cut algorithm for solving a nonconvex quadratically constrained quadratic programming (QCQP) problem with a nonempty bounded feasible domain. The problem is first transformed into a linear conic programming problem, and then approximated by semidefinite programming problems over different intervals. In order to improve the lower bounds, polar cuts, generated from corresponding cut-generation problems, are embedded in a branch-and-cut algorithm to yield a globally--optimal solution (with respect to feasibility and optimality, respectively) in a finite number of iterations. To enhance the computational speed, an adaptive branch-and-cut rule is adopted. Numerical experiments indicate that the number of explored nodes required for solving QCQP problems can be significantly reduced by employing the proposed polar cuts. [ABSTRACT FROM PUBLISHER]

A special reformulation-linearization technique based linear conic relaxation is proposed for discrete quadratic programming (DQP). We show that the proposed relaxation is tighter than the traditional positive semidefinite programming relaxation. More importantly, when the proposed relaxation problem has an optimal solution with rank one or two, optimal solutions to the original DQP problem can be explicitly generated. This rank-two property is further extended to binary quadratic optimization problems and linearly constrained DQP problems. Numerical results indicate that the proposed relaxation is capable of providing high quality and robust lower bounds for DQP. [ABSTRACT FROM PUBLISHER]

Support vector machines have been proven to be useful for regression analysis and forecasting. When stochastic uncertainty is involved in the datasets, robust support vector regression (SVR) models are useful. In this study, we proposed a kernel-free quadratic surface support vector regression (QSSVR) model based on optimal margin distribution (OMD). This model minimizes the variance of the functional margins of all data points to achieve better generalization capability. When the data points exhibit stochastic uncertainty (without the assumption of any specific distribution), the covariance information of noise is employed to construct a robust OMD-based QSSVR (RQSSVR-OMD) model, with a set of probabilistic constraints to ensure its worst-case performance. Moreover, the probabilistic constraints in the proposed model are proven to be equivalently reformulated as second-order cone constraints for efficient implementation. Extensive computational experiments on public benchmark datasets were conducted to demonstrate the superior performance of the proposed RQSSVR-OMD model over other well-established SVR models in terms of forecasting accuracy and time. The proposed model was also validated to successfully handle real-life uncertain battery data for battery power-consumption forecasting. [ABSTRACT FROM AUTHOR]

Cubic L 1 spline fits have shown some favorable shape-preserving property for geometric data. To quantify the shape-preserving capability, we consider the basic shape of two parallel line segments in a given window. When one line segment is sufficiently longer than the other, the spline fit can preserve its linear shape in at least half of the window. We propose to use the minimum of such length difference as a shape-preserving metric because it represents the extra information that the spline fits need to preserve the shape. We analytically calculate this metric in a 3-node window for second-derivative-based, first-derivative-based and function-value-based spline fits. In a 5-node window, we compute this metric numerically. In both cases, the shape-preserving metric is rather small, which explains the observed strong shape-preserving capability of spline fits. Moreover, the function-value-based spline fits are indicated to preserve shape better than the other two types of spline fits. This study initiates a quantitative research on shape preservation of L 1 spline fits. [ABSTRACT FROM AUTHOR]

In this paper, we develop a semidefinite relaxation-based branch-and-bound algorithm that exploits the chordal sparsity patterns of the max-cut problem. We first study how the chordal sparsity pattern affects the hardness of a max-cut problem. To do this, we derive a polyhedral relaxation based on the clique decomposition of the chordal sparsity patterns and prove some sufficient conditions for the tightness of this polyhedral relaxation. The theoretical results show that the max-cut problem is easy to solve when the sparsity pattern embedded in the problem has a small treewidth and the number of vertices in the intersection of maximal cliques is small. Based on the theoretical results, we propose a new branching rule called hierarchy branching rule, which utilizes the tree decomposition of the sparsity patterns. We also analyze how the proposed branching rule affects the chordal sparsity patterns embedded in the problem, and explain why it can be effective. The numerical experiments show that the proposed algorithm is superior to those known algorithms using classical branching rules and the state-of-the-art solver BiqCrunch on most instances with sparsity patterns arisen in practical applications. [ABSTRACT FROM AUTHOR]

Based on the annual data set, which involves a balanced panel of four port cities exports to 32 countries and Hong Kong, China over the period of 1997 2008, this paper is the first attempt to extend the conventional gravity model to measure the effect of transport costs and port efficiency on China's exports. Our results can be summarized in three main findings: 1) the improvement of port efficiency and reduction of road transport costs play a vital role in China's export competitiveness in the global market. The coefficient estimates on them are relatively large, around 0.72 and -0.89, respectively; 2) the effect of transport costs and port efficiency on China-to-Asia exports significantly exceeds that on China-to-Europe and China-to-America exports; 3) the overall estimated elasticity of road, railroad and port measure is 1.66, which is almost three times that of the average wage of port cities. The empirical results provide strong evidences that improvement in the quality of transportation services in port cities of China exhibit significant economic effects not only on how much but also how trade grows. [ABSTRACT FROM AUTHOR]

Multi-class classification is an important and challenging research topic with many real-life applications. The problem is much harder than the classical binary classification, especially when the given data set is imbalanced. Hidden nonlinear patterns in the data set can further complicate the task of multi-class classification. In this paper, we propose a kernel-free least squares twin support vector machine for multi-class classification. The proposed model employs a special fourth order polynomial surface, namely the double well potential surface, and adopts the "one-verses-all" classification strategy. An ℓ 2 regularization term is added to accommodate data sets with different levels of nonlinearity. We provide some theoretical analysis of the proposed model. Computational results using artificial data sets and public benchmarks clearly show the superior performance of the proposed model over other well-known multi-class classification methods, in particular for imbalanced data sets. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we develop a compressed version of the primal-dual interior point method for generating bivariate cubic splines. Discretization of the underlying optimization model, which is a nonsmooth convex programming problem, leads to an overdetermined linear system that can be handled by interior point methods. Taking advantage of the special matrix structure of the cubic spline problem, we design a compressed primal-dual interior point algorithm. Computational experiments indicate that this compressed primal-dual method is robust and is much faster than the ordinary (uncompressed) primal-dual interior point algorithm. [Copyright &y& Elsevier]

Abstract: The results in this paper quantify the ability of cubic splines to preserve the shape of nonparametric data. The data under consideration include multiscale data, that is, data with abrupt changes in spacing and magnitude. A simplified dual-to-primal transformation for a geometric programming model for cubic splines is developed. This transformation allows one to establish in a transparent manner relationships between the shape-preserving properties of a cubic spline and the solution of the dual geometric-programming problem. Properties that have often been associated with shape preservation in the past include preservation of linearity and convexity/concavity. Under various circumstances, cubic splines preserve linearity and convexity/concavity of data. When four consecutive data points lie on a straight line, the cubic spline is linear in the interval between the second and third data points. Cubic splines of convex/concave data preserve convexity/concavity if the first divided differences of the data do not increase/decrease too rapidly. When cubic splines do not preserve convexity/concavity, they still do not cross the piecewise linear interpolant and, therefore, they do not have extraneous oscillation. [Copyright &y& Elsevier]

Abstract: Bivariate cubic L 1 splines provide C 1-smooth, shape-preserving interpolation of arbitrary data, including data with abrupt changes in spacing and magnitude. The minimization principle for bivariate cubic L 1 splines results in a nondifferentiable convex optimization problem. This problem is reformulated as a generalized geometric programming problem. A geometric dual with a linear objective function and convex cubic constraints is derived. A linear system for dual-to-primal conversion is established. The results of computational experiments are presented. [Copyright &y& Elsevier]

Current research on auctions is mainly based on the game theory. In the first part of this paper, we propose a fuzzy set based formulation of auctions. We define fuzzy sets to represent the seller and buyers’ valuations, bid possibilities and win possibilities. Analyzing the properties of these fuzzy sets, we study fuzzy versions of discriminating and nondiscriminating bidding strategies for each bidder in a single-object auction by using Bellman and Zadeh''s concept of confluence of fuzzy decisions instead of the game theoretic Nash equilibrium. The monotonically decreasing membership function replaces of the corresponding distribution density function in Milgrom and Weber''s auction theory. In the second part of the paper, we develop a soft computing approach to maximize the seller''s revenue in multiple-object auctions through the use of object sequencing. The proposed sequencing approach is based on “fuzzy rule quantification”. Numerical experience shows the potential of the proposed approach for application in practical auctions. [Copyright &y& Elsevier]

Evaluating the performance of activities or organizations by traditional data envelopment analysis (DEA) models requires crisp input/output data. However, in real-world problems inputs and outputs are often imprecise. This paper develops DEA models using imprecise data represented by fuzzy sets (i.e., “fuzzy DEA” models). It is shown that fuzzy DEA models take the form of fuzzy linear programming which typically are solved with the aid of some methods to rank fuzzy sets. As an alternative, a possibility approach is introduced in which constraints are treated as fuzzy events. The approach transforms fuzzy DEA models into possibility DEA models by using possibility measures of fuzzy events (fuzzy constraints). We show that for the special case, in which fuzzy membership functions of fuzzy data are of trapezoidal types, possibility DEA models become linear programming models. A numerical experiment is used to illustrate the approach and compare the results with those obtained with alternative approaches. [Copyright &y& Elsevier]

In this paper, we investigate the universal approximation property of Radial Basis Function (RBF) networks. We show that RBFs are not required to be integrable for the REF networks to be universal approximators. Instead, RBF networks can uniformly approximate any continuous function on a compact set provided that the radial basis activation function is continuous almost everywhere, locally essentially bounded, and not a polynomial. The approximation in Lp(μ)(1≤p<∞) space is also discussed. Some experimental results are reported to illustrate our findings. [Copyright &y& Elsevier]

The problem of finding an x∈Rn such that Ax&les;b and x&ges;0 arises in numerous contexts. We propose a new optimization method for solving this feasibility problem. After converting Ax&les;b into a system of equations by introducing a slack variable for each of the linear inequalities, the method imposes an entropy function over both the original and the slack variables as the objective function. The resulting entropy optimization problem is convex and has an unconstrained convex dual. If the system is consistent and has an interior solution, then a closed-form formula converts the dual optimal solution to the primal optimal solution, which is a feasible solution for the original system of linear inequalities. An algorithm based on the Newton method is proposed for solving the unconstrained dual problem. The proposed algorithm enjoys the global convergence property with a quadratic rate of local convergence. However, if the system is inconsistent, the unconstrained dual is shown to be unbounded. Moreover, the same algorithm can detect possible inconsistency of the system. Our numerical examples reveal the insensitivity of the number of iterations to both the size of the problem and the distance between the initial solution and the feasible region. The performance of the proposed algorithm is compared to that of the surrogate constraint algorithm recently developed by Yang and Murty. Our comparison indicates that the proposed method is particularly suitable when the number of constraints is larger than that of the variables and the initial solution is not close to the feasible region. [Copyright &y& Elsevier]

This paper studies a new class of optimization problems which have multiple objective functions subject to a set of fuzzy relation equations. Since the feasible domain of such a problem is in general non-convex and the objective functions are not necessarily linear, traditional optimization methods may become ineffective and inefficient. Taking advantage of the special structure of the solution set, a reduction procedure is developed to simplify a given problem. Moreover, a genetic-based algorithm is proposed to find the “Pareto optimal solutions”. The major components of the proposed algorithm together with some encouraging test results are reported. [ABSTRACT FROM AUTHOR]

Simulation optimization deals with finding the values of input parameters of a complex simulated system which result in desired output. Traditional techniques may require an enormous amount of simulation runs to evaluate the system. To alleviate this problem, the proposed work provides the means of incorporating knowledge, expressed in natural language, that is often available among analysts and decision makers. Using convenient linguistic representations, the proposed mechanism can satisfy vaguely stated goals to a high degree (e.g. “high utilization” or “low inventory”). This mechanism is also able to generate an approximate Pareto optimal set in the presence of multiple goals. The optimization strategy used here depends on a fuzzy controller guided by a set of rules derived from statistical concepts, response surface models, and experts’ knowledge. To illustrate this approach we present computational experiments on the design of a flow line manufacturing system (in terms of a tandem of queues with blocking) with one and two goals. The actions derived from the controller, using approximate reasoning, are able to generate a high quality solution in only a few iterations. The results are compared extensively with those obtained with a genetic-based approach. [ABSTRACT FROM AUTHOR]

Presents a genetics-based inexact approach proposed for the imitation of the human decision procedure for production planning. Statistical data used in this study.

In recent years, risk management has attracted a great deal of attention from both researchers and practitioners. Complexity and uncertainty in many practical problems require new methods and tools. Risk management can be used as a tool for greater rewards, not just for control against loss. Risks are studied from different silo disciplinary perspectives, with a discussion of how various methods and tools are used to optimize risk management. Enterprise risk management has become an important topic in today's more complex, interrelated global business environment, replete with threats from natural, political, economic and technical sources. [ABSTRACT FROM AUTHOR]

Optimal reliability design (ORD) problem is challenging and fundamental to the study of system reliability. For a system with n components/stages where each of them can be set in m possible reliability levels, state-of-the-art linear reformulation models of ORD problem require O(nm) binary variables, O(mn) continuous variables together with either O(mn) inequality constraints or O(nm) equality constraints. Using the special property of prime factorization and adopting the logarithmic expression technique, in this article, we propose a novel linear reformulation model of the ORD problem requiring O(nm) binary variables, O(mn/n!) continuous variables, and very few linear constraints. This theoretic reduction in variables and constraints can lead to significant savings in computational efforts. Our numerical experiments further confirm the drastic reduction in computational time for solving ORD problems in large size. [ABSTRACT FROM AUTHOR]

The following article has been included in a multiple retraction: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.abstractIn 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.abstractCanonical 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.abstractCanonical 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.abstractDouble 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.abstractDouble 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.abstractAnalytic 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.abstractTriality 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.abstractGlobal 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.abstractUnified Canonical Duality Methodology for Global Optimization; Vittorio Latorre, David Y Gao and N. Ruan.http://mms.sagepub.com/content/early/2015/07/06/1081286515591305.abstractA 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.abstractCanonical 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.abstractGlobal 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.abstractOn Minimal Distance between Two Non-Convex Surfaces; Daniel Morales-Silva, David Y Gao.http://mms.sagepub.com/content/early/2015/07/27/1081286515592949.abstractThe 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. [ABSTRACT FROM AUTHOR]

Semi-supervised support vector machine is a popular method in the research area of machine learning. Considering a large amount of unlabeled data points in real-life world, the semi-supervised support machine has the ability of good generalization for dealing with nonlinear classification problems. In this paper, a proximal quadratic surface support vector machine model is proposed for semi-supervised binary classification. The main advantage of our new model is that the proximal quadratic surfaces are constructed directly for nonlinear classification instead of using the kernel function, which avoids the tasks of choosing kernels and tuning their parameters. We reformulate this proposed model as an unconstrained mixed-integer quadratic programming problem. Semi-definite relaxation is then adopted, and a primal alternating direction method is further proposed for fast computation. We test the proposed method on some artificial and public benchmark data sets. Preliminary results indicate that our method outperforms some well-known methods for semi-supervised classification in terms of the efficiency and classifying accuracy. [ABSTRACT FROM AUTHOR]

The problem of geometric programming subject to max-product fuzzy relation constraints with discrete variables is studied. The major difficulty in solving this problem comes from nonconvexity caused by these product terms in the general geometric function and the max-product relation constraints. We proposed a 0-1 mixed integer linear programming model and adopted the branch-and-bound scheme to solve the problem. Numerical experiments confirm that the proposed solution method is effective. [ABSTRACT FROM AUTHOR]

Collaborative vehicle routing of multiple logistics providers is an important component of horizontal logistic collaboration that generates economic and societal benefits. Existing research on collaborative vehicle routing is limited to the homogeneous setting where the logistics providers transport the same product. To better address the need of a general modeling framework and fast computational methods for the growth of collaboration among logistics providers carrying various products, we investigate a heterogeneous multi-depot collaborative vehicle routing problem (HMCVRP) in this paper. The key operational and computational challenge of realizing the collaborative route planning is to properly select transfer points for product transshipment between vehicles of different depots. We propose a Benders-based branch-and-cut algorithm with the technique of combinatorial Benders' cuts to solve a mixed-integer programming formulation of HMCVRP. Numerical experiments indicate that the proposed algorithm significantly outperforms the CPLEX solver using the commonly adopted big-M transformation-based method. Additional computational study further reveals the importance of the locations of depots and having a well-designed cost savings allocation mechanism in practice. • The heterogeneous multi-depot collaborative vehicle routing problem is studied. • Collaboration is realized via transshipment at the customer locations and depots. • A Benders-based branch-and-cut algorithm is developed to solve this problem. • Algorithm is enhanced with Combinatorial benders' cuts and enhancement strategies. • We provide some important operational insights of heterogeneous collaboration. [ABSTRACT FROM AUTHOR]

Cones of nonnegative quadratic functions are keys to the understanding of quadratic optimization problems, since any quadratically constrained quadratic programming problem can be reformulated as a linear conic programming problem over such a cone. This paper proposes an adaptive computable approximation scheme to cones of nonnegative quadratic functions and uses it for solving linear conic programming problems over such a cone. We study some basic properties of cones of nonnegative quadratic functions and present a class of simple cones with computable linear matrix inequalities representations. Building on these simple cones, we design a computable approximation scheme for handling a general cone of nonnegative quadratic functions. When the scheme is applied for solving linear conic programming problems over a cone of nonnegative quadratic functions, we incorporate an adaptive approach to enhance the performance of the proposed computable approximation scheme. The computational performance and theoretic convergence proof of the proposed adaptive computable approximation scheme are shown for solving box-constrained quadratic programming problems. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a “two-group knapsack game”. A number of investors colligate into two groups to bid on a common pool of potential projects. Each investor has his/her own budget limit and a cost estimation for undertaking each possible project. Each group represents a power by its market share. Associated with each project, there is a potential profit that can be realized. Investors in the same group hold an internal agreement of putting the group’s collective interest ahead of the individual’s interest and not bidding on the same project by more than one investor in the group. The profit of a particular project can be wholly taken by the sole bidder or shared proportionally by two bidders in different groups according to their group power. The objective of each group may be based not only on its own group profit but also on the other group’s profit. Assuming that each investor acts in a selfish manner with the best response to optimize its group’s objective subject to the budget constraint, we show that a pure Nash equilibrium exists under certain conditions. We also have some interesting findings of the “price of anarchy” (the ratio of the worst Nash equilibrium to the social optimum) associated with a simplified version of the two-group knapsack game with three investors. [Copyright &y& Elsevier]

The following article has been included in a multiple retraction: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.abstractIn 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.abstractCanonical 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.abstractCanonical 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.abstractDouble 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.abstractDouble 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.abstractAnalytic 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.abstractTriality 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.abstractGlobal 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.abstractUnified Canonical Duality Methodology for Global Optimization; Vittorio Latorre, David Y Gao and N. Ruan.http://mms.sagepub.com/content/early/2015/07/06/1081286515591305.abstractA 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.abstractCanonical 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.abstractGlobal 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.abstractOn Minimal Distance between Two Non-Convex Surfaces; Daniel Morales-Silva, David Y Gao.http://mms.sagepub.com/content/early/2015/07/27/1081286515592949.abstractThe 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. [ABSTRACT FROM AUTHOR]

In this article, we study the problem of best L approximation of Heaviside-type functions from Chebyshev and weak-Chebyshev spaces. We extend the Hobby-Rice theorem (Proc. Am. Math. Soc., 16, 665-670, 1965) into an appropriate framework and prove the unicity of best L approximation of Heaviside-type functions from an even-dimensional Chebyshev space under some assumptions on the dimension of the subspaces composed of the odd and even functions. We also apply the results to compute best L approximations of Heaviside-type functions by polynomials and Hermite polynomial splines with fixed knots. [ABSTRACT FROM AUTHOR]

• Districting consideration is successfully integrated with the multi-customer JRP. • This study is vital to a company outsourcing logistics to a 3PL service provider. • Our numerical experiments are based on a real-life example in a bank system. • An effective GA-based algorithm is proposed for solving the problem. • Managerial insights provide baseline information before contract negotiation. This paper studies a multi-customer joint replenishment problem with districting consideration (MJRPDC) which is of particular importance to a company that outsources its transportation and delivery operations to a third-party logistics (3PL) service provider. To solve the problem, we first propose an innovative search algorithm for solving the traditional multi-customer joint replenishment problem in a given zone. Then we design a GA-based framework to handle the corresponding districting problem based on the performance of each district evaluated by using the proposed search algorithm. The proposed methodologies are demonstrated by using an example of solving MJRPDC for a bank. [ABSTRACT FROM AUTHOR]