American Institute of Mathematical Sciences

Recently, Hu, Huang and Chen [Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math. 230 (2009): 69-82] introduced a family of generalized NCP-functions, which include many existing NCP-functions as special cases. They obtained several favorite properties of the functions; and by which, they showed that a derivative-free descent method is globally convergent under suitable assumptions. However, no result on convergent rate of the method was reported. In this paper, we further investigate some properties of this family of generalized NCP-functions. In particular, we show that, under suitable assumptions, the iterative sequence generated by the descent method discussed in their paper converges globally at a linear rate to a solution of the nonlinear complementarity problem. Some preliminary numerical results are reported, which verify the theoretical results obtained.

In this paper we give necessary and sufficient optimality conditions for a fractional vector optimization problem over cones involving support functions in objective as well as constraints, using cone-convex functions. We also associate Mond-Weir type and Schaible type duals with the primal problem and establish weak and strong duality results under cone-convexity, pseudoconvexity and quasiconvexity assumptions. A number of previously studied problems appear as special cases.

In this paper we attempt to develop a parametric simplex algorithm for solving biobjective convex separable piecewise linear programming problems. The algorithm presented in this paper can be regarded as an extension of the parametric simplex algorithm for solving biobjective linear programming problems to the piecewise linear case. Analogous to the linear case, this parametric simplex algorithm provides a decomposition of parametric space. A numerical example is presented to illustrate the implementation of the algorithm.

We consider a single machine scheduling problem in which the processing time of a job is a nondecreasing function of its starting time. The jobs have a common due date. The objective is to minimize the weighted number of tardy jobs. The problem is NP-hard. We present a pseudo-polynomial dynamic programming algorithm and a fully polynomial-time approximation scheme (FPTAS).

In this paper, a scaled method that combines the conjugate gradient with moving asymptotes is presented for solving the large-scaled nonlinear unconstrained optimization problem. A diagonal matrix is obtained by the moving asymptote technique, and a scaled gradient is determined by multiplying the gradient with the diagonal matrix. The search direction is either a scaled conjugate gradient direction or a negative scaled gradient direction under different conditions. This direction is sufficient descent if the step size satisfies the strong Wolfe condition. A global convergence analysis of this method is also provided. The numerical results show that the scaled method is efficient for solving some large-scaled nonlinear problems.

Semi-supervised learning is an attractive method in classification problems when insufficient training information is available. In this investigation, a new semi-supervised classifier is proposed based on the concept of maximum vector-angular margin, (called S$^3$MAMC), the main goal of which is to find an optimal vector $c$ as close as possible to the center of the dataset consisting of both labeled samples and unlabeled samples. This makes S$^3$MAMC better generalization with smaller VC (Vapnik-Chervonenkis) dimension. However, S$^3$MAMC formulation is a non-convex model and therefore it is difficult to solve. Following that we present two optimization algorithms, mixed integer quadratic program (MIQP) and DC (difference of convex functions) program algorithms, to solve the S$^3$MAMC. Compared with the supervised learning methods, numerical experiments on real and synthetic databases demonstrate that the S$^3$MAMC can improve generalization when the labelled samples are relatively few. In addition, the S$^3$MAMC has competitive experiment results in generalization compared to the traditional semi-supervised classification methods.

The problem of solving regulator-conjugate matrix equations is considered in this paper. The regulator-conjugate matrix equations are a class of nonhomogeneous equations. Utilizing several complex matrix operations and the concepts of controllability-like and observability-like matrices, a special solution to this problem is constructed, which includes solving an ordinary algebraic matrix. Combined with our recent results on Sylvester-conjugate matrix equations, complete solutions to regulator-conjugate matrix equations can be obtained by superposition principle. The correctness and effectiveness are verified by a numerical example.

This paper concerns the distribution-free, multi-period newsboy problem in which the newsboy has to decide the order quantity of the newspaper in the subsequent period without knowing the distribution of the demand. The Weak Aggregating Algorithm (WAA) developed in learning and prediction with expert advices makes decision only based on historical information and provides theoretical guarantee for the decision-making method. Based on the advantage of WAA and stationary expert advices, this paper continues providing distribution-free methods for the extended multi-period newsboy problems in which the shortage cost and the integral order quantities are considered. In particular, we provide an alternative proof for the theoretical result which guarantees the cumulative gain our proposed method achieves is as large as that of the best stationary expert advice. Numerical examples are provided to illustrate the effectiveness of our proposed methods.

Hybridizing the three–term conjugate gradient method proposed by Zhang et al. and the nonlinear conjugate gradient method proposed by Dai and Liao based on the scaled memoryless BFGS update, a one–parameter class of four–term conjugate gradient methods is proposed. It is shown that the suggested class of conjugate gradient methods possesses the sufficient descent property, without convexity assumption on the objective function. A brief global convergence analysis is made for uniformly convex objective functions. Results of numerical comparisons are reported. They demonstrate efficiency of a method of the proposed class in the sense of the Dolan–Moré performance profile.

Optimization involving eigenvalues arises in a large spectrum of applications in various domains, such as physics, engineering, statistics and finance. In this paper, we consider the arbitrary eigenvalue minimization problems over an affine family of symmetric matrices, which is a special class of eigenvalue function--D.C. function \begin{document} $λ^_{l}$ \end{document}. An explicit proximal bundle approach for solving this class of nonsmooth, nonconvex (D.C.) optimization problem is developed. We prove the global convergence of our method, in the sense that every accumulation point of the sequence of iterates generated by the proposed algorithm is stationary. As an application, we use the proposed bundle method to solve some particular eigenvalue problems. Some encouraging preliminary numerical results indicate that our method can solve the test problems efficiently.

In this paper, a novel single machine scheduling problem with deterioration depending on waiting times is investigated. Firstly, a new deterioration model for the problem is presented. Secondly, according to characteristics of the problem, dominance properties and lower bounds are proposed and integrated into the Branch and Bound algorithm (B & B) to solve the small-medium scale problems. Thirdly, for solving a large-scale problem, the Rules Guided Nested Partitions method (RGNP) is proposed. The numerical experiments show that when the size of the problem is no more than 17 jobs, the B & B algorithm can obtain the optimal solutions in a reasonable time. The RGNP method can also obtain an average error percentage for near-optimal solutions less than 0.048 within 0.2s. The analysis shows the efficiency of RGNP, and, hence, it can be used for solving large size problems.

In this paper, we address the scheduling problem with positional-dependent processing times in a disruptive environment, in which there is a possibility that some of the machines become unavailable for a certain period of time with a certain probability due to a disruption at a particular time. By positional-dependent processing times, we mean that the actual processing time of a job depends on its processing position on a machine. Since some machines may be unavailable for a certain period of time, both non-resumable and resumable cases are considered. The objective is to minimize the expected total completion time. For various cases, we provide the complexity results and present efficient pseudo-polynomial time algorithms for the corresponding problems.

In this paper, we consider the multiple common due-dates assignment and machine scheduling with linear deteriorating jobs and optimal maintenance activity. The linear deteriorating jobs means job processing times are an increasing function of their starting times. The maintenance activity requires a fixed time interval. During the time interval, the machine is turned off and no job is processed. Once completing the maintenance, the machine will revert to its initial condition. The objective is to schedule the jobs, the due dates and the maintenance activity, so as to minimize the total cost including earliness, tardiness, and the due dates. We provide some properties of optimal sequence and introduce an efficient \begin{document}$O({n^{\rm{2}}}\log n)$\end{document} algorithm to solve the problem.

In this paper, we consider a perturbed compound Poisson model with varying premium rates. The surplus process is observed at a sequence of review times. The effective premium rate is adjusted according to the surplus increment between the inter-review times. We study the Gerber-Shiu functions by Laplace transform method. When the claim size density is a combination of exponentials, the explicit expressions for the Laplace transforms of ruin time are derived.

This paper studies optimal reinsurance and investment strategies that maximize expected utility of the terminal wealth for an insurer in a stochastic market. The insurer's preference is represented by a two-piece utility function which can be regarded as a generalization of traditional concave utility functions. We employ martingale approach and convex optimization method to transform the dynamic maximization problem into an equivalent static optimization problem. By solving the optimization problem, we derive explicit expressions of the optimal reinsurance and investment strategy and the optimal wealth process.

In this paper, we propose a Hidden Markov Model (HMM) which incorporates the threshold effect of the observation process. Simulated examples are given to show the accuracy of the estimated model parameters. We also give a detailed implementation of the model by using a dataset of crude oil price in the period 1986-2011. The prediction of crude oil spot price is an important and challenging issue for both government policy makers and industrial investors as most of the world's energy comes from the consumption of crude oil. However, many random events and human factors may lead the crude oil price to a strongly fluctuating and highly non-linear behavior. To capture these properties, we modulate the mean and the variance of log-returns of commodity prices by a finite-state Markov chain. The $h$-day ahead forecasts generated from our model are compared with regular HMM and the Autoregressive Moving Average model (ARMA). The results indicate that our proposed HMM with threshold effect outperforms the other models in terms of predicting ability.

This paper considers a two-supplier one-retailer coordinated supply chain system with auction and contracting mechanism incorporating participants' risk attitudes. The risk attitude is quantified using the value-at-risk (VaR) measure and the retailer faces a stochastic linear price-dependent demand function. In the supply chain, the suppliers (providing identical products) compete with each other in order to win the ordering contract of the retailer. Several auction and contracting mechanisms are developed and compared. It can be analytically shown that the retail price of the risk-averse system is higher than that of the risk-neutral system, but the order quantity is lower than that of the risk-neutral system.

In this paper, following the framework of robust optimization, we consider robust optimal solutions for a fractional optimization problem in the face of data uncertainty both in the objective and constraints. To this end, by using the properties of the subdifferential sum formulae, we first introduce some robust basic subdifferential constraint qualifications, and then obtain some completely characterizations of the robust optimal solutions of this uncertain fractional optimization problem. We show that our results encompass as special cases some optimization problems considered in the recent literature. Moreover, as applications, the proposed approach is applied to investigate weakly robust efficient solutions for multi-objective fractional optimization problems in the face of data uncertainty both in the objective and constraints.

In this paper, we consider a robust sensor scheduling problem which estimates the state of an uncertain process based on measurements obtained by a given set of noisy sensors, where the measurements of sensors are subject to external interference uncertainties. We formulate this problem into a minimax optimal control problem, which is equivalent to a semi-infinite programming problem with a dynamic system. A discretization method is used to solve this problem, where the computation is very large scale in general. We propose an approximation method to reduce the computational complexity. For illustration, two numerical examples are solved.

In this study, an optimal algorithm is presented for the obstacle neutralization problem (ONP). ONP is a recently introduced path planning problem wherein an agent needs to swiftly navigate from a source to a destination through an arrangement of obstacles in the plane. The agent has a limited neutralization capability in the sense that the agent can safely pass through an obstacle upon neutralization at a cost added to the traversal length. The goal of an agent is to find the sequence of obstacles to be neutralized en route minimizing the overall traversal length subject to the neutralization limit. Our optimal algorithm consists of two phases. In the first phase an upper bound of the problem is obtained using a suboptimal algorithm. In the second phase, starting from the bound obtained from phase Ⅰ, a $k$-th shortest path algorithm is exploited to find the optimal solution. The performance of the algorithm is presented with computational experiments conducted both on real and synthetic naval minefield data. Results are promising in the sense that the proposed method can be applied in online applications.

In this paper, we present a new nonmonotone memory gradient algorithm for unconstrained optimization problems. An attractive property of the proposed method is that the search direction always provides sufficient descent step at each iteration. This property is independent of the line search used. Under mild assumptions, the global and local convergence results of the proposed algorithm are established respectively. Numerical results are also reported to show that the proposed method is suitable to solve large-scale optimization problems and is more stable than other similar methods in practical computation.

We study a supply chain in which a rational manufacturer relies on an overconfident sales agent to sell the products. The actual sales outcome is determined by the sales agent's selling effort, the product price and a random market condition, and the sales agent is overconfident in their estimation of sales outcome. Apart from them, both of the sales agent's degree of overconfidence and selling effort are his private information. We consider the salesforce incentive to motivate the sales agent and screen his real degree of overconfidence using a principle-agent method under dual information asymmetry, then the manufacturer uses the information to realize her joint decision on pricing and production. Furthermore, we derive the optimal compensation contract as well as the optimal pricing and production, and compare it to the symmetric overconfidence scenario. Finally, some interesting insights are found: when the manufacturer is uncertain about the sales agent's the degree of overconfidence, her expected profit decreases while the sales agent with private information exerts less effort but obtains higher income, which implies the value of information; the manufacturer should hire a more overconfident sales agent, while a higher commission rate is not guaranteed. These results suggest that the manufacturer should not only focus on hiring the overconfident sales agent but also on disclosing the degree of overconfidence.

Flow lines in which workstations and buffers are linked along a single flow path one after another are widely used for modeling manufacturing systems. In this paper we consider the flow lines with multiple independent unreliable machines at each workstation and blocking. The processing times, time to failure and time to repair of each machine are assumed to exponentially distributed and blocking after service blocking protocol is also assumed. An approximate analysis for throughput in the flow lines is presented. The method developed here is based on the decomposition method using the subsystems with three workstations including virtual station and two buffers between workstations. Some numerical examples are presented for accuracy of approximation.

This paper investigates a distributed fault-tolerant consensus tracking algorithm for a group non-identical motors with unmeasured angular speed and unknown failures. First, the failures are modeled by nonlinear functions, and sliding mode observer is designed to estimate the angular speed and nonlinear failures. Then, in order to achieve the desired results, a novel distributed fault-tolerant algorithm is constructed based on the estimated angular speed and reconstructed failures. Theoretical analysis illustrates the stability and globally exponentially asymptotically convergence of the proposed observer and controller. The numerical simulations verify the high estimation accuracy, effectiveness and robustness of the proposed methods. The semi-physical experiments based on RT-LAB real-time simulator further test the system and controller with accurate performance in real-time.

We study a stochastic inventory model with a fixed setup cost and zero order lead time. In a finite-horizon lost sales model, when demand has a Polya frequency distribution (P F_n), we show that there are no more than a pre-determined number of minima of the cost function. Consequently, depending on the relative cost of lost sales and inventory holding cost, there can be as few as one local minimum. These properties have structural implications for the optimal policies and cost functions. A necessary condition for the results to hold for the backordered model has been explained. We further conduct a numerical study to validate our structural results.

In this paper, we propose an optimal trade-off model for portfolio selection with sensitivity of parameters, which are estimated from historical data. Mathematically, the model is a quadratic programming problem, whose objective function contains three terms. The first term is a measurement of risk. And the later two are the maximum and minimum sensitivity, which are non-convex and non-smooth functions and lead to the whole model to be an intractable problem. Then we transform this quadratic programming problem into an unconstrained composite problem equivalently. Furthermore, we develop a modified accelerated gradient (AG) algorithm to solve the unconstrained composite problem. The convergence and the convergence rate of our algorithm are derived. Finally, we perform both the empirical analysis and the numerical experiments. The empirical analysis indicates that the optimal trade-off model results in a stable return with lower risk under the stress test. The numerical experiments demonstrate that the modified AG algorithm outperforms the existed AG algorithm for both CPU time and the iterations, respectively.

In this paper, by using the nonlinear scalarization method and under some new assumptions, which do not involve any information on the solution set, we establish the continuity of solution mappings of parametric generalized non-weak vector Ky Fan inequality with moving cones. The results are new and improve corresponding ones in the literature. Some examples are given to illustrate our results.

We consider the problem of scheduling a set of \begin{document} $n$ \end{document} jobs with arbitrary job sizes, processing times and release times on a set of \begin{document} $m$ \end{document} parallel batch machines with non-identical capacities; the objective is to minimize the makespan. We first present an algorithm to compute a lower bound for the optimal makespan. Based on different rules of batching the jobs and assigning the batches to the machines, several heuristics are proposed to solve the problem. The performance of the proposed heuristics is evaluated by computational experiments. The proposed heuristics are compared against the lower bound and against each other. Our results show that the one of the proposed algorithms outperforms all the other heuristics.

We investigate the infinite-time ruin probability of a renewal risk model with exponential Lévy process investment and dependent claims and inter-arrival times. Assume that claims and corresponding inter-arrival times form a sequence of independent and identically distributed copies of a random pair \begin{document} $(X,T)$ \end{document} with dependent components. When the product of the claims and the discount factors of the corresponding inter-arrival times are heavy tailed, we establish an asymptotic formula for the infinite-time ruin probability without any restriction on the dependence structure of \begin{document} $(X,T)$ \end{document}.

In this paper, a superlinearly convergent hybrid algorithm is proposed for solving nonlinear programming. First of all, an improved direction is obtained by a convex combination of the solution of an always feasible quadratic programming (QP) subproblem and a mere feasible direction, which is generated by a reduced system of linear equations (SLE). The coefficient matrix of SLE only consists of the constraints and their gradients corresponding to the working set. Then, the second-order correction direction is obtained by solving another reduced SLE to overcome the Maratos effect and obtain the superlinear convergence. In particular, the convergence rate is proved to be locally one-step superlinear under a condition weaker than the strong second-order sufficient condition and without the strict complementarity. Moreover, the line search in our algorithm can effectively combine the initialization processes with the optimization processes, and the line search conditions are weaker than the previous work. Finally, some numerical results are reported.

In this paper, we consider single machine scheduling problems with controllable processing time (resource allocation), truncated job-dependent learning and deterioration effects. The goal is to find the optimal sequence of jobs and the optimal resource allocation separately for minimizing a cost function containing makespan (total completion time, total absolute differences in completion times) and/or total resource cost. For two different processing time functions, i.e., a linear and a convex function of the amount of a common continuously divisible resource allocated to the job, we solve them in polynomial time respectively.

In the supply chain management, configuration of supply chain is the most important decision in the long term and supplier selection and order allocation are the most important decision in the medium-term that are considered separately. Considering these together can overcome the sub-optimality. This paper deals with an integrated model that has two phases. In the first phase, we present a framework for supplier selection criteria in Closed Loop Supply Chain (CLSC). In addition, we define two performance levels for each supplier based on the quantity and capability of purchasing from it to be closer to real world problem. The output of this phase is the score of each supplier in each criterion in each level. In the second phase, we propose a nonlinear multi-objective mixed integer model that determines the number and location of all facilities (strategic decision), flow in each echelon of CLSC (tactical decision) and supplier selection and order allocation (hybrid decision). The objective functions maximize profit and scores of suppliers and minimize total pollution. To solve the model, we have created a transformation based on the piecewise linearization method. The mathematical programming model illustrated by a real numerical example.

The routing and wavelength assignment (RWA) problem in wave-length-division multiplexing optical networks is critically important for increasing the efficiency of wavelength-routed all-optical networks. Given the physical network structure and a set of connection requests, the problem aims to establishing routes for the requests and assigning a wavelength to each of them, subject to the wavelength continuous and wavelength clash constraints. The objective is to minimize he number of required wavelengths to satisfy all requests. The RWA problem was proven to be NP-hard and has been investigated for decades. In this paper, an algorithm based on the maximum edge-disjoint paths is proposed to tackle the RWA problem. Its performance is verified by experiments on some realistic network topologies. Compared with the state-of-the-art bin-packing based methods and the particle swarm optimization algorithm, the proposed method can find the best solution among all testing instances in reasonable computing time.

For quality improvement purposes often times, a manufacturing unit has to change certain parts of equipment. Any such changes in the assembly line manufacturing system or production process involves a cost known as the setup cost. Minimizing the setup cost and improving the product quality is of prime importance in today's competitive business arena. This paper develops the effects of setup cost reduction and quality improvement in a two-echelon supply chain model with deterioration. The objective is to minimize the total cost of the entire supply chain model (SCM) by simultaneously optimizing setup cost, process quality, number of deliveries, and lot size. Numerical examples are provided to illustrate the model.

The current main method to analyze the staged venture investment is some game models, which finally get the optimal contract between the venture entrepreneurs and the venture capitalists by constructing the participation constraint and the incentive constraint. But this method only considers the probability of the success of the project, and ignores whether the project itself is enforceable or not. This paper introduces the concept of the posterior probability, extends the Bergemann and Hege model from the single period to the multi period. Then by using the posterior probability and the successful chance of the project, this paper analyzes the numerous factors which influence the optimal design of the contract under three conditions, such as the fixed and the floating investment in multi-stage and the time when the successful result is related to the current investment quota. What's more, it dose not only give the optimal stop point but compares it in case of the information symmetry and the contrary condition in the floating multi-stage investment. At the same time, it pays attention to the importance of the posterior probability in the present multi-stage venture investment researches. Last but not the least, it provides a reference for the related researches and makes great significance to the venture investment practice.

This paper studies pricing and remanufacturing decisions for two substitutable products in a supply chain with two manufacturers and one common retailer. The two manufacturers produce two substitutable products and sell them to the retailer. Specifically, the first manufacturer is a traditional manufacturer who produces the new product directly from raw material, while the second manufacturer has incorporated a remanufacturing process for used product into his original production system, so that he can manufacture a new product directly from raw material, or remanufacture part or whole of a returned unit into a new product. We establish seven game models by considering the chain members' horizontal and vertical competitions, and obtain the corresponding closed-form expressions for equilibrium solution. Then, the equilibrium characteristics with respect to the second manufacturer's remanufacturing decision and all channel members' pricing decisions are explored, the sensitivity analysis of equilibrium solution is conducted for some model parameters, and the maximal profits and equilibrium solutions obtained in different game models are compared by numerical analyses. Based on these results, some interesting and valuable economic and managerial insights are established.