ISSN:

1547-5816

eISSN:

1553-166X

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## Journal of Industrial & Management Optimization

2016 , Volume 12 , Issue 1

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2016, 12(1): 1-15
doi: 10.3934/jimo.2016.12.1

*+*[Abstract](270)*+*[PDF](390.2KB)**Abstract:**

In this paper, we consider the class of stochastic generalized Nash equilibrium problems (SGNEP). Such problems have a wide range of applications and have attracted significant attention recently. First, using the first order optimality condition of SGNEP and the nonlinear complementary function, we present an expected residual minimization (ERM) model for the case when the involved functions are not continuously differentiable. Then, we introduce a smoothing function, depending on a smoothing parameter, to yield a smooth approximate ERM model. We further show that the solutions of this smooth ERM model converge to the solutions of the original ERM model as the smoothing parameter tends to zero. Since the ERM formulation contains an expectation, we further propose a sample average approximate problem for the ERM model. Moreover, we show that the global optimal solutions of these approximate problems converge to the global optimal solutions of the ERM problem with probability one. Here, convergence can be achieved in two ways. One is to fix the smoothing parameter, the other is to let the smoothing parameter tend to zero as the sample increases.

2016, 12(1): 17-29
doi: 10.3934/jimo.2016.12.17

*+*[Abstract](316)*+*[PDF](358.1KB)**Abstract:**

In this paper, we present new state and partial state feedback laws as global stabilizers of the well-known frictionless ball and beam system. Dealing with nonlinear terms in the manner different from the ones in the literature, we have achieved a new, simple state-dependent saturation control law. The key technique is to assign a suitable state-dependent saturation level function and jointly use the computation techniques of linear gains. Then, combining such a state feedback law with a homogeneous observer, we again obtain a new partial state feedback design.

2016, 12(1): 31-43
doi: 10.3934/jimo.2016.12.31

*+*[Abstract](683)*+*[PDF](364.4KB)**Abstract:**

This paper investigates the asymptotic behavior of the random-time ruin probability in a time-dependent renewal risk model with pairwise quasi-asymptotically independent and subexponential claims, where the time-dependence structure is constructed between a claim size and its inter-arrival time, and described by a conditional tail probability of the claim size given the inter-arrival time before the claim occurs. In particular, the results we obtained are also valid for the finite-time ruin probability.

2016, 12(1): 45-71
doi: 10.3934/jimo.2016.12.45

*+*[Abstract](217)*+*[PDF](476.3KB)**Abstract:**

This study focuses on discount-offering and demand-rejection decisions for a retailer selling substitutable products. Those products are assumed to be generated different unit net revenues. We first consider a basic model with only two substitutable products and formulate the retailer's objective function by means of dynamic programming model. We show that the retailer can employ simple offering- and rejection-decision rules that are derived in accordance with the stocking quantities of the products. Moreover, it is identified that the retailer's offering decision can also be made according to the number of remaining selling periods. We conduct some numerical examples to testify how those decision rules to vary with regard to different parameters. Further, we extend the basic model to the case with three partially substitutable products. It is shown that the simple offering-decision rules are existed only under some restrictive conditions, and especially, it is difficult to analytically develop any simple rejection-decision rules.

2016, 12(1): 73-82
doi: 10.3934/jimo.2016.12.73

*+*[Abstract](317)*+*[PDF](384.9KB)**Abstract:**

In this paper, we consider a fractional optimal control problem governed by system of linear differential equations, where its cost function is expressed as the ratio of convex and concave functions. The problem is a hard nonconvex optimal control problem and application of Pontriyagin's principle does not always guarantee finding a global optimal control. Even this type of problems in a finite dimensional space is known as NP hard. This optimal control problem can, in principle, be solved by Dinkhelbach algorithm [10]. However, it leads to solving a sequence of hard D.C programming problems in its finite dimensional analogy. To overcome this difficulty, we introduce a reachable set for the linear system. In this way, the problem is reduced to a quasiconvex maximization problem in a finite dimensional space. Based on a global optimality condition, we propose an algorithm for solving this fractional optimal control problem and we show that the algorithm generates a sequence of local optimal controls with improved cost values. The proposed algorithm is then applied to several test problems, where the global optimal cost value is obtained for each case.

2016, 12(1): 83-102
doi: 10.3934/jimo.2016.12.83

*+*[Abstract](269)*+*[PDF](498.5KB)**Abstract:**

How to manage the social security trust funds is a topic of wide interests both academically and professionally. In the setting of portfolio selection with social security funds investment, we propose a modified safety-first (MSF) rule with portfolio selection including risk-free saving. We first demonstrate under some mild assumptions that the solution to the MSF model for an individual investor can be expressed by explicitly analytical formula and the necessary and sufficient conditions for their existence are obtained. We then derive the safety-first efficient frontiers in both the space of expected return and insured return level and the space of standard deviation and expected return, with numerical examples illustrated. By comparing the results of the MSF model with those of the mean-variance (M-V) model, some novel insights into the differences between them are further gained.

2016, 12(1): 103-116
doi: 10.3934/jimo.2016.12.103

*+*[Abstract](243)*+*[PDF](391.2KB)**Abstract:**

Based on an equivalent reformulation of the central path, we obtain a modified-Newton step for linear optimization. Using this step, we propose an infeasible interior-point algorithm. The algorithm uses only one full-modified-Newton step search in each iteration. The complexity bound of the algorithm is the best known for infeasible interior-point algorithm.

2016, 12(1): 117-140
doi: 10.3934/jimo.2016.12.117

*+*[Abstract](189)*+*[PDF](528.9KB)**Abstract:**

In a distribution network, materials or products that go through a decomposition process can be considered as flows entering a specialized node, called D-node, which distributes each decomposed flow along an outgoing arc. Flows on each arc emanating from a D-node have to obey a pre-specified proportional relationship, in addition to the capacity constraints. The solution procedures for calculating optimal flows over distribution networks in literature often assumes D-nodes to be disjoint, whereas in reality D-nodes may often connect to each other and complicate the problem. In this paper, we propose a polynomial-time network compaction scheme that compresses a distribution network into an equivalent one of smaller size, which can then be directly solved by conventional solution methods in related literature. In order to provide test cases of distribution networks containing D-nodes for computational tests in related research, we implement a random network generator that produces a connected and acyclic distribution network in a compact form. Mathematical properties together with their proofs are also discussed to provide more insights in the design of our generator.

2016, 12(1): 141-167
doi: 10.3934/jimo.2016.12.141

*+*[Abstract](290)*+*[PDF](1410.3KB)**Abstract:**

From the perspective of business management, system supervisors are usually more concerned with the cost of a system rather than its reliability. This study determines the optimal component allocation based on the cost criterion for a computer system subject to a reliability threshold in which the computer system is represented as a network composed of a set of links and a set of vertices. The component allocation means allocating some from the set of components to the network's links, where the cost of allocating a component is counted in terms of the length. Any computer network associated with a component allocation is called a multistate computer network (MCN) because each component has multiple states with a probability distribution. Associated with a component allocation, the system reliability is the probability that the specific units of data are successfully transmitted through the MCN. An optimization algorithm, which integrates tabu search and minimal paths, is proposed to solve the problem under consideration. Several benchmark computer networks are utilized to demonstrate the computational efficiency of the proposed algorithm compared with several popular meta-heuristic algorithms.

2016, 12(1): 169-186
doi: 10.3934/jimo.2016.12.169

*+*[Abstract](256)*+*[PDF](451.3KB)**Abstract:**

This work concerns the robust stabilization of a class of ``Quasi-Lipschitz" nonlinear uncertain systems governed by stochastic Differential Equations (SDE) subject to both multiplicative and additive stochastic noises modeled by a vector Brownian motion. The state-vector is admitted to be non-completely available, and be estimated by a Luenberger-type filter. The stabilization around the origin is realized by a linear feedback proportional to the current state-estimates. First, the class of feedback matrices and filter matrix-gains, providing the boundedness of the stochastic trajectories with probability one in a vicinity of the origin, is specified. Then a corresponding ellipsoid, containing these trajectories, is found. Its ``size" (the trace of the ellipsoid matrix) is derived as a function of the applied gain matrices. To make this ellipsoid ``as small as possible" the corresponding constrained optimization problem is suggested to be solved. These constraints are given by a system of Matrix Inequalities (MI's) which under a specific change of variables may be converted into a conventional system of Bilinear Matrix Inequalities (BMI's). The last may be resolved by the standard MATLAB toolboxes such as ``penbmiTL, Tomlab toolbox". Finally, a numerical example, containing the arctangent-type nonlinearities, is presented to illustrate the effectiveness of the suggested methodology.

2016, 12(1): 187-209
doi: 10.3934/jimo.2016.12.187

*+*[Abstract](335)*+*[PDF](580.1KB)**Abstract:**

This paper studies dynamic asset allocation with stochastic interest rates and inflation rates under the continuous-time mean-variance model in a more general market that may be incomplete. First, by the Lagrange method and the dynamic programming approach, we derive the associated Hamilton-Jacobi-Bellman equation and solve it explicitly. Then, closed form expressions for the efficient strategy and the efficient frontier are derived by applying the Lagrange dual theory. In addition, we state a necessary and sufficient condition under which the efficient frontier is a straight line in the standard deviation-mean plane, and some degenerate cases are discussed. Finally, empirical analysis based on real data from the Chinese market is presented to illustrate applications of the results obtained in this paper.

2016, 12(1): 211-228
doi: 10.3934/jimo.2016.12.211

*+*[Abstract](242)*+*[PDF](456.9KB)**Abstract:**

How to equitably distribute a common fixed cost among decision making units (DMUs) of an organization is a typical problem in organization management. Based on the data envelopment analysis technique, this paper proposes a new proportional sharing model to determine a unique fixed cost allocation under two assumptions: efficiency invariance and zero slack. It is noteworthy that the fixed cost allocation determined by our proportional sharing model is a feasible solution to the model proposed by Cook and Zhu [Cook and Zhu,

*Allocation of shared costs among decision making units: A DEA approach*, Computers & Operations Research,

**32**(2005) 2171-2178]. To ensure the uniqueness of the fixed cost allocation, three algorithms are proposed under the new model. Different from current fixed cost allocation methods under the efficiency invariance assumption, our approach can generate a fixed cost allocation that is unique, partially dependent of DMUs' inputs and units-invariant, and thus is more effective. Numerical examples are used to illustrate the validity and superiorities of our approach.

2016, 12(1): 229-249
doi: 10.3934/jimo.2016.12.229

*+*[Abstract](366)*+*[PDF](441.3KB)**Abstract:**

Due to the non-separability of the variance operator, the optimal investment policy of the multi-period mean-variance model in Markovian markets doesn't satisfy the time consistency. We propose a new weak time consistency in stochastic markets and show that the pre-commitment optimal policy satisfies the weak time consistency at any intermediate period as long as the investor's wealth is no more than a specific threshold. When the investor's wealth exceeds the threshold, the weak time consistency no longer holds. In this case, by modifying the pre-commitment optimal policy, we derive a wealth interval, from which we determine a more efficient revised policy. The terminal wealth obtained under this revised policy can achieve the same mean as, but not greater variance than those of the terminal wealth obtained under the pre-commitment optimal policy; a series of superior investment policies can be obtained depending on the degree the investor wants the conditional variance to decrease. It is shown that, in the above revising process, a positive cash flow can be taken out of the market. Finally, an empirical example illustrates our theoretical results. Our results generalize existing conclusions for the multi-period mean-variance model in deterministic markets.

2016, 12(1): 251-268
doi: 10.3934/jimo.2016.12.251

*+*[Abstract](348)*+*[PDF](1594.7KB)**Abstract:**

Canonical correlation analysis(CCA) is a well-known technique for simultaneously reducing two relevant data sets, and finding maximal correlation between them. However, it fails to preserve the local structure of each data set, as well as the global discriminant ability, which are important in real applications. In this paper, a new CCA model, called discriminant minimum class locality preserving canonical correlation analysis(called as DMPCCA) is proposed. The proposed method introduces locall structure information and global discriminant information into the classical CCA and considers a optimal combination of intra-class locality preserving, global discriminant ability and the maximal correlation between two sets. The experiments on data visualization, web image retrieval and face recognition validate the effectiveness of the proposed method.

2016, 12(1): 269-283
doi: 10.3934/jimo.2016.12.269

*+*[Abstract](279)*+*[PDF](398.2KB)**Abstract:**

In this paper, we study the polyhedral cone constrained homogeneous quadratic programming problem and provide an equivalent linear conic reformulation. Based on a union of second-order cones which covers the polyhedral cone, a sequence of computable linear conic programming problems are constructed to approximate the linear conic reformulation. The convergence of the sequential solutions is guaranteed as the number of second-order cones increases such that the union of the second-order cones gets close to the polyhedral cone. In order to relieve the computational burden and improve the efficiency, an adaptive scheme and valid inequalities derived by the reformulation-linearization technique are added to the proposed algorithm. Finally, the numerical results demonstrate the effectiveness of the algorithm.

2016, 12(1): 285-301
doi: 10.3934/jimo.2016.12.285

*+*[Abstract](262)*+*[PDF](1143.7KB)**Abstract:**

Support vector machines (SVMs) with positive semidefinite kernels yield convex quadratic programming problems. SVMs with indefinite kernels yield nonconvex quadratic programming problems. Most optimization methods for SVMs rely on the convexity of objective functions and are not efficient for solving such nonconvex problems. In this paper, we propose a subgradient-based neural network (SGNN) for the problems cast by SVMs with indefinite kernels. It is shown that the state of the proposed neural network has finite length, and as a consequence it converges toward a singleton. The coincidence between the solution and the

*slow solution*of SGNN is also proved starting from the initial value of SGNN. Moreover, we employ the Łojasiewicz inequality to exploit the convergence rate of trajectory of SGNN. The obtained results show that each trajectory is either exponentially convergent, or convergent in finite time, toward a singleton belonging to the set of constrained critical points through a quantitative evaluation of the Łojasiewicz exponent at the equilibrium points. This method is easy to implement without adding any new parameters. Three benchmark data sets from the University of California, Irvine machine learning repository are used in the numerical tests. Experimental results show the efficiency of the proposed neural network.

2016, 12(1): 303-315
doi: 10.3934/jimo.2016.12.303

*+*[Abstract](468)*+*[PDF](378.7KB)**Abstract:**

This paper studies the robust finite-time $H_\infty$ control for a class of nonlinear systems with time-varying delay and disturbances via output feedback. Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel delay-dependent conditions for the existence of output feedback controllers are established in terms of linear matrix inequalities (LMIs). The proposed conditions allow us to design the output feedback controllers which robustly stabilize the closed-loop system in the finite-time sense. An application to $H_\infty$ control of uncertain linear systems with interval time-varying delay is also given. A numerical example is given to illustrate the efficiency of the proposed method.

2016, 12(1): 317-336
doi: 10.3934/jimo.2016.12.317

*+*[Abstract](352)*+*[PDF](421.4KB)**Abstract:**

In this paper, we propose an alternating-direction-type numerical method to solve a class of inverse semi-definite quadratic programming problems. An explicit solution to one direction subproblem is given and the other direction subproblem is proved to be a convex quadratic programming problem over positive semi-definite symmetric matrix cone. We design a spectral projected gradient method for solving the quadratic matrix optimization problem and demonstrate its convergence. Numerical experiments illustrate that our method can solve inverse semi-definite quadratic programming problems efficiently.

2016, 12(1): 337-355
doi: 10.3934/jimo.2016.12.337

*+*[Abstract](416)*+*[PDF](1300.4KB)**Abstract:**

Supply chain members coordinate with each other in order to obtain more profit. The major mechanisms for coordination among supply chain echelons are pricing, inventory management, and ordering decisions. Regarding to these mechanisms, supply chain participants have conflicts of interest. This paper concerns coordination of enterprise decisions including pricing, advertising, ordering, and inventory decisions in a multi-echelon supply chain consisting of multiple suppliers, one manufacturer, and multiple retailers. In the current study, a novel inventory model is presented for both the manufacturer, and the retailers who are able to determine the number of orders for each product. Moreover, each supply chain member has equal power and make their decisions simultaneously. The proposed model considers the relationships among three echelon supply chain members based on a non-cooperative Nash game with pricing and inventory decisions. An iterative solution algorithm is proposed to find Nash equilibrium point of the game. Several numerical examples are presented to study the application of the model as well as the effectiveness of the algorithm. Finally, a comprehensive sensitivity analysis is performed and some important managerial insights are highlighted.

2016, 12(1): 357-373
doi: 10.3934/jimo.2016.12.357

*+*[Abstract](399)*+*[PDF](702.5KB)**Abstract:**

In this paper, using the concept of Fisher discriminant analysis and a new fuzzy membership function, a kernel-free fuzzy quadratic surface support vector machine model is proposed for binary classification. The membership function is specially designed to consider not only the ``quadratic-margin distance'' between a training point and its related ``quadratic center surface'' but also the affinity among training points. A decomposition algorithm is designed to solve the proposed model. Computational results on artificial and four real-world classifying data sets indicate that the proposed model outperforms fuzzy support vector machine models with Gaussian or Quadratic kernel and soft quadratic surface support vector machine model, especially, when the data sets contain a large amount of outliers and noises.

2016, 12(1): 375-387
doi: 10.3934/jimo.2016.12.375

*+*[Abstract](260)*+*[PDF](398.9KB)**Abstract:**

In this paper, a new definition of the filled function is given. Based on the new definition, a new class of filled functions is constructed, and the properties of the new filled functions are analysed and discussed. Moreover, according to the new class of filled functions, a criterion is given to decide whether the point we have obtained is an approximate global optimal solution. Finally, a global optimization algorithm based on the new class of filled functions is presented. The implementation of the algorithm on several test problems is reported with numerical results.

2016, 12(1): 389-402
doi: 10.3934/jimo.2016.12.389

*+*[Abstract](301)*+*[PDF](342.7KB)**Abstract:**

In this paper, we consider the problem of minimizing a nonsmooth convex objective which is the sum of a proper, nonsmooth, closed, strongly convex extend real-valued function with a proper, nonsmooth, closed, convex extend real-valued function which is a composition of a proper closed convex function and a nonzero affine map. We first establish its dual problem which consists of minimizing the sum of a smooth convex function with a closed proper nonsmooth convex function. Then we apply first order proximal gradient methods on the dual problem, where an error is present in the calculation of the gradient of the smooth term. Further we present a dual proximal-gradient methods with approximate gradient. We show that when the errors are summable although the dual lowest objective function sequence generated by the proximal-gradient method with the errors converges to the optimal value with the rate $O(\frac{1}{k})$, the rate of convergence of the primal sequence is of the order $O(\frac{1}{\sqrt{k}}$).

2016 Impact Factor: 0.994

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