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1547-5816
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1553-166X
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Journal of Industrial & Management Optimization
January 2013 , Volume 9 , Issue 1
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2013, 9(1): 1-12
doi: 10.3934/jimo.2013.9.1
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Abstract:
The generalized minimax theorems of two real set-valued mappings, which generalize and improve the corresponding conclusions of D. Kuroiwa (Appl. Math. Lett., 1996, 9: 97-101.) and Li et.al.(J. Optim. Theory Appl., 2000, 106: 183-200; J. Math. Anal. Appl., 2003, 281: 707-723.), and the generalized minimax theorems of two set-valued mappings valued in Fréchet spaces are proved. Some examples are given also to show the results.
The generalized minimax theorems of two real set-valued mappings, which generalize and improve the corresponding conclusions of D. Kuroiwa (Appl. Math. Lett., 1996, 9: 97-101.) and Li et.al.(J. Optim. Theory Appl., 2000, 106: 183-200; J. Math. Anal. Appl., 2003, 281: 707-723.), and the generalized minimax theorems of two set-valued mappings valued in Fréchet spaces are proved. Some examples are given also to show the results.
2013, 9(1): 13-30
doi: 10.3934/jimo.2013.9.13
+[Abstract](2086)
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Abstract:
This paper considers a multi-period inventory problem with partially observed supply capacity in the lost sales case. Partially observed supply means that exact available supply in a period is observed only when the order quantity is not less than the supply capacity. Then, these observations are used to update the supply capacity distribution from one period to the next. For this inventory problem with partially observed supply information and random demand, we establish the inventory model according to a known Markov decision process(MDP) space. The existence of an optimal policy for this inventory problem is proved. Finally, some numerical examples considering Poisson distributed demand are given to verify the ability to find an optimal order quantity.
This paper considers a multi-period inventory problem with partially observed supply capacity in the lost sales case. Partially observed supply means that exact available supply in a period is observed only when the order quantity is not less than the supply capacity. Then, these observations are used to update the supply capacity distribution from one period to the next. For this inventory problem with partially observed supply information and random demand, we establish the inventory model according to a known Markov decision process(MDP) space. The existence of an optimal policy for this inventory problem is proved. Finally, some numerical examples considering Poisson distributed demand are given to verify the ability to find an optimal order quantity.
2013, 9(1): 31-56
doi: 10.3934/jimo.2013.9.31
+[Abstract](3353)
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Abstract:
In this paper, production-distribution planning in construction supply chain management is investigated. A bi-level model for a production-distribution planning problem under a fuzzy random environment is presented. Construction projects, the leader in the hierarchy, not only control the allocation of items to each depot but are also responsible for ordering items from the manufacturing company. The manufacturing company, the follower in the hierarchy, reacts to these orders by deciding which manufacturing plants and production lines are going to be used. The model presented in this paper is able to handle such practical issues and is solved using an improved Artificial Bee Colony algorithm based on a fuzzy random simulation. A case study is presented to illustrate the effectiveness of the proposed approach. Conclusions and future research directions are discussed.
In this paper, production-distribution planning in construction supply chain management is investigated. A bi-level model for a production-distribution planning problem under a fuzzy random environment is presented. Construction projects, the leader in the hierarchy, not only control the allocation of items to each depot but are also responsible for ordering items from the manufacturing company. The manufacturing company, the follower in the hierarchy, reacts to these orders by deciding which manufacturing plants and production lines are going to be used. The model presented in this paper is able to handle such practical issues and is solved using an improved Artificial Bee Colony algorithm based on a fuzzy random simulation. A case study is presented to illustrate the effectiveness of the proposed approach. Conclusions and future research directions are discussed.
2013, 9(1): 57-74
doi: 10.3934/jimo.2013.9.57
+[Abstract](2629)
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Abstract:
In this paper, some characterizations for the solution sets of a class of set-valued vector mixed variational inequalities to be nonempty and bounded are presented in real reflexive Banach spaces. An equivalence relation between the solution sets of the vector mixed variational inequalities and the weakly efficient solution sets of the vector optimization problems is shown under some suitable assumptions. By using some known results for the vector optimization problems, several characterizations for the solution sets of the vector mixed variational inequalities are obtained in real reflexive Banach spaces. Furthermore, some stability results for the vector mixed variational inequality are given when the mapping and the constraint set are perturbed by two different parameters. Finally, the upper semicontinuity and the lower semicontinuity of the solution sets are given under some suitable assumptions which are different from the ones used in [7, 11, 22]. Some examples are also given to illustrate our results.
In this paper, some characterizations for the solution sets of a class of set-valued vector mixed variational inequalities to be nonempty and bounded are presented in real reflexive Banach spaces. An equivalence relation between the solution sets of the vector mixed variational inequalities and the weakly efficient solution sets of the vector optimization problems is shown under some suitable assumptions. By using some known results for the vector optimization problems, several characterizations for the solution sets of the vector mixed variational inequalities are obtained in real reflexive Banach spaces. Furthermore, some stability results for the vector mixed variational inequality are given when the mapping and the constraint set are perturbed by two different parameters. Finally, the upper semicontinuity and the lower semicontinuity of the solution sets are given under some suitable assumptions which are different from the ones used in [7, 11, 22]. Some examples are also given to illustrate our results.
2013, 9(1): 75-97
doi: 10.3934/jimo.2013.9.75
+[Abstract](2211)
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Abstract:
This paper studies a class of nonlinear Lagrangian algorithms for solving unconstrained minimax problems, which will provide an approach to constructing a concrete and novel Lagrangian algorithm and simplify the relevant theory analysis. A class of nonlinear Lagrangians is constructed and a set of mild conditions on them are proposed to guarantee the convergence theory of the corresponding algorithms. The unified convergence analysis framework for the class of algorithms is established. It is shown that the sequence solutions obtained by the class of algorithms are Q-linearly convergent when the controlling parameter is less than a threshold under some assumptions and the error bounds of the sequence solutions are obtained at the same time. The properties of dual problem framework based on the unified nonlinear Lagrangians are analyzed, which the classical Lagrangian lacks. Furthermore, it is presented that the proposed class of nonlinear Lagrangians contains four well-known nonlinear Lagrangians for unconstrained minimax problems appearing in the literatures. At last, numerical results for ten typical test problems are reported, based on the four nonlinear Lagrangians in the proposed class.
This paper studies a class of nonlinear Lagrangian algorithms for solving unconstrained minimax problems, which will provide an approach to constructing a concrete and novel Lagrangian algorithm and simplify the relevant theory analysis. A class of nonlinear Lagrangians is constructed and a set of mild conditions on them are proposed to guarantee the convergence theory of the corresponding algorithms. The unified convergence analysis framework for the class of algorithms is established. It is shown that the sequence solutions obtained by the class of algorithms are Q-linearly convergent when the controlling parameter is less than a threshold under some assumptions and the error bounds of the sequence solutions are obtained at the same time. The properties of dual problem framework based on the unified nonlinear Lagrangians are analyzed, which the classical Lagrangian lacks. Furthermore, it is presented that the proposed class of nonlinear Lagrangians contains four well-known nonlinear Lagrangians for unconstrained minimax problems appearing in the literatures. At last, numerical results for ten typical test problems are reported, based on the four nonlinear Lagrangians in the proposed class.
2013, 9(1): 99-115
doi: 10.3934/jimo.2013.9.99
+[Abstract](2265)
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Abstract:
In this paper, we deal with the semi-infinite complementarity problems (SICP), in which several important issues are covered, such as solvability, semismoothness of residual functions, and error bounds. In particular, we characterize the solution set by investigating the relationship between SICP and the classical complementarity problem. Furthermore, we show that the SICP can be equivalently reformulated as a typical semi-infinite min-max programming problem by employing NCP functions. Finally, we study the concept of error bounds and introduce its two variants, $\varepsilon$-error bounds and weak error bounds, where the concept of weak error bounds is highly desirable in that the solution set is not restricted to be nonempty.
In this paper, we deal with the semi-infinite complementarity problems (SICP), in which several important issues are covered, such as solvability, semismoothness of residual functions, and error bounds. In particular, we characterize the solution set by investigating the relationship between SICP and the classical complementarity problem. Furthermore, we show that the SICP can be equivalently reformulated as a typical semi-infinite min-max programming problem by employing NCP functions. Finally, we study the concept of error bounds and introduce its two variants, $\varepsilon$-error bounds and weak error bounds, where the concept of weak error bounds is highly desirable in that the solution set is not restricted to be nonempty.
2013, 9(1): 117-129
doi: 10.3934/jimo.2013.9.117
+[Abstract](3285)
+[PDF](422.7KB)
Abstract:
In this paper, we present a multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints, which can be viewed as an extension of multivariate spectral gradient method for solving unconstrained optimization problems. The proposed method does not need the computation of the derivative as well as the solution of some linear equations. Under some suitable conditions, we can establish its global convergence results. Preliminary numerical results show that the proposed method is efficient and promising.
In this paper, we present a multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints, which can be viewed as an extension of multivariate spectral gradient method for solving unconstrained optimization problems. The proposed method does not need the computation of the derivative as well as the solution of some linear equations. Under some suitable conditions, we can establish its global convergence results. Preliminary numerical results show that the proposed method is efficient and promising.
2013, 9(1): 131-142
doi: 10.3934/jimo.2013.9.131
+[Abstract](2875)
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Abstract:
In this paper, we define the concepts of $LU$ optimal solution to interval-valued programming problem. By considering the concepts of $LU$ optimal solution, the Fritz John type and Kuhn-Tucker type necessary and sufficient optimality conditions for nondifferentiable interval programming are derived. Further, we establish the dual problem and prove the duality theorems.
In this paper, we define the concepts of $LU$ optimal solution to interval-valued programming problem. By considering the concepts of $LU$ optimal solution, the Fritz John type and Kuhn-Tucker type necessary and sufficient optimality conditions for nondifferentiable interval programming are derived. Further, we establish the dual problem and prove the duality theorems.
2013, 9(1): 143-151
doi: 10.3934/jimo.2013.9.143
+[Abstract](2337)
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Abstract:
In this paper, some scalar characterizations of approximate weakly efficient solutions and approximate Henig efficient solutions for vector equilibrium problems are derived without imposing any convexity assumption on objective functions and feasible set. Meanwhile, the linear scalar characterization of approximate weakly efficient solutions is also established under the conditions of generalized convexity. As an application of the results in this paper, scalar characterizations of weakly efficient solution, Henig efficient solution and super efficient solution for vector equilibrium problems are obtained.
In this paper, some scalar characterizations of approximate weakly efficient solutions and approximate Henig efficient solutions for vector equilibrium problems are derived without imposing any convexity assumption on objective functions and feasible set. Meanwhile, the linear scalar characterization of approximate weakly efficient solutions is also established under the conditions of generalized convexity. As an application of the results in this paper, scalar characterizations of weakly efficient solution, Henig efficient solution and super efficient solution for vector equilibrium problems are obtained.
2013, 9(1): 153-169
doi: 10.3934/jimo.2013.9.153
+[Abstract](2106)
+[PDF](436.9KB)
Abstract:
This paper is devoted to the study of the proximal point algorithm for solving monotone and nonmonotone nonlinear complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. The motivations of this paper are twofold. One is analyzing the proximal point algorithm based on the generalized Fischer-Burmeister function which includes the Fischer-Burmeister function as special case, another one is trying to see if there are relativistic change on numerical performance when we adjust the parameter in the generalized Fischer-Burmeister.
This paper is devoted to the study of the proximal point algorithm for solving monotone and nonmonotone nonlinear complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. The motivations of this paper are twofold. One is analyzing the proximal point algorithm based on the generalized Fischer-Burmeister function which includes the Fischer-Burmeister function as special case, another one is trying to see if there are relativistic change on numerical performance when we adjust the parameter in the generalized Fischer-Burmeister.
2013, 9(1): 171-189
doi: 10.3934/jimo.2013.9.171
+[Abstract](2762)
+[PDF](488.2KB)
Abstract:
A type of inverse linear second-order cone programming problems is discussed, in which the parameters in both the objective function and the constraint set of a given linear second-order cone programming need to be adjusted as little as possible so that a known feasible solution becomes optimal. This inverse problem can be formulated as a minimization problem with second-order cone complementarity constraints. With the help of the smoothed Fischer-Burmeister function over second-order cones, we construct a smoothing approximation of the formulated problem whose feasible set and optimal solution set are demonstrated to be continuous and outer semicontinuous respectively at the perturbed parameter $\varepsilon=0$. A damped Newton method is employed to solve the perturbed problem and its global convergence and local quadratic convergence rate are shown. Finally, the numerical results are reported to show the effectiveness of the damped Newton method for solving the inverse linear second-order cone programming.
A type of inverse linear second-order cone programming problems is discussed, in which the parameters in both the objective function and the constraint set of a given linear second-order cone programming need to be adjusted as little as possible so that a known feasible solution becomes optimal. This inverse problem can be formulated as a minimization problem with second-order cone complementarity constraints. With the help of the smoothed Fischer-Burmeister function over second-order cones, we construct a smoothing approximation of the formulated problem whose feasible set and optimal solution set are demonstrated to be continuous and outer semicontinuous respectively at the perturbed parameter $\varepsilon=0$. A damped Newton method is employed to solve the perturbed problem and its global convergence and local quadratic convergence rate are shown. Finally, the numerical results are reported to show the effectiveness of the damped Newton method for solving the inverse linear second-order cone programming.
2013, 9(1): 191-204
doi: 10.3934/jimo.2013.9.191
+[Abstract](2407)
+[PDF](365.9KB)
Abstract:
A problem of minimization of $L_1$-penalized conditional value-at-risk (CVaR) is considered. It is shown that there exists a non-negative threshold value of the penalty parameter such that the optimal value of the penalized problem is unbounded if the penalty parameter is less than the threshold value, and it is bounded if the penalty parameter is greater or equal than this value. It is established that the threshold value can be found via the solution of a linear programming problem, and, therefore, readily computable. Theoretical results are illustrated by numerical examples.
A problem of minimization of $L_1$-penalized conditional value-at-risk (CVaR) is considered. It is shown that there exists a non-negative threshold value of the penalty parameter such that the optimal value of the penalized problem is unbounded if the penalty parameter is less than the threshold value, and it is bounded if the penalty parameter is greater or equal than this value. It is established that the threshold value can be found via the solution of a linear programming problem, and, therefore, readily computable. Theoretical results are illustrated by numerical examples.
2013, 9(1): 205-225
doi: 10.3934/jimo.2013.9.205
+[Abstract](1865)
+[PDF](1412.9KB)
Abstract:
This paper focuses on empirical design and performs real data test of a novel algorithm that contributes to the purpose of solving a specific SIP problem arising from a classical wideband active sonar echo location system in noisy environment. The algorithm is achieved by firstly isolating potential contact signals of interest embedded in the scattered returns through the first-stage denoising using an adaptive noise canceling (ANC) neuro-fuzzy scheme. The ANC output is then feed into an iterative target motion analysis (TMA) scheme composed of the second-stage denoising and optimal motion estimation. In the first-stage denoising, the adaptive neuro-fuzzy inference system (ANFIS) is the core processor of ANC for tracking both the linear and nonlinear relations among complex contact signals. The second-stage denoising is appealed for further noise compression and is accomplished via trimmed-mean (TM) levelization and discrete wavelet denoising (WDeN). The two-stage comprehensive denoising techniques yield fine tuned signals for the system deconvolution based on solving a semi-infinite programming (SIP) problem. These two schemes form an ANC-TMA(CWT) algorithm for rapid processing of target echoes and provide a higher degree of signal detection capability with an increased robustness against false signal detections. Advantages and simulation results are discussed in terms of detection performance and computational time consumption.
This paper focuses on empirical design and performs real data test of a novel algorithm that contributes to the purpose of solving a specific SIP problem arising from a classical wideband active sonar echo location system in noisy environment. The algorithm is achieved by firstly isolating potential contact signals of interest embedded in the scattered returns through the first-stage denoising using an adaptive noise canceling (ANC) neuro-fuzzy scheme. The ANC output is then feed into an iterative target motion analysis (TMA) scheme composed of the second-stage denoising and optimal motion estimation. In the first-stage denoising, the adaptive neuro-fuzzy inference system (ANFIS) is the core processor of ANC for tracking both the linear and nonlinear relations among complex contact signals. The second-stage denoising is appealed for further noise compression and is accomplished via trimmed-mean (TM) levelization and discrete wavelet denoising (WDeN). The two-stage comprehensive denoising techniques yield fine tuned signals for the system deconvolution based on solving a semi-infinite programming (SIP) problem. These two schemes form an ANC-TMA(CWT) algorithm for rapid processing of target echoes and provide a higher degree of signal detection capability with an increased robustness against false signal detections. Advantages and simulation results are discussed in terms of detection performance and computational time consumption.
2013, 9(1): 227-241
doi: 10.3934/jimo.2013.9.227
+[Abstract](3045)
+[PDF](426.9KB)
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In this paper, both the constrained Levenberg-Marquardt method and the projected Levenberg-Marquardt method are presented for nonlinear equations $F(x)=0$ subject to $x\in X$, where $X$ is a nonempty closed convex set. The Levenberg-Marquardt parameter is taken as $\| F(x_k) \|_2^\delta$ with $\delta\in (0, 2]$. Under the local error bound condition which is weaker than nonsingularity, the methods are shown to have the same convergence rate, which includes not only the convergence results obtained in [12] for $\delta=2$ but also the results given in [7] for unconstrained nonlinear equations.
In this paper, both the constrained Levenberg-Marquardt method and the projected Levenberg-Marquardt method are presented for nonlinear equations $F(x)=0$ subject to $x\in X$, where $X$ is a nonempty closed convex set. The Levenberg-Marquardt parameter is taken as $\| F(x_k) \|_2^\delta$ with $\delta\in (0, 2]$. Under the local error bound condition which is weaker than nonsingularity, the methods are shown to have the same convergence rate, which includes not only the convergence results obtained in [12] for $\delta=2$ but also the results given in [7] for unconstrained nonlinear equations.
2013, 9(1): 243-253
doi: 10.3934/jimo.2013.9.243
+[Abstract](2596)
+[PDF](445.0KB)
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This paper presents an outcome-space outer approximation algorithm for solving the problem of minimizing the product of two convex functions over a compact convex set in $\mathbb{R}^n$. The computational experiences are reported. The proposed algorithm is convergent.
This paper presents an outcome-space outer approximation algorithm for solving the problem of minimizing the product of two convex functions over a compact convex set in $\mathbb{R}^n$. The computational experiences are reported. The proposed algorithm is convergent.
2013, 9(1): 255-274
doi: 10.3934/jimo.2013.9.255
+[Abstract](3030)
+[PDF](455.8KB)
Abstract:
In the last decade, as calibrations of the classical traffic equilibrium problems, various models of traffic equilibrium problems with nonadditive route costs have been proposed. For solving such models, this paper develops a self-adaptive projection-auxiliary problem method for monotone variational inequality (VI) problems. It first converts the original problem where the feasible set is the intersection of a linear manifold and a simple set to an augmented VI with simple set, which makes the projection easy to implement. The self-adaptive strategy avoids the difficult task of choosing `suitable' parameters, and leads to fast convergence. Under suitable conditions, we prove the global convergence of the method. Some preliminary computational results are presented to illustrate the ability and efficiency of the method.
In the last decade, as calibrations of the classical traffic equilibrium problems, various models of traffic equilibrium problems with nonadditive route costs have been proposed. For solving such models, this paper develops a self-adaptive projection-auxiliary problem method for monotone variational inequality (VI) problems. It first converts the original problem where the feasible set is the intersection of a linear manifold and a simple set to an augmented VI with simple set, which makes the projection easy to implement. The self-adaptive strategy avoids the difficult task of choosing `suitable' parameters, and leads to fast convergence. Under suitable conditions, we prove the global convergence of the method. Some preliminary computational results are presented to illustrate the ability and efficiency of the method.
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