ISSN:
1547-5816
eISSN:
1553-166X
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Journal of Industrial & Management Optimization
April 2008 , Volume 4 , Issue 2
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2008, 4(2): 213-225
doi: 10.3934/jimo.2008.4.213
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Abstract:
This paper presents a canonical duality approach to solve an integer quadratic programming problem, in which the objective function is quadratic and each variable may assume the value of one of $p~( \ge 3)$ integers. We first transform the problem into a $\{-1,1\}$ integer quadratic programming problem and then derive its ''canonical dual''. It is shown that, under certain conditions, this nonconvex multi-integer programming problem is equivalent to a concave maximization dual problem over a convex feasible domain. A global optimality condition is derived and some computational examples are provided to illustrate this approach.
This paper presents a canonical duality approach to solve an integer quadratic programming problem, in which the objective function is quadratic and each variable may assume the value of one of $p~( \ge 3)$ integers. We first transform the problem into a $\{-1,1\}$ integer quadratic programming problem and then derive its ''canonical dual''. It is shown that, under certain conditions, this nonconvex multi-integer programming problem is equivalent to a concave maximization dual problem over a convex feasible domain. A global optimality condition is derived and some computational examples are provided to illustrate this approach.
2008, 4(2): 227-246
doi: 10.3934/jimo.2008.4.227
+[Abstract](2479)
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Abstract:
This paper considers the computational issue of the optimal stopping problem for the stochastic functional differential equation treated in [6] The finite difference method developed by Barles and Souganidis [3] is used to obtain a numerical approximation for the viscosity solution of the infinite dimensional Hamilton-Jacobi-Bellman variational inequality (HJBVI) associated with the optimal stopping problem. The convergence results are then established.
This paper considers the computational issue of the optimal stopping problem for the stochastic functional differential equation treated in [6] The finite difference method developed by Barles and Souganidis [3] is used to obtain a numerical approximation for the viscosity solution of the infinite dimensional Hamilton-Jacobi-Bellman variational inequality (HJBVI) associated with the optimal stopping problem. The convergence results are then established.
2008, 4(2): 247-270
doi: 10.3934/jimo.2008.4.247
+[Abstract](2653)
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Abstract:
We investigate a nonsmooth Newton's method for the numerical solution of discretized optimal control problems subject to pure state constraints and mixed control-state constraints. The infinite dimensional problem is discretized by application of a general one-step method to the differential equation. By use of the Fischer-Burmeister function the first order necessary conditions for the discretized problem are transformed into an equivalent nonlinear and nonsmooth equation. This nonlinear and nonsmooth equation is solved by a globally convergent nonsmooth Newton's method. Numerical examples for the minimum energy problem and the optimal control of a robot conclude the article.
We investigate a nonsmooth Newton's method for the numerical solution of discretized optimal control problems subject to pure state constraints and mixed control-state constraints. The infinite dimensional problem is discretized by application of a general one-step method to the differential equation. By use of the Fischer-Burmeister function the first order necessary conditions for the discretized problem are transformed into an equivalent nonlinear and nonsmooth equation. This nonlinear and nonsmooth equation is solved by a globally convergent nonsmooth Newton's method. Numerical examples for the minimum energy problem and the optimal control of a robot conclude the article.
2008, 4(2): 271-285
doi: 10.3934/jimo.2008.4.271
+[Abstract](1853)
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Abstract:
In this paper, we first present a concise representation of multivariate polynomial, based on which we deduce the calculation formulae of its derivatives using tensor. Then, we propose a solution method to determine a global descent direction for the minimization of general normal polynomial. At a local and non-global maximizer or saddle point, we could use this method to get a global descent direction of the objective function. By using the global descent direction, we can transform an $n$-dimensional optimization problem into a one-dimensional one. Based on some efficient algorithms for one dimensional global optimization, we develop an algorithm to compute the global minimizer of normal multivariate polynomial. Numerical examples show that the proposed algorithm is promising.
In this paper, we first present a concise representation of multivariate polynomial, based on which we deduce the calculation formulae of its derivatives using tensor. Then, we propose a solution method to determine a global descent direction for the minimization of general normal polynomial. At a local and non-global maximizer or saddle point, we could use this method to get a global descent direction of the objective function. By using the global descent direction, we can transform an $n$-dimensional optimization problem into a one-dimensional one. Based on some efficient algorithms for one dimensional global optimization, we develop an algorithm to compute the global minimizer of normal multivariate polynomial. Numerical examples show that the proposed algorithm is promising.
2008, 4(2): 287-298
doi: 10.3934/jimo.2008.4.287
+[Abstract](2250)
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Abstract:
In this paper, a class of nondifferentiable multiobjective fractional programs is studied, in which every component of the objective function contains a term involving the support function of a compact convex set. Kuhn-Tucker necessary and sufficient optimality conditions, duality and saddle point results for weakly efficient solutions of the nondifferentiable multiobjective fractional programming problems are given. The results presented in this paper improve and extend some the corresponding results in the literature.
In this paper, a class of nondifferentiable multiobjective fractional programs is studied, in which every component of the objective function contains a term involving the support function of a compact convex set. Kuhn-Tucker necessary and sufficient optimality conditions, duality and saddle point results for weakly efficient solutions of the nondifferentiable multiobjective fractional programming problems are given. The results presented in this paper improve and extend some the corresponding results in the literature.
2008, 4(2): 299-312
doi: 10.3934/jimo.2008.4.299
+[Abstract](3213)
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Abstract:
This paper deals with gradient methods for minimizing $n$-dimen-sional strictly convex quadratic functions. Two new adaptive stepsize selection rules are presented and some key properties are proved. Practical insights on the effectiveness of the proposed techniques are given by a numerical comparison with the Barzilai-Borwein (BB) method, the cyclic/adaptive BB methods and two recent monotone gradient methods.
This paper deals with gradient methods for minimizing $n$-dimen-sional strictly convex quadratic functions. Two new adaptive stepsize selection rules are presented and some key properties are proved. Practical insights on the effectiveness of the proposed techniques are given by a numerical comparison with the Barzilai-Borwein (BB) method, the cyclic/adaptive BB methods and two recent monotone gradient methods.
2008, 4(2): 313-327
doi: 10.3934/jimo.2008.4.313
+[Abstract](1977)
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Abstract:
In this paper, we study the parametric well-posedness for vector equilibrium problems and propose a generalized well-posed concept for equilibrium problems with equilibrium constraints (EPEC in short) in topological vector spaces setting. We show that under suitable conditions, the well-posedness defined by approximating solution nets is equivalent to the upper semicontinuity of the solution mapping of perturbed problems. Further, since optimization problems and variational inequality problems are special cases of equilibrium problems, related variational problems can be adopted under some equivalent conditions. Finally, we also study the relationship between well-posedness and parametric well-posedness.
In this paper, we study the parametric well-posedness for vector equilibrium problems and propose a generalized well-posed concept for equilibrium problems with equilibrium constraints (EPEC in short) in topological vector spaces setting. We show that under suitable conditions, the well-posedness defined by approximating solution nets is equivalent to the upper semicontinuity of the solution mapping of perturbed problems. Further, since optimization problems and variational inequality problems are special cases of equilibrium problems, related variational problems can be adopted under some equivalent conditions. Finally, we also study the relationship between well-posedness and parametric well-posedness.
2008, 4(2): 329-337
doi: 10.3934/jimo.2008.4.329
+[Abstract](1876)
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Abstract:
A new measure for network performance evaluation called topology lifetime was introduced in [4, 5]. This measure is based on the notion of unexpected traffic growth and can be used for comparison of topologies. We discuss some advantages and disadvantages of the approach of [4] and suggest some modifications to this approach. In particular we discuss how to evaluate the influence of a subgraph to the lifetime measure and introduce the notion of the order of a path. This notion is useful if we consider a possible extension to the set of working paths in order to support the traffic for the time that is needed for installation of new facilities.
A new measure for network performance evaluation called topology lifetime was introduced in [4, 5]. This measure is based on the notion of unexpected traffic growth and can be used for comparison of topologies. We discuss some advantages and disadvantages of the approach of [4] and suggest some modifications to this approach. In particular we discuss how to evaluate the influence of a subgraph to the lifetime measure and introduce the notion of the order of a path. This notion is useful if we consider a possible extension to the set of working paths in order to support the traffic for the time that is needed for installation of new facilities.
2008, 4(2): 343-351
doi: 10.3934/jimo.2008.4.343
+[Abstract](1729)
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Abstract:
In this paper, we study multi-parametric sensitivity analysis for programming problems with the piecewise linear fractional objective function in the tolerance region. We construct critical regions for simultaneous and independent perturbations in the objective function coefficients and in the right-hand-side vector of the given problem. Necessary and sufficient conditions are derived to classify perturbation parameters as 'focal' and 'non-focal'. Non-focal parameters can be deleted from the analysis, because of their low sensitivity in practice. Theoretical results are illustrated with the help of a numerical example.
In this paper, we study multi-parametric sensitivity analysis for programming problems with the piecewise linear fractional objective function in the tolerance region. We construct critical regions for simultaneous and independent perturbations in the objective function coefficients and in the right-hand-side vector of the given problem. Necessary and sufficient conditions are derived to classify perturbation parameters as 'focal' and 'non-focal'. Non-focal parameters can be deleted from the analysis, because of their low sensitivity in practice. Theoretical results are illustrated with the help of a numerical example.
2008, 4(2): 353-362
doi: 10.3934/jimo.2008.4.353
+[Abstract](2204)
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Abstract:
A filled function method is presented in this paper to solve constrained nonlinear integer programming problems. It is shown that for a given non-global local minimizer, a better local minimizer can be obtained by local search staring from an improved initial point which is obtained by locally solving a box-constrained integer programming problem. Several illustrative numerical examples are reported to show the efficiency of the present method.
A filled function method is presented in this paper to solve constrained nonlinear integer programming problems. It is shown that for a given non-global local minimizer, a better local minimizer can be obtained by local search staring from an improved initial point which is obtained by locally solving a box-constrained integer programming problem. Several illustrative numerical examples are reported to show the efficiency of the present method.
2008, 4(2): 363-384
doi: 10.3934/jimo.2008.4.363
+[Abstract](2359)
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Abstract:
Polyhedral discrepancies are relevant for the quantitative stability of mixed-integer two-stage and chance constrained stochastic programs. We study the problem of optimal scenario reduction for a discrete probability distribution with respect to certain polyhedral discrepancies and develop algorithms for determining the optimally reduced distribution approximately. Encouraging numerical experience for optimal scenario reduction is provided.
Polyhedral discrepancies are relevant for the quantitative stability of mixed-integer two-stage and chance constrained stochastic programs. We study the problem of optimal scenario reduction for a discrete probability distribution with respect to certain polyhedral discrepancies and develop algorithms for determining the optimally reduced distribution approximately. Encouraging numerical experience for optimal scenario reduction is provided.
2008, 4(2): 385-391
doi: 10.3934/jimo.2008.4.385
+[Abstract](2044)
+[PDF](120.5KB)
Abstract:
In this paper, a pair of higher order symmetric dual models for multiobjective nonlinear programming is introduced. The weak, strong and converse duality theorems are proven for the formulated higher order symmetric dual programs under invexity conditions.
In this paper, a pair of higher order symmetric dual models for multiobjective nonlinear programming is introduced. The weak, strong and converse duality theorems are proven for the formulated higher order symmetric dual programs under invexity conditions.
2008, 4(2): 393-406
doi: 10.3934/jimo.2008.4.393
+[Abstract](1904)
+[PDF](222.5KB)
Abstract:
In this paper, we compare between the forward-backward splitting method and the extra-gradient method for solving general variational inequalities. It is known that both of these methods are predictor-corrector methods. They use different search directions in the correction-step. Our analysis explains theoretically why the extra-gradient methods would be better than the forward-backward splitting methods for general variational inequalities. We suggest some new step selection procedure independent of the Lipschitz constant. This is a very desirable circumstance when the operator approximates a differential operator. We prove its convergence in Hilbert spaces of any dimension. Our proof is simple as compared with other methods.
In this paper, we compare between the forward-backward splitting method and the extra-gradient method for solving general variational inequalities. It is known that both of these methods are predictor-corrector methods. They use different search directions in the correction-step. Our analysis explains theoretically why the extra-gradient methods would be better than the forward-backward splitting methods for general variational inequalities. We suggest some new step selection procedure independent of the Lipschitz constant. This is a very desirable circumstance when the operator approximates a differential operator. We prove its convergence in Hilbert spaces of any dimension. Our proof is simple as compared with other methods.
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