# American Institute of Mathematical Sciences

ISSN:
1547-5816

eISSN:
1553-166X

All Issues

## Journal of Industrial & Management Optimization

April 2007 , Volume 3 , Issue 2

A special issue dedicated to the memory of Professor Alexander Rubinov

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2007, 3(2): i-ii doi: 10.3934/jimo.2007.3.2i +[Abstract](643) +[PDF](36.2KB)
Abstract:
This special issue of ''Journal of Industrial and Management Optimization'' is dedicated to the memory of Professor Alexander Rubinov.
2007, 3(2): 165-171 doi: 10.3934/jimo.2007.3.165 +[Abstract](1608) +[PDF](129.8KB)
Abstract:
We present a class of gap functions for the quasi-variational inequality problem (QVIP). We show the equivalence between the optimization reformulation with the gap function and the original QVIP. We also give conditions under which the gap function is continuous and directionally differentiable.
2007, 3(2): 173-191 doi: 10.3934/jimo.2007.3.173 +[Abstract](1075) +[PDF](313.5KB)
Abstract:
Quadratic assignment problems are one of the famous strongly NP-hard combinatorial optimization problems. Because of the wide area of applications and the difficulties existing in finding optimal solutions, these problems are one of the most common objects for application of optimization methods. This paper examines the use of a generalized version of the modified subgradient algorithm for solving quadratic assignment problems. Promising computational results are reported for benchmark problem instances taken from the quadratic assignment problem library - QAPLIB.
2007, 3(2): 193-208 doi: 10.3934/jimo.2007.3.193 +[Abstract](1223) +[PDF](299.4KB)
Abstract:
This paper examines methods of pointwise construction of aggregation operators via optimal interpolation. It is shown that several types of application-specific requirements lead to interpolatory type constraints on the aggregation function. These constraints are translated into global optimization problems, which are the focus of this paper. We present several methods of reduction of the number of variables, and formulate suitable numerical algorithms based on Lipschitz optimization.
2007, 3(2): 209-222 doi: 10.3934/jimo.2007.3.209 +[Abstract](985) +[PDF](179.1KB)
Abstract:
In this paper, we apply a smoothing approach to a minimization problem with a max-min constraint (i.e., a min-max-min problem). More specifically, we first rewrite the min-max-min problem as an optimization problem with several min-constraints and then approximate each min-constraint function by a smooth function. As a result, the original min-max-min optimization problem can be solved by solving a sequence of smooth optimization problems. We investigate the relationship between the global optimal value and optimal solutions of the original min-max-min optimization problem and that of the approximate smooth problem. Under some conditions, we show that the limit points of the first-order (second-order) stationary points of the smooth optimization problems are first-order (second-order) stationary points of the original min-max-min optimization problem.
2007, 3(2): 223-232 doi: 10.3934/jimo.2007.3.223 +[Abstract](815) +[PDF](155.8KB)
Abstract:
In this paper we propose an exact method for the 0-1 polynomial knapsack problem. The algorithm seeks the exact optimal solution of the problem by a back-tracking branch-and-bound procedure. The upper bounds are computed by a Lagrangian dual search where the Lagrangian relaxations are solved by the maximum-flow method. Heuristic procedures are derived to search for feasible solutions and thus to improve the performance of the algorithm. Promising computational results are reported for test problems with both single constraint and multiple constraints.
2007, 3(2): 233-255 doi: 10.3934/jimo.2007.3.233 +[Abstract](1058) +[PDF](294.8KB)
Abstract:
We investigate a multifunction $x$→Ñ f $(x)$ derived via normal cones to the level sets Š $(x)$ := { $x^$' | $f(x^$') $< f(x)$} for an important class of pseudo--convex functions. It is shown that $x$→Ñ f $(x)$ is simultaneously both a maximally cyclically pseudo--monotone and a maximally pseudo-monotone relation within neighbourhoods on which $f$ is nonconstant. The relevance of this work to the problem of construction of a utility function from observations of revealed preferences of a consumer is discussed.
2007, 3(2): 257-277 doi: 10.3934/jimo.2007.3.257 +[Abstract](903) +[PDF](305.4KB)
Abstract:
Many optimization problems related to integrated oil and gas production systems are nonconvex and multimodal. Additionally, apart from the innate nonsmoothness of many optimization problems, nonsmooth functions such as minimum and maximum functions may be used to model flow/pressure controllers and cascade mass in the gas gathering and blending networks. In this paper we study the application of different versions of the derivative free Discrete Gradient Method (DGM) as well as the Lipschitz Global Optimizer (LGO) suite to production optimization in integrated oil and gas production systems and their comparison with various local and global solvers used with the General Algebraic Modeling System (GAMS). Four nonconvex and nonsmooth test cases were constructed from a small but realistic integrated gas production system optimization problem. The derivation of the system of equations for the various test cases is also presented. Results demonstrate that DGM is especially effective for solving nonsmooth optimization problems and its two versions are capable global optimization algorithms. We also demonstrate that LGO solves successfully the presented test (as well as other related real-world) problems.
2007, 3(2): 279-292 doi: 10.3934/jimo.2007.3.279 +[Abstract](859) +[PDF](153.9KB)
Abstract:
We consider the management of urban stormwater in two connected dams. Stormwater generated by local rainfall flows into a large capture dam and is subsequently pumped to a smaller supply dam. We assume random supply and constant demand. A simple management policy is to pump as much water as possible each day from the capture dam to the supply dam without allowing the supply dam to overflow. If there is insufficient water in the supply dam to meet the desired daily demand then all water in the supply dam is used. The policy defines a large block transition matrix with repeating entries. We use Matrix Analytic Methods to decompose the event space for the various state transitions and subsequently construct simplified equations for the invariant state probability vector. In this way we enable an elementary solution procedure to find the invariant probability. This paper extends previous work by Piantadosi [12] by allowing higher levels of demand and prefaces more recent work by Piantadosi et al [13].
2007, 3(2): 293-304 doi: 10.3934/jimo.2007.3.293 +[Abstract](1342) +[PDF](206.4KB)
Abstract:
This paper presents a canonical duality theory for solving nonconvex polynomial programming problems subjected to box constraints. It is proved that under certain conditions, the constrained nonconvex problems can be converted to the so-called canonical (perfect) dual problems, which can be solved by deterministic methods. Both global and local extrema of the primal problems can be identified by a triality theory proposed by the author. Applications to nonconvex integer programming and Boolean least squares problems are discussed. Examples are illustrated. A conjecture on NP-hard problems is proposed.
2007, 3(2): 305-312 doi: 10.3934/jimo.2007.3.305 +[Abstract](1352) +[PDF](121.5KB)
Abstract:
In this paper we use a doubly stochastic matrix to define a copula that preserves the given marginal distributions and matches a known grade correlation coefficient in such a way that the entropy of the doubly stochastic matrix is maximized. We will describe briefly how this work can be applied to the modelling of daily rainfall.
2007, 3(2): 313-320 doi: 10.3934/jimo.2007.3.313 +[Abstract](866) +[PDF](124.2KB)
Abstract:
We consider the management of water in two connected dams. Stormwater generated by rainfall flows into a large capture dam and is subsequently pumped to a smaller supply dam. We use a discrete state space and assume random supply and daily demand. A simple management policy is to pump as much water as possible each day from the capture dam to the supply dam without allowing the supply dam to overflow. We shall refer to this policy as pump-to-fill. We will show that pump-to-fill minimizes overflow from the system and maximizes the amount of demand met thus providing the optimal pumping policy between a pair of discrete dams in series with a general input and demand process.
2007, 3(2): 321-334 doi: 10.3934/jimo.2007.3.321 +[Abstract](1098) +[PDF](279.4KB)
Abstract:
In this survey article we give the basic description of the interpolation based derivative free optimization methods and their variants. We review the recent contributions dealing with maintaining the geometry of the interpolation set, the management of the trust region radius and the stopping criteria. Derivative free algorithms developed for problems with some structure, like for partially separable functions, are discussed. Two different versions of derivative free algorithms are applied for the optimization of the configuration of the geometry of a stirrer. Numerical results are presented to show the applicability of the algorithms to practical problems.
2007, 3(2): 335-356 doi: 10.3934/jimo.2007.3.335 +[Abstract](799) +[PDF](242.3KB)
Abstract:
Disruptions to commercial airline schedules are frequent and can inflict significant costs. In this paper we continue a line of research initiated by Vranas, Bertsimas and Odoni [15,16], that aims to develop techniques facilitating rapid return to normal operations whenever disruptions occur. Ground Holding is a technique that has been successfully employed to combat disruptions at North American airports. However, this alone is insufficient to cope with the problem. We develop an adaptive optimization model that allows the implementation of other tactics, such as flight cancellations, airborne holding and diversions. While the approach is generic, our model incorporates features of Sydney airport in Australia, such as a night curfew from 11:00pm to 6:00am. For an actual day when there was a significant capacity drop, we demonstrate that our model clearly outperforms the actions that were initiated by the air traffic controllers at Sydney.
2007, 3(2): 357-379 doi: 10.3934/jimo.2007.3.357 +[Abstract](1003) +[PDF](299.6KB)
Abstract:
There are a few areas of science and technology which are only as challenging, emerging and promising as computational biology. This area is looking for its mathematical foundations, for methods of prediction while guaranteeing robustness, and it is of a rigorous interdisciplinary nature. In this paper, we deepen and extend the approach of learning gene-expression patterns in the framework of gene-environment networks by optimization, especially, generalized semi-infinite optimization (GSIP). With respect to research done previously, we additionally imply the fact that there are measurement errors in the microarray technology and in the environmental data likewise; moreover, the effects which exists among the genes and environmental items can seldom be precisely quantified. Furthermore, we present the well-established matrix algebra for our extended model space, and we indicate further new approaches.
Based on data from DNA microarray experiments, nonlinear ordinary differential equations are extracted by least-squares and, then, time-discretized dynamical systems are derived. Using a combinatorial algorithm which constructs and observes polyhedra sequences, the region of parametric stability is detected. This supports the testing of the quality of data fitting. For the parameter estimation we apply a GSIP problem; we characterize its structural stability.
Hopefully, this pioneering study will serve and lead to a more realistic understanding and forecast in biomedicine, food engineering, and biotechnology. The inclusion of error and imprecision intervals may lead to a more careful evaluation of the experimental data in the forthcoming years, especially, when the microarray technology becomes more and more refined.
2007, 3(2): 381-398 doi: 10.3934/jimo.2007.3.381 +[Abstract](1252) +[PDF](264.3KB)
Abstract:
We use a primal-dual scheme to devise a new update rule for a penalty function method applicable to general optimization problems, including nonsmooth and nonconvex ones. The update rule we introduce uses dual information in a simple way. Numerical test problems show that our update rule has certain advantages over the classical one. We study the relationship between exact penalty parameters and dual solutions. Under the differentiability of the dual function at the least exact penalty parameter, we establish convergence of the minimizers of the sequential penalty functions to a solution of the original problem. Numerical experiments are then used to illustrate some of the theoretical results.
2007, 3(2): 399-413 doi: 10.3934/jimo.2007.3.399 +[Abstract](1058) +[PDF](234.3KB)
Abstract:
Deterministic long run average problems of optimal control are ''asymptotically equivalent" to infinite-dimensional linear programming problems ( LPP ) and the latter are approximated by finite dimensional LPP. The solutions of this finite dimensional LPP can be used for numerical analysis of periodic optimization problems. In the present paper we establish the convergence of controls constructed on the basis of the solution of the finite dimensional LPP to the optimal control of a periodic optimization problem. Results are illustrated with a numerical example.

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