Zbl 0973.92022
Maniatis, T.A.; Nikita, K.S.; Uzunoglu, N.K.
Diffraction tomography based on Gaussian basis functions expansion of the
scatterer and nonlinear optimization. (English)
[CA] Dassios, G. (ed.) et al., Scattering theory and biomedical engineering modelling
and applications. Proceedings of the 4th international workshop, Perdika, Greece,
October 8-10, 1999. Singapore: World Scientific. 242-249 (2000). [ISBN 981-02-4391X]
Summary: A diffraction tomography method for medical imaging applications using
ultrasonic or electromagnetic radiation is presented. The Lippmann-Schwinger
Scattering Integral Equation (SIE), used to describe the associated inverse scattering
problem, is discretized using a spectral domain moment technique, where the unknown
internal field is expanded in a limited number of plane waves while the scatterer is
expressed in terms of Gaussian basis functions. The inverse scattering problem is
formulated as a problem of nonlinear optimization that is solved using either the
modified gradient method or the quasi-Newton method. Numerical results indicate that
the quasi-Newton method provides faster convergence and better accuracy for the given
inverse scattering problem formulation.
Zbl 0922.90106
Adjiman, C.S.; Schweiger, C.A.; Floudas, C.A.
Mixed-integer nonlinear optimization in process synthesis. (English)
[CA] Du, Ding-Zhu (ed.) et al., Handbook of combinatorial optimization. Vol. 1.
Boston: Kluwer Academic Publishers. 1-76 (1998). [ISBN 0-7923-5018-9/hbk; ISBN 07923-5019-7/set]
The use of networks allows the representation of a variety of important engineering
problems. The treatment of a particular class of network applications, the process
synthesis problem, is exposed in this paper. Process Synthesis seeks to develop
systematically process flowsheets that convert raw materials into desired products. In
recent years, the optimization approach to process synthesis has shown promise in
tackling this challenge. It requires the development of a network of interconnected units,
the process superstructure, that represents the alternative process flowsheets. The
mathematical modeling of the superstructure has a mixed set of binary and continuous
variables and results in a mixed-integer optimization model. Due to the nonlinearity of
chemical models, these problems are generally classified as Mixed-Integer Nonlinear
Programming (MINLP) problems. \par A number of local optimization algorithms,
developed for the solution of this class of problems, are presented in this paper:
Generalized Benders Decomposition (GBD), Outer Approximation (OA), Generalized
Cross Decomposition (GCD), Branch-and-Bound (BB), Extended Cutting Plane (ECP),
and Feasibility Approach (FA). Some recent developments for the global optimization
of nonconvex MINLPs are then introduced. In particular, two branch-and-bound
approaches are discussed: the Special structure Mixed Integer Nonlinear $\alpha$BB
(SMIN-$\alpha$BB), where the binary variables should participate linearly or in mixedbilinear terms, and the General structure Mixed Integer Nonlinear $\alpha$BB (GMIN$\alpha$BB), where the continuous relaxation of the binary variables must lead to a
twice-differentiable problem. Both algorithms are based on the $\alpha$BB global
optimization algorithm for nonconvex continuous problems.\par Once the theoretical
issues behind local and global optimization algorithms for MINLPs have been exposed,
attention is directed to their algorithmic development and implementation. The
framework MINOPT is discussed as a computational tool for the solution of process
synthesis problems. It is an implementation of a number of local optimization
algorithms for the solution of MINLPs. The use of MINOPT is illustrated through the
solution of a variety of process network problems. The synthesis problem for a heat
exchanger network is then presented to demonstrate the global optimization SMIN$\alpha$BB algorithm.
A Novel Continuous-Time Modeling and Optimization Framework for Well Platform
Planning Problems
Xiaoxia Lin
Department of Chemical Engineering, Princeton University, Princeton, NJ 08544-5263, USA
Christodoulos A. Floudas
Department of Chemical Engineering, Princeton University, Princeton, NJ 08544-5263, USA.
floudas@titan.princeton.edu
Abstract
The long-term planning problem for integrated gas field development is investigated. The key
decisions involve both design of the production and transportation network structure and
operation of the gas fields over time. A novel continuous-time modeling and optimization approach
is proposed, which introduces the concept of event points and allows the well platforms to come
online at potentially any time within the continuous horizon under consideration. A two-level
formulation and solution framework is developed to take into account complicated economic
calculations and results in mixed-integer nonlinear programming (MINLP) problems. As compared
with the discrete-time model, the proposed approach leads to more compact mathematical models
and significant reduction of the size of the resulting MINLP problems. Even though, the proposed
approach in its current form cannot guarantee convergence to the optimal solution, computational
results show that this approach can reduce the computational efforts required substantially and
solve problems that are intractable for the discrete-time model.
Keywords
long-term planning, gas field development, continuous-time formulation, mixed-integer nonlinear
programming (MINLP)
Article ID: 5113332
Zbl 1030.92015
Ferris, Michael C.; Lim, Jinho; Shepard, David M.
Radiosurgery treatment planning via nonlinear programming. (English)
[J] Ann. Oper. Res. 119, 247-260 (2003). [ISSN 0254-5330]
Summary: The Gamma Knife is a highly specialized treatment unit that provides an
advanced stereotactic approach to the treatment of tumors, vascular malformations, and
pain disorders within the head. Inside a shielded treatment unit, multiple beams of
radiation are focussed into an approximately spherical volume, generating a high dose
shot of radiation. The treatment planning process attempts to cover the tumor with
sufficient dosage without overdosing normal tissue or surrounding sensitive structures.
An optimization problem is formulated that determines where to center the shots, for
how long to expose each shot on the target, and what size focussing helmets should be
used.\par We outline a new approach that models the dose distribution nonlinearly, and
use a smoothing approach to treat discrete problem choices. The resulting nonlinear
program is not convex and several heuristic approaches are used to improve solution
time and quality. The overall approach is fast and reliable; we give several results
obtained from use in a clinical setting.
MSC 2000:
*92C50 Medical appl. of mathematical biology
90C90 Appl. of mathematical programming
90C30 Nonlinear programming
Zbl 1016.49030
Gerdts, Matthias
A moving horizon technique for the simulation of automobile test-drives. (English)
[J] ZAMM, Z. Angew. Math. Mech. 83, No.3, 147-162 (2003). [ISSN 0044-2267; ISSN
1521-4001]
Summary: The test-drive of an automobile along a given test-course can be modeled by
formulation of a suitable optimal control problem. For the numerical solution the
optimal control problem is discretized by a direct shooting method and transformed into
a finite dimensional nonlinear optimization problem. With increasing length of the testcourse, the dimension of the nonlinear optimization problem increases as well and its
numerical solution becomes very difficult due to stability reasons. Therefore a moving
horizon technique with reduced range of vision for the test-driver is introduced. Instead
of treating the complete test-course, a comparatively short local sector is considered on
which a corresponding local optimal control problem can be solved comfortably. The
local solutions are then combined by suitable transient conditions. A numerical example
with a realistic car model is given.
MSC 2000:
*49N90 Applications of optimal control and differential games
90C90 Appl. of mathematical programming
49M37 Methods of nonlinear programming type
90C30 Nonlinear programming
Optimizing color picture tubes by highcost nonlinear programming
Dick den Hertog
a
b
,a
and Peter Stehouwer
,
,b
Tilburg University, CentER, P.O. Box 90153, 5000 LE, Tilburg, The Netherlands
Centre for Quantitative Methods, P.O. Box 414, 5600 AK, Eindhoven, The Netherlands
Available online 8 February 2002.
Abstract
In this paper we discuss the application of optimization techniques for the design of several parts
of the color picture tube, the key component of television sets and computer monitors. These
projects have been carried out for Philips Display Components in Eindhoven. Philips developed
several computer simulation models of picture tube parts. Designers use these models to simulate
the physical behavior of a particular part design. Depending on the amount of detail in the model,
the running time of a typical simulation ranges from one up to 10 hours. Tube designers are
confronted with the problem of finding settings for a large number of design parameters that are
optimal with respect to several simulated tube characteristics. This problem can be modeled as a
so-called high-cost nonlinear programming problem. This paper reports on the successful
application of our four-step compact model approach to solve this problem. The presented results
are based on four projects in which we optimized picture tube parts. Among the realized benefits
for Philips are a design improvement of 30% and a time-to-market reduction of 50–60%.
Author Keywords: Nonlinear programming; Design of computer experiments; Compact
modeling; Engineering design; Simulation
Zbl pre01878837
Kaiser, M.J.
The polygonal containment model. (English)
[J] Appl. Math. Lett. 15, No.1, 89-94 (2002). [ISSN 0893-9659]
Summary: The polygonal containment problem is to position two structures of convex
forms under rigid motion mechanics such that the forms of one structure are completely
contained within the elements of the associated structure. In this note, the polygonal
containment problem is formalized in terms of a generalized containment model and
solved as a constrained nonlinear program.
MSC 2000:
*90C90 Appl. of mathematical programming
90C30 Nonlinear programming
Optimum shapes of reservoirs
H. Mehrazin
Department of Civil Engineering, Faculty of Engineering, University of Teheran, P.O. Box 11365,
Tehran 4563, Iran
Available online 7 December 2000.
Abstract
The aim of this paper is to find the optimal shape of reservoirs and to minimize the whole lateral
surface area for a known volume V. The minimization is studied for some geometrical shapes
generated by rotation. It is proved that the lateral surface area of a hollow ellipsoid with negligible
or non negligible thickness is minimized when it is transformed to a sphere and a formula is
proved for the calculation of lateral surface area of ellipsoid. It is then proved that the
minimization of the whole lateral surface area with a constant volume results to a differential
equation which cannot be resolved in general case; but only for some particular boundary values
by which we obtain the greatest ratio V/Sm for the sphere.
Author Keywords: Optimization; Lateral surface; Ellipsoid; Sphere
Zbl 0996.49023
Kleis, D.; Sachs, E.W.
Optimal control of the sterilization of prepackaged food. (English)
[J] SIAM J. Optim. 10, No.4, 1180-1195 (2000). [ISSN 1052-6234; ISSN 1095-7189]
Summary: To model the process of sterilization by heating in the food industry, we
derive an optimal control problem with state and control constraints governed by a
nonlinear heat equation. A discretized form of the problem can then be expressed as a
large-scale continuous optimization problem and solved by a special sequential
quadratic programming method. The model provides useful insights -- for example,
when maximizing the retention of vitamins, the computed optimal control differs from
the one typically used in industry -- and can be generalized.
MSC 2000:
*49N90 Applications of optimal control and differential games
90C90 Appl. of mathematical programming
90C55 Methods of successive quadratic programming type
90C30 Nonlinear programming
49M37 Methods of nonlinear programming type
Zbl 0979.90123
Aliyu, M.D.S.
A vertex algorithm for collision detection. (English)
[J] Eur. J. Oper. Res. 120, No.1, 174-180 (2000). [ISSN 0377-2217]
Summary: An algorithm for detecting the collision of moving objects is presented. The
algorithm applies to polyhedral objects that can be represented as convex hulls of finite
number of vertices in two- or three-dimensional space. This is then extended to a threeor four-dimensional space (respectively) to represent the objects and their motion.
Nonlinear programming techniques are then employed to detect possible interference.
The algorithm detects in one step whether or not the objects will interfere during their
motion which may involve pure translations or rotations or both. A subalgorithm for
computing the minimum distance between the objects is also presented. This can be
used to solve the interference detection problem or for more general applications.
MSC 2000:
*90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
90C30 Nonlinear programming
90C90 Appl. of mathematical programming
93C85 Automated control systems
Zbl 0949.90027
Ahmadi, Reza H.; Kouvelis, Panagiotis
Design of electronic assembly lines: An analytical framework and its application.
(English)
[J] Eur. J. Oper. Res. 115, No.1, 113-137 (1999). [ISSN 0377-2217]
Summary: The design of component assembly lines in Printed Circuit Board (PCB)
manufacturing environments is a challenging problem faced by many firms in the
electronics industry. The main design approaches to such component assembly lines are
the mini-line, flexible flow line, and hybrid line designs. In this paper, we discuss the
operational trade-offs associated with these design alternatives and present a
mathematical programming framework that captures relevant system design issues.
Each of the design alternatives can be viewed as a special case of the stated
mathematical programming model. We develop effective algorithms to solve these
mathematical programs. We have used the framework in a specific PCB manufacturing
environment to advise managers on the best configuration of their lines. The models
were used as sensitivity analysis tools. The results of our computational experiments,
combined with qualitative comparisons of different design approaches developed by a
crossfunctional team (engineers, manufacturing and product managers), have led to the
development of a set of managerial guidelines for the selection of the design plan for
component assembly lines in the studied environment.
MSC 2000:
*90B30 Production models
90C90 Appl. of mathematical programming
90C30 Nonlinear programming
Zbl 0945.90062
Makowski, Marek
Generation and analysis of a nonlinear optimization problem: European ozone
model case study. (English)
[CA] Polis, Michael P. (ed.) et al., Systems modelling and optimization. Proceedings of
the 18th IFIP TC7 conference, Detroit, MI, USA, July 22-25, 1997. Boca Raton, FL:
Chapman \& Hall/ CRC. Chapman Hall/CRC Res. Notes Math. 396, 477-485 (1999).
[ISBN 0-8493-0607-8]
Summary: The paper presents an outline of the Ozone model being developed at IIASA
and used for analysis of various policy measures aimed at reduction of air pollution in
order to improve the corresponding air quality in Europe.\par Generation of the Ozone
model requires statistical analysis of a large amount of data, and analysis of the model
requires solution of large scale nonlinear optimization problems. A model generator has
been developed not only for providing various optimization solvers with the model
specification but also for performing data consistency check and in order to allow for
implementation two techniques (soft constraints and regularization) necessary for a
more complete model analysis.
MSC 2000:
*90C30 Nonlinear programming
90C90 Appl. of mathematical programming
Zbl 0910.90206
Lindberg, Per Olov; Wolf, Andreas
Optimization of the short term operation of a cascade of hydro power stations.
(English)
[CA] Hager, William H. (ed.) et al., Optimal control: theory, algorithms, and
applications. Proceedings of a conference, University of Florida, Gainesville, FL, USA,
February 27--March 1, 1997. Dordrecht: Kluwer Academic Publishers. Appl. Optim.
15, 326-345 (1998). [ISBN 0-7923-5067-7/hbk]
We study the operation of a cascade of hydropower stations for given time varying
prices and given inflows. The objective is to maximize the value of the power output,
under constraints on final reservoir contents. The flow of the river is modeled through
nonlinear partial differential equations (PDEs), the St. Venant equations, which are
solved iteratively. For the stations, empirical but smooth efficiency curves are
employed. Exploiting the structure of the problem, a descent method of reduced
gradient type is developed. Convexification is used to avoid getting trapped in local
minima. Further scaling of the problem has been applied with rather dramatic success.
MATLAB computations on a small but realistic problem are presented.
MSC 2000:
*90B90 Case-oriented studies in OR
90C90 Appl. of mathematical programming
90C30 Nonlinear programming
Zbl 0907.90115
Çetinkaya, S.; Parlar, M.
Nonlinear programming analysis to estimate implicit inventory backorder costs.
(English)
[J] J. Optimization Theory Appl. 97, No.1, 71-92 (1998). [ISSN 0022-3239]
We use nonlinear programming to provide an alternative treatment of the economic
order quantity problem with planned backorders. Many businesses, such as capitalgoods firms that deal with expensive products and some service industries that cannot
store their services, operate with substantial backlogs. In practical problems, it is usually
very difficult to estimate accurately the values of the two types of backorder costs, i.e.,
the time-dependent unit backorder cost and the unit backorder cost. We redefine the
original problem without including these backorder costs and construct a nonlinear
programming problem with two service measure constraints which may be easier to
specify than the backorder costs. We find that, with this different formulation of our
new problem, we obtain results which give implicit estimates of the backorder costs.
The alternative formulation provides an easier-to-use model and managerially
meaningful results. Next, we show that, for a wide range of parameter values, it usually
suffices to consider only one type of backorder cost, or equivalently, only one type of
service measure constraint. Finally, we develop expressions which bracket the optimal
values of the decision variables in a narrow range and provide a simple method for
computing the optimal solution. In the most complicated case, this method requires
finding the unique root of a polynomial.
MSC 2000:
*90B05 Inventory management
90C90 Appl. of mathematical programming
90C30 Nonlinear programming
Zbl 0924.90075
Bretthauer, Kurt M.; Côté, Murray J.
Nonlinear programming for multiperiod capacity planning in a manufacturing
system. (English)
[J] Eur. J. Oper. Res. 96, No.1, 167-179 (1997). [ISSN 0377-2217]
We present nonlinear programming methods for capacity planning in a manufacturing
system that consists of a set of machines or work stations producing multiple products.
We model the facility as an open network of queues where capacity at each work station
in the system may be changed in each of a finite number of time periods. To determine
the timing and size of capacity changes, we present two nonlinear programming models
and methods for solving the resulting problems. One model involves minimizing total
capacity costs such that plant congestion is controlled via upper limits on work-inprocess. The other model involves minimizing a weighted sum of product lead times
subject to budget constraints on capacity costs. We present solution methods for
continuous and discrete capacity options and convex and nonconvex (e.g., economies of
scale) capacity cost functions. We use branch-and-bound and outer approximation
techniques to determine globally optimal solutions to the nonconvex problems.
Computational testing of the algorithms is reported.
MSC 2000:
*90B30 Production models
90C90 Appl. of mathematical programming
90C30 Nonlinear programming
Zbl 0890.90125
Amos, F.; Rönnqvist, M.; Gill, G.
Modelling the pooling problem at the New Zealand Refining Company. (English)
[J] J. Oper. Res. Soc. 48, No.8, 767-778 (1997). [ISSN 0160-5682]
Pooling is usually present throughout an oil refinery right from the processing of raw
crudes through to the blending of petroleum products. Pooling occurs when two or more
crudes, each with specific properties such as cost, sulphur content and unique
distillation yields, are processed through distilling units simultaneously to yield
downstream fractions. The decisions required in this problem are to select the quantities
of each crude to be processed in each crude distiller and to select the best points which
produce the desired fractions while minimising total cost of crude. Cut points are
temperatures in the distillers at which different output streams are separated. In the
proposed model, we introduce the use of cumulative functions for the distillation yields
which enables a detailed description of the process in a mathematical model.
Preliminary numerical results at the New Zealand Refining Company show that the
nonlinear model accurately describes the pooling problem and simultaneously is
efficiently solvable.
MSC 2000:
*90B90 Case-oriented studies in OR
90C90 Appl. of mathematical programming
90C30 Nonlinear programming
Zbl 0866.90044
Filar, J.A.; Gaertner, P.S.; Janssen, M.A.
An application of optimization to the problem of climate change. (English)
[CA] Floudas, C. A. (ed.) et al., State of the art in global optimization: computational
methods and applications. Papers of the conference, Princeton, NJ, USA, April 28--30,
1995. Dordrecht: Kluwer Academic Publishers. Nonconvex Optim. Appl. 7, 475-498
(1996). [ISBN 0-7923-3838-3]
The objective of this paper is to demonstrate a methodology whereby reductions of
greenhouse gas emissions can be allocated on a regional level with minimal deviation
from the ``business as usual emission scenario''. The methodology developed employs a
two stage optimization process utilizing techniques of mathematical programming. The
stage one process solves a world emission reduction problem producing an optimal
emission reduction strategy for the world by maximizing an economic utility function.
Stage two addresses a regional emission reduction allocation problem via the solution of
an auxiliary optimization problem minimizing disruption from the above business as
usual emission strategies. Our analysis demonstrates that optimal CO$_2$ emission
reduction strategies are very sensitive to the targets placed on CO$_2$ concentrations,
in every region of the world. It is hoped that the optimization analysis will help
decision-makers narrow their debate to realistic environmental targets.
MSC 2000:
*91B76 Environmental economics
90C90 Appl. of mathematical programming
90C30 Nonlinear programming
Zbl 0805.94011
Chang, Yaw O.; Karlof, John K.
Large scale geometric programming: An application in coding theory. (English)
[J] Comput. Oper. Res. 21, No.7, 747-755 (1994). [ISSN 0305-0548]
An algorithm is presented that solves the initial vector problem for group codes for the
Gaussian channel. The Gaussian channel is a communications model in which messages
are represented by vectors in $\bbfR\sp n$ and transmitted over a noisy channel. In this
paper, the set of messages, called a code, is generated by a group of permutation
matrices and the problem of finding the code with the largest minimum distance
between codewords for a particular permutation group is called the initial vector
problem. This problem is an especially large scale nonlinear programming problem. We
transform it to a generalized geometric programming problem and use the double
condensation method to develop an algorithm for its solution.
MSC 2000:
*94A24 Coding theorems (Shannon theory)
90C90 Appl. of mathematical programming
90C30 Nonlinear programming
94A40 Channel models