Nonlinear Mathematical Optimization Research Articles

A model for allocating CDC’s HIV prevention resources in the United States

The Division of HIV/AIDS Prevention (DHAP) at the Centers for Disease Control and Prevention has an annual budget of approximately $325 million for funding HIV prevention programs in the U.S. The purpose of this paper is to thoroughly describe the methods used to develop a national HIV resource allocation model intended to inform DHAP on allocation strategies that might improve the overall effectiveness of HIV prevention efforts. The HIV prevention resource allocation problem consists of choosing how to apportion prevention resources among interventions and populations so that HIV incidence is minimized, given a budget constraint. We developed an epidemic model that projects HIV infections over time given a specific allocation scenario. The epidemic model is then embedded in a nonlinear mathematical optimization program to determine the allocation scenario that minimizes HIV incidence over a 5-year horizon. In our model, we consider the general U.S. population and specific at-risk populations. The at-risk populations include 15 subgroups structured by gender, race/ethnicity and HIV transmission risk group. HIV transmission risk groups include high-risk heterosexuals, men who have sex with men and injection drug users. We consider HIV screening interventions and interventions to reduce HIV-related risk behaviors. The output of the model is the optimal funding scenario indicating the amounts to be allocated to all combinations of populations and interventions. For illustrative purposes only, we provide a sample application of the model. In this example, the optimal allocation scenario is compared to the current baseline funding scenario to highlight how the current allocation of funds could be improved. In the baseline allocation, 29% of the annual budget is aimed at the general population, while the model recommends targeting 100% of the budget to the at-risk populations with no allocation targeted to the general population. Within the allocation to behavioral interventions the model recommends an increase in targeting diagnosed positives. Also, the model allocation suggests a greater focus on MSM and IDUs with a 72% of the annual budget allocated to them, while the baseline allocation for MSM and IDUs totals 37%. Incorporating future epidemic trends in the decision-making process informs the selection of populations and interventions that should be targeted. Improving the use of funds by targeting the interventions and population subgroups at greatest risk may lead to improved HIV outcomes. These models can also direct research by pointing to areas where the development of cost-effective interventions can have the most impact on the epidemic.

Thermodynamic analysis of combustion processes and pollutants emission using nonlinear optimization approach

AbstractMathematical formulation and modeling of combustion processes is an important tool in the understanding of this phenomenon. Determination of equilibrium temperature and composition is often the first stage in calculation of combustion characteristics. There are number of different techniques for simulation of combustion process. In this study a basic model has been developed based on the minimization of Gibb's free energy to simulate the combustion processes. A nonlinear mathematical optimization has been developed based on Lagrange multipliers and solved using Quasi‐Newton method written in MathCAD environment. The effect of various parameters such as initial temperature, pressure, and equivalence ratio on the equilibrium temperature and composition has been investigated. In contrast with the equilibrium constant method, obtained results out of this work show that the maximum flame temperature for normal paraffins occurs around the equivalence ratio of 1.05. In addition, from environmental point of view, effects of different conditions on the emission of pollutants in a combustion process have been studied. The simulation results showed that the NO and NO2 emission rates pass through a maximum point and there is an optimum point where the NOCO emission could be minimized. Copyright © 2010 Curtin University of Technology and John Wiley & Sons, Ltd.

Optimization of Project Time-Cost Trade-Off Problem with Discounted Cash Flows

Traditional time-cost trade-off (TCTO) analysis assumes constant value of activities' cost along the project time span. However, the value of money decreases with time and, therefore, discounted cash flows should be considered when solving TCTO optimization problem. Optimization problems in project management have been traditionally solved by two distinctive approaches: heuristic methods and optimization techniques. Although heuristic methods can handle large-size projects, they do not guarantee optimal solutions. A nonlinear mathematical optimization model for project TCTO problem is developed, which minimizes project direct cost and takes into account discounted cash flows. Costs of activities are assumed to be incurred at their finish times. The model guarantees the optimal solution, in which precise discrete activity time-cost function is used. The model input includes precedence relationship between project activities, discrete utility data for project activities, and discount rate. Details of model formulation are illustrated by an example project. The results show that selected activities' durations and costs and consequently optimal project duration differ from traditional analysis if discounted cash flow is considered. The new approach provides project practitioners with a way for considering net present value in time-cost decisions so that the best option can be identified.

The economics of wind power with energy storage

We develop a nonlinear mathematical optimization program for investigating the economic and environmental implications of wind penetration in electrical grids and evaluating how hydropower storage could be used to offset wind power intermittence. When wind power is added to an electrical grid consisting of thermal and hydropower plants, it increases system variability and results in a need for additional peak-load, gas-fired generators. Our empirical application using load data for Alberta's electrical grid shows that costs of wind-generated electricity vary from $37 per MWh to $68/MWh, and depend primarily on the wind profiles of installed turbines. Costs of reducing CO 2 emissions are estimated to be $41–$56 per t CO 2. When pumped hydro storage is introduced in the system or the capacity of the water reservoirs is enhanced, the hydropower facility could provide most of the peak load requirements obviating the need to build large peak-load gas generators.

Improving Two-Temperature Method Retrievals Based on a Nonlinear Optimization Approach

The two-temperature method (TTM) is known to be sensitive to noise, and therefore, land-surface temperature (LST) and emissivity (LSE) retrievals based on TTM are in general not reliable when obtained by algebraic procedures. Accordingly, the added value of using TTM together with a nonlinear mathematical optimization approach is investigated, focusing on the effect that an increase in the temperature difference as well as in the number of observations might have on LST and LSE retrievals. TTM has provided values of LST and LSE with a bias (root mean square error) ranging from 0.1-0.4 K (2.1-2.8 K) and from 0.005-0.010 (0.040-0.055), respectively. Obtained results were almost the same for both well-determined and overdetermined cases, as well as for the considered temperature differences, suggesting that increasing the number of observations and the temperature difference does not lead to significant improvements on the results. On the other hand, it was found out that a greater temperature difference between the first and the last observation acts like a natural constraint by restricting the solutions to a narrower region. In this case, the estimated LST and LSE values do not strongly depend upon the initial guess, and therefore, the use of several initial guess vectors may be avoided, turning TTM computationally more efficient.

Numerical Analyses of a Directed Graph Formulation of the Multistage Distribution Expansion Problem

In this paper, the numerical analysis of a multistage formulation of the distribution expansion problem developed is presented. The approximate objective function is employed without sacrificing optimality. A complete nonlinear mathematical programming optimization is employed to identify the global optimum. The algorithm is applied here to two practical test cases. Details of the input data and configuration of test systems as well as the solutions are listed and discussed. Computational considerations are addressed.

Multi-Machine Power Systems Stabilizing Control Using Output Feedback Excitation System

Electric network is now confronted with market deregulation or overload. Power System Stabilizer (PSS) is used to enhance capacity limits and improve transient stability, and it also make important role in the renewed system. This paper proposes design of excitation control with Fletcher-Powell method, which is non-linear mathematical optimization technique to realize output feedback control. The design procedure and the effectiveness are discussed through 5-machine system models. Scheme of decomposing control matrices for the large network is also discussed for successful application of the method.

Design of concrete mixes for minimum brittleness

Abstract Designers of concrete mixes are often faced with competing requirements such as high strength and high ductility and have to rely on empirical and heuristic approaches. In the present study, this is overcome by incorporating the characteristic length 1ch and the brittleness number B into the design criteria and using well-established, mostly nonempirical, formulae relating the effective properties (elastic modulus, tensile strength, and fracture energy) to the microstructural parameters (water/cement ratio, aggregate size, and volume fractions of coarse and fine aggregates). Nonlinear mathematical optimization problems, equivalent to the design criteria, are formulated and solved with a view to: (1) maximizing 1ch for a prescribed minimum direct tensile strength f′t; (2) maximizing lch and f′t simultaneously for a prescribed compressive strength f′c; (3) minimizing B for a prescribed f′c.

Concurrent tool and production system design for a surface broach cutting tool: a knowledge-based systems approach

This paper presents the economics of broaching operations, as analysed within the framework of concurrent design of the broach tool and its operating conditions. The tool is specified by its cutting teeth—tooth structure, tooth pitch and the number of such teeth—which in turn determine the length of the tool. Its operation is specified by the depth of cut per tooth and by the speed of broaching. Design constraints include broaching machine capabilities and surface finish requirements. A nonlinear mathematical optimization model, which incorporates all design variables and constraints, is developed and used to analyse broach design and operation for an illustrative case.

Robustness characteristics of optimum structural/control design

An optimization program has been developed to design a minimum-weight flexible structure with constraints imposed on the closed-loop damping parameters, eigenvalue distribution, and the robustness bounds of the control system to retain stability under structured uncertainties. The formulation of approximate analytical expressions for the derivatives of the robustness measure is presented. The optimization problem was solved by using a nonlinear mathematical optimization technique. The numerical results are presented for the Draper 1 tetrahedral truss structure to satisfy different design requirements. These results indicate that the constraints on the damping parameters have to be imposed in addition to the robustness measure in order to improve the dynamic response. HE traditional approach to the design of a structure and control system has been sequential. The structure is first designed to satisfy the constraints on the allowable stresses, displacements, frequencies, etc., and then the control system is designed on this fixed structure. The control design has to satisfy its design requirements such as performance, control effort, robustness, stability, etc. This sequential approach generally does not lead to an optimum performance of the structure and control system for a specific mission. The opti- mum structural design and optimum control design have de- veloped into mature fields. However, because of the stringent requirements in the design of future aerospace structures, there is currently considerable interest in developing al- gorithms for the simultaneous design of the structure and control system. The mathematical model representing a physical system in- evitably contains inaccuracies. This would compromise the performance of the control system, making the system un- stable if one does not allow for these modeling errors. The modeling uncertainties are generally divided into two broad categories. The first is the parametric or structured uncertain- ties that result due to the variations in the real and analytical frequencies of the structure. The second is due to the unmod- eled higher frequencies. The robust-control system is designed to maintain closed-loop stability and performance in the pres- ence of unknown or known structured and unstructured un- certainties. The order of the finite element model of a space structure is generally too high and this necessitates the use of a reduced-order model constructed by truncating the higher vibration modes to design a controller. This reduction in the dimensionality of the controller sometimes leads to the spill- over effect when it is used on a higher order structural model. There have been various approaches proposed by different investigators in order to reduce or prevent the spillover effect. The objective of this paper was to develop an optimization algorithm to design a structure and control system by taking into consideration a robustness parameter that defines the structural uncertainties in a closed-loop system. The linear quadratic regulator (LQR) with a full-order controller is used

Three-dimensional imaging for brachytherapy planning.

A computerized 3-D brachytherapy treatment planning program has been developed which simulates catheter trajectories, radioisotope seed positioning and percentage dosimetry zones about a catheter/seed array in relationship to CT or MRI images. The high activity 125I seed is used in this program. A visual and a non-linear mathematical optimization program will also be described.

Use of Robustness Constraints in the Optimum Design of Space Structures

This paper describes an optimization procedure to design a minimum weight structure and optimum control system with constraints on the eigenvalues and robustness parameter for structured uncertainties in the closed-loop plant matrix. The design variables are the cross-sectional areas of the members and the elements of the state and control weighting matrices. The optimization problem was solved by using a nonlinear mathematical optimization technique based on the interior penalty functions. The numeri cal results are presented for the ACOSS-FOUR structure. This example illustrates how the use of optimization techniques can reduce the weight of the nominal structure, make the control system more robust and Improve the response due to the external disturbance.

Improving the active vibrational control of large space structures through structural modifications

This study provides a method of vibration control of large space structures by the simultaneous and integrated structure and control design to reduce structural response from disturbances encountered. The formulation of a design scheme is obtained by structural modification of some nominal finite element model, which is controlled in optimal fashion by a linear regulator, to increase the active modal damping factor beyond that of the nominal structure. The structural modifications are determined by calculating the sensitivity of the closed-loop eigenvalues to structural stiffness variation. Because an optimal active control method is used, the sensitivity of the Riccati matrix to structural modifications are provided. Improvements in modal damping factors are specified and a method for determination of structural modification provided. The structural modifications, i.e. changes in the cross-sectional areas of the members, are achieved by using a nonlinear mathematical optimization technique. Application of the algorithm is illustrated by structural modification of a nominal ACOSS-FOUR model with different constraints on the closed-loop eigenvalues.

Optimal structural modifications to enhance the active vibration control of flexible structures

This study provides a method of vibration control of large space structures by simultaneously integrating the structure and control design to reduce the structural response from a disturbance. The formulation of the design scheme is obtained by the structural modification of some nominal finite element model, which is controlled in an optimal fashion by a linear regulator, to increase the active modal damping factor beyond that of the nominal structure. The structural modifications are achieved by using a nonlinear mathematical optimization technique. The objective function is the weight of the structure with a constraint on the damping parameter of the closedloop system. The application of the algorithm is illustrated by designing an ACOSS-FOUR model with different constraint values.

Improved synthesis capability for “ T” ring-stiffened cylindrical shells under hydrostatic pressure

The new “Direct Search-Feasible Direction” (DSFD) nonlinear mathematical programming optimization algorithm is applied to the design of stiffened submersible shells. An automated design capability for this problem (SBSHL6) is described wherein the program will generate the least weight design by locating the optimal, or near optimal, values of skin thickness, web thickness and height, flange thickness and width, and stiffener spacing given the design parameters such as shell size, immersion pressure, shell eccentricity, materials properties, and minimum natural frequency. Constraint equations control, general, panel (between stiffener), web, and flange instability, skin and stiffener yielding, and minimum natural frequency. The DSFD procedure appears capable of reliably locating optimal designs whereas earlier attempts in investigations using other optimization methods, including the popular SUMT procedure, failed to provide optimal solutions to the same problem. This and an earlier detailed comparison study strongly suggest that SUMT is not a reliable procedure for structural optimization while DSFD seems to provide reasonably reliable performance. Designs generated by SBSHL6 are presented and compared with those of the earlier studies. The results of a series of synthesis runs from widely separated starting points are also presented. The designs developed by SBSHL6 are substantially lighter than those reported earlier. The multipath runs for each set of parameters studied all converged to similar designs of essentially identical weights demonstrating program reliability.