Showing papers in "Optimization and Engineering in 2007"

A tutorial on geometric programming

Abstract: A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. Recently developed solution methods can solve even large-scale GPs extremely efficiently and reliably; at the same time a number of practical problems, particularly in circuit design, have been found to be equivalent to (or well approximated by) GPs. Putting these two together, we get effective solutions for the practical problems. The basic approach in GP modeling is to attempt to express a practical problem, such as an engineering analysis or design problem, in GP format. In the best case, this formulation is exact; when this is not possible, we settle for an approximate formulation. This tutorial paper collects together in one place the basic background material needed to do GP modeling. We start with the basic definitions and facts, and some methods used to transform problems into GP format. We show how to recognize functions and problems compatible with GP, and how to approximate functions or data in a form compatible with GP (when this is possible). We give some simple and representative examples, and also describe some common extensions of GP, along with methods for solving (or approximately solving) them.

Decoupled approach to multidisciplinary design optimization under uncertainty

Abstract: We propose solution methods for multidisciplinary design optimization (MDO) under uncertainty. This is a class of stochastic optimization problems that engineers are often faced with in a realistic design process of complex systems. Our approach integrates solution methods for reliability-based design optimization (RBDO) with solution methods for deterministic MDO problems. The integration is enabled by the use of a deterministic equivalent formulation and the first order Taylor’s approximation in these RBDO methods. We discuss three specific combinations: the RBDO methods with the multidisciplinary feasibility method, the all-at-once method, and the individual disciplinary feasibility method. Numerical examples are provided to demonstrate the procedure.

Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations

Abstract: We present a technique for the rapid and reliable optimization of systems characterized by linear-functional outputs of coercive elliptic partial differential equations with affine (input) parameter dependence. The critical ingredients are: reduced-basis approximation to effect significant reduction in state-space dimensionality; a posteriori error bounds to provide rigorous error estimation and control; “offline/online” computational decompositions to permit rapid evaluation of output bounds, output bound gradients, and output bound Hessians in the limit of many queries; and reformulation of the approximate optimization statement to ensure (true) feasibility and control of suboptimality. To illustrate the method we consider the design of a three-dimensional thermal fin: Given volume and power objective-function weights, and root temperature “not-to-exceed” limits, the optimal geometry and heat transfer coefficient can be determined—with guaranteed feasibility—in only 2.3 seconds on a 500 MHz Pentium machine; note the latter includes only the online component of the calculations. Our method permits not only interactive optimal design at conception and manufacturing, but also real-time reliable adaptive optimal design in operation.

A globally convergent algorithm for transportation continuous network design problem

Abstract: The continuous network design problem (CNDP) is characterized by a bilevel programming model, in which the upper level problem is generally to minimize the total system cost under limited expenditure, while at the lower level the network users make choices with regard to route conditions following the user equilibrium principle. In this paper, the bilevel programming model for CNDP is transformed into a single level convex programming problem by virtue of an optimal-value function tool and the relationship between System Optimum (SO) and User Equilibrium (UE). By exploring the inherent nature of the CNDP, the optimal-value function for the lower level user equilibrium problem is proved to be continuously differentiable and its derivative in link capacity enhancement can be obtained efficiently by implementing user equilibrium assignment subroutine. However, the reaction (or response) function between the upper and lower level problem is implicit and its gradient is difficult to obtain. Although, here we approximately express the gradient with the difference concept at each iteration, based on the method of successive averages (MSA), we propose a globally convergent algorithm to solve the single level convex programming problem. Comparing with widely used heuristic algorithms, such as sensitivity analysis based (SAB) method, the proposed algorithm needs not strong hypothesis conditions and complex computation for the inverse matrix. Finally, a numerical example is presented to compare the proposed method with some existing algorithms.

Deterministic global optimization approach to steady-state distribution gas pipeline networks

Abstract: Natural gas is normally transported through a vast network of pipelines. A pipeline network is generally established either to transmit gas at high pressure from coastal supplies to regional demand points (transmission network) or to distribute gas to consumers at low pressure from the regional demand points (distribution network). In this study, the distribution network is considered. The distribution network differs from the transmission one in a number of ways. Pipes involved in a distribution network are often much smaller and the network is simpler, having no valves, compressors or nozzles. In this paper, we propose the problem of minimizing the cost of pipelines incurred by driving the gas in a distribute non-linear network under steady-state assumptions. In particular, the decision variables include the length of the pipes’ diameter, pressure drops at each node of the network, and mass flow rate at each pipeline leg. We establish a mathematical optimization model of this problem, and then present a global approach, which is based on the GOP primal-relaxed dual decomposition method presented by Visweswaran and Floudas (Global optimization in engineering design. Kluwer book series in nonconvex optimization and its applications. Kluwer, Netherlands, 1996), to the optimization model. Finally, results from application of the approach to data from gas company are presented.

On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs

Abstract: We consider equivalent reformulations of nonlinear mixed 0–1 optimization problems arising from a broad range of recent applications of topology optimization for the design of continuum structures and composite materials. We show that the considered problems can equivalently be cast as either linear or convex quadratic mixed 0–1 programs. The reformulations provide new insight into the structure of the problems and may provide a foundation for the development of new methods and heuristics for solving topology optimization problems. The applications considered are maximum stiffness design of structures subjected to static or periodic loads, design of composite materials with prescribed homogenized properties using the inverse homogenization approach, optimization of fluids in Stokes flow, design of band gap structures, and multi-physics problems involving coupled steady-state heat conduction and linear elasticity. Several numerical examples of maximum stiffness design of truss structures are presented.

A heuristic for optimizing stochastic activity networks with applications to statistical digital circuit sizing

Abstract: A deterministic activity network (DAN) is a collection of activities, each with some duration, along with a set of precedence constraints, which specify that ac- tivities begin only when certain others have finished. One critical performance mea- sure for an activity network is its makespan, which is the minimum time required to complete all activities. In a stochastic activity network (SAN), the durations of the activities and the makespan are random variables. The analysis of SANs is quite involved, but can be carried out numerically by Monte Carlo analysis. This paper concerns the optimization of a SAN, i.e., the choice of some design variables that affect the probability distributions of the activity durations. We con- centrate on the problem of minimizing a quantile (e.g., 95%) of the makespan, sub- ject to constraints on the variables. This problem has many applications, ranging from project management to digital integrated circuit (IC) sizing (the latter being our motivation). While there are effective methods for optimizing DANs, the SAN op-

Direct transcription solution of optimal control problems with higher order state constraints: theory vs practice

Abstract: Theoretical results from optimal control theory and numerical analysis play an important role in understanding and numerically solving optimal control problems with higher order state inequality constraints. However, the application of these theoretical results to numerical practice can sometimes be misleading. In this paper we examine a specific example which arose in the development of an industrial grade optimal control package that uses a direct transcription approach. This example illustrates several aspects of direct transcription codes that require a reinterpretation of the numerical and analytical theory.

Application of a sensitivity equation method to the k – ε model of turbulence

Abstract: In this paper, we present some examples of sensitivity analysis for flows modeled by the standard k–e model of turbulence with wall functions. The flow and continuous sensitivity equations are solved using an adaptive finite element method. Our examples emphasize a number of applications of sensitivity analysis: identification of the most significant parameters, analysis of the flow model, assessing the influence of closure coefficients, calculation of nearby flows, and uncertainty analysis. The sensitivity parameters considered are closure coefficients of the turbulence model and constants appearing in the wall functions.

A non-linear goal programming model and solution method for the multi-objective trip distribution problem in transportation engineering

Abstract: Trip distribution is one of the important stages in transportation planning model, by which decision-makers can estimate the number of trips among zones. As a basis, the gravity model is commonly used. To cope with complicated situations, a multiple objective mathematical model was developed to attain a set of conflict goals. In this paper, a goal programming model is proposed to enhance the developed multiple objective model to optimize three objectives simultaneously, i.e. (1) maximization of the interactivity of the system, (2) minimization of the generalized costs and (3) minimization of the deviation from the observed year. A genetic algorithm (GA) is developed to solve the proposed non-linear goal programming model. As with other genetic algorithms applied to real-world problems, the GA procedure contains representation, initialization, evaluation, selection, crossover, and mutation. The modification of crossover and mutation to satisfy the doubly constraints is described. A set of Hong Kong data has been used to test the effectiveness and efficiency of the proposed mode. Results demonstrate that decision-makers can find the flexibility and robustness of the proposed model by adjusting the weighting factors with respect to the importance of each objective.

Characteristic functions of directed graphs and applications to stochastic equilibrium problems

Abstract: In this paper we introduce the notions of characteristic and potential functions of directed graphs and study their properties. The main motivation for our research is the stochastic equilibrium traffic assignment problem, in which the drivers choose their routes with some probabilities. Since the number of the strategies in this game is very big, we need to find an efficient way of computation of the expected arc flows in the network. We show that the characteristic functions of the graphs are very useful in this respect. Using this technique we can form and solve numerically the equilibrium traffic assignment problem in a reasonable computational time. As a byproduct of our results we show that the spectral radius of a matrix with non-negative elements admits a convex parametrization as a function of its entries.

A mixed logic dynamical modeling formulation and optimal control of intelligent robots

Abstract: By considering the dexterous hand manipulation problem as a hybrid system, we propose a mixed logic dynamical (MLD) modeling formulation which encapsulates phases of continuous motion, switching between types of motion, and occurrence of impacts. We first formulates the multi-contact manipulation system into a general nonlinear dynamical equation subject to (in)equality and complementarity constraints, then transform the constrained system to a MLD system model. Based on the derived MLD model, dexterous hand manipulation can be realized optimally via mixed integer quadric programming (MIQP) algorithm. This modeling formulation and an optimization approach are applied to a whole body manipulation task as an example.

A generalized two point ellipsoidal anisotropic ray tracing for converted waves

Abstract: Rocks can be anisotropic due to a variety of reasons. When estimating rock velocities from seismic data, failure to introduce anisotropy into earth models could generate distortions in the final images that can have enormous economic impact. To estimate anisotropic earth velocities by tomographic methods, it is necessary to trace rays or to solve the wave equation in models where anisotropy has been properly considered. Thus, in this work we present a 3-D generalized ellipsoidal travel time formulation that allow us to trace rays in an anisotropic medium. We propose to trace rays in anisotropic media by solving a set of nonlinear optimization problems, where the group velocities for P and S wave propagation modes are 3-D ellipsoidal approximations that have been recently obtained. Moreover, we prove that this 3-D ellipsoidal anisotropic ray tracing formulation is a convex nonlinear optimization problem, and therefore any solution of the problem is a global minimum. Each optimization problem is solved by the global spectral gradient method, which requires first order information and has low computation and low storage requirements. Our approach for tracing rays in anisotropic media is a generalization in the sense that handles titled axis of symmetry and, close to the axis of symmetry, it is an accurate formulation for 2-D transversely isotropic media and 3-D orthorhombic media, depending on the input parameters. Moreover, this formulation gives the exact ray trajectories in 2-D and 3-D homogeneous isotropic media. The simplicity of the formulation and the low computational cost of the optimization method allow us to present a variety of numerical results that illustrate the behavior and computational advantages of the approach, and the difficulties when working in anisotropic media.

Magnetic resonance tissue quantification using optimal bSSFP pulse-sequence design

Abstract: We propose a merit function for the expected contrast to noise ratio in tissue quantifications, and formulate a nonlinear, nonconvex semidefinite opti- mization problem to select locally-optimal balanced steady-state free precession (bSSFP) pulse-sequence design variables. The method could be applied to other pulse sequence types, arbitrary numbers of tissues, and numbers of images. To solve the problem we use a mixture of a grid search to get good starting points, and a sequential, semidefinite, trust-region method, where the subproblems contain only linear and semidefinite constraints. We give the results of numerical experiments for the case of three tissues and three, four or six images, in which we observe a better increase in contrast to noise than would be obtained by averaging the results of repeated experiments. As an illustration, we show how the pulse sequences designed numerically could be applied to the problem of quantifying intraluminal lipid deposits in the carotid artery.

Production-inventory policy for a deteriorating item with a single vendor-buyer system

Abstract: In this paper, production-inventory models for a deteriorating item in a sin- gle vendor-buyer system has been developed with constant production and demand rate. Shortages at the buyer (when it is allowed) depends on time. The models have been formulated as cost minimization problem via both integrated and non-integrated approaches and solved using genetic algorithms developed to solve the single and multiobjective production inventory problems. Numerical illustrations of the models have been presented and the sensitivity analysis with respect to rates of production, demand and deterioration are performed.

Modeling leakage power reduction in VLSI as optimization problems

Abstract: Reducing power dissipation is one of the most important issues in VLSI design today. Scaling causes subthreshold leakage currents to become a large component of total power dissipation. Multi-Threshold CMOS (MTCMOS) technology has emerged as a promising technique to reduce leakage power. This paper first introduces how to model the sleep transistor sizing problem in the MTCMOS circuits as a Bin-Packing Problem (BPP). The gate-clustering BPP and the First-Fit (FF) techniques are also introduced to further improve the solution quality. To take the circuit’s routing complexity into consideration which is critical for Deep Sub-Micron (technologies that are 0.25 μm and below) (DSM) implementations, a Set-Partitioning Problem (SPP) is then formed. However, this highly constrained model limits it’s application for large circuit design. A Set-Covering (SCP) model is therefore investigated to efficiently solve the problem.

Robust envelope-constrained filter with orthonormal bases and semi-definite and semi-infinite programming

Abstract: In this paper, the equivalence relation between a semi-infinite quadratically constrained convex quadratic programming problem and a combined semi-definite and semi-infinite programming problem is considered. Then, an efficient and reliable discretization algorithm for solving a general class of combined semi-definite and semi-infinite programming problems is developed. Both the continuous-time envelope-constrained optimal equalization filter and the corresponding robust envelope-constrained filter for a communication channel are solved by using the proposed algorithm.