Showing papers on "Nonlinear programming published in 2004"
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TL;DR: In this article, the focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them, and a comprehensive introduction to the subject is given. But the focus of this book is not on the optimization problem itself, but on the problem of finding the appropriate technique to solve it.
Abstract: Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.
33,341 citations
TL;DR: Application of heuristic programming methods using evolutionary and genetic algorithms are described, along with application of neural networks and fuzzy rule-based systems for inferring reservoir system operating rules, to assess the state of the art in optimization of reservoir system management and operations.
Abstract: With construction of new large-scale water storage projects on the wane in the U.S. and other developed countries, attention must focus on improving the operational effectiveness and efficiency of existing reservoir systems for maximizing the beneficial uses of these projects. Optimal coordination of the many facets of reservoir systems requires the assistance of computer modeling tools to provide information for rational management and operational decisions. The purpose of this review is to assess the state-of-the-art in optimization of reservoir system management and operations and consider future directions for additional research and application. Optimization methods designed to prevail over the high-dimensional, dynamic, nonlinear, and stochastic characteristics of reservoir systems are scrutinized, as well as extensions into multiobjective optimization. Application of heuristic programming methods using evolutionary and genetic algorithms are described, along with application of neural networks and fuzzy rule-based systems for inferring reservoir system operating rules.
1,484 citations
TL;DR: In this article, a new nonlinear process monitoring technique based on kernel principal component analysis (KPCA) is developed, which can efficiently compute principal components in high-dimensional feature spaces by means of integral operators and nonlinear kernel functions.
927 citations
TL;DR: The development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs is addressed and novel relaxation schemes, range reduction tests, and branching strategies are developed which are incorporated into the prototypical branch-and-bound algorithm.
Abstract: This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototypical branch-and-bound algorithm. In the theoretical/algorithmic part of the paper, we begin by developing novel strategies for constructing linear relaxations of mixed-integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties. We then use Lagrangian/linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming. This theory leads to new range reduction schemes, including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch-and-bound tree. Finally, we incorporate these relaxation and reduction strategies in a branch-and-bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems. In the computational part of the paper, we describe our implementation discussing, wherever appropriate, the use of suitable data structures and associated algorithms. We present computational experience with benchmark separable concave quadratic programs, fractional 0–1 programs, and mixed-integer nonlinear programs from applications in synthesis of chemical processes, engineering design, just-in-time manufacturing, and molecular design.
579 citations
TL;DR: The proposed method out-performs and provides quality solutions compared to other existing techniques for EDP considering valve-point effects are shown in general.
538 citations
27 Aug 2004
TL;DR: In this article, the idea of continuous designs for moving sensors is proposed to optimize sensor trajectories based on the optimal experimental design problem in the presence of correlated observations and maximizing an Observability Measure.
Abstract: INTRODUCTION The Optimum Experimental Design Problem in Context A General Overview of Literature KEY IDEAS OF IDENTIFICATION AND EXPERIMENTAL DESIGN System Description Parameter Identification Measurement Location Problem Main Impediments Deterministic Interpretation of the FIM Calculation of Sensitivity Coefficients A Final Introductory Note LOCALLY OPTIMAL DESIGNS FOR STATIONARY SENSORS Linear-in-Parameters Lumped Models Construction of Minimax Designs Continuous Designs in Measurement Optimization Clusterization-Free Designs Nonlinear Programming Approach A Critical Note on Some Deterministic Approach Modifications Required by Other Settings Summary LOCALLY OPTIMAL STRATEGIES FOR SCANNING AND MOVING OBSERVATIONS Optimal Activation Policies for Scanning Sensors Adapting the Idea of Continuous Designs for Moving Sensors Optimization of Sensor Trajectories Based on Optimal-Control Techniques Concluding Remarks MEASUREMENT STRATEGIES WITH ALTERNATIVE DESIGN OBJECTIVES Optimal Sensor Location for Prediction Sensor Location for Model Discrimination Conclusions ROBUST DESIGNS FOR SENSOR LOCATION Sequential Designs Optimal Designs in the Average Sense Optimal Designs in the Minimax Sense Robust Sensor Location Using Randomized Algorithms Concluding Remarks TOWARDS EVEN MORE CHALLENGING PROBLEMS Measurement Strategies in the Presence of Correlated Observations Maximization of an Observability Measure Summary APPLICATIONS FROM ENGINEERING Electrolytic Reactor Calibration of Smog Prediction Models Monitoring of Groundwater Resources Quality Diffusion Process With Correlated Observational Errors Vibrating H-Shaped Membrane CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS APPENDICES List of Symbols Mathematical Background On Statistical Properties of Estimators Analysis of the Largest Eigenvalue Differentiation of Nonlinear Operators Accessory Results for PDE's Interpolation of Tabulated Sensitivity Coefficients Differentials of Section 4.3.3 Solving Sensor Location Problems Using Maple and MATLAB
494 citations
TL;DR: This work explicitly characterize the robust counterpart of a linear programming problem with uncertainty set described by an arbitrary norm as well as providing guarantees for constraint violation under probabilistic models that allow arbitrary dependencies in the distribution of the uncertain coefficients.
489 citations
01 Jan 2004
TL;DR: In this article, a linear least-squares solution is proposed for the iterative closest point (ICP) problem when the relative orientation between the two input surfaces is small.
Abstract: The Iterative Closest Point (ICP) algorithm that uses the point-toplane error metric has been shown to converge much faster than one that uses the point-to-point error metric. At each iteration of the ICP algorithm, the change of relative pose that gives the minimal point-to-plane error is usually solved using standard nonlinear least-squares methods, which are often very slow. Fortunately, when the relative orientation between the two input surfaces is small, we can approximate the nonlinear optimization problem with a linear least-squares one that can be solved more efficiently. We detail the derivation of a linear system whose least-squares solution is a good approximation to that obtained from a nonlinear optimization.
399 citations
01 Jan 2004
TL;DR: Daniele et al. as mentioned in this paper studied the strong solvability of a unilateral boundary value problem for nonlinear discontinuous operators in the plane, and the existence of solutions to vector optimization problems.
Abstract: Preface. On the numerical solution of finite-dimensional variational inequalities by an interior point method S. Bellavia, M.G. Gasparo. Fixed points in ordered Banach spaces and applications to elliptic boundary-value problems G. Bonanno, S. Marano. A theorem of the alternative for linear control systems P. Cubiotti. Variational inequalities for static equilibrium market. Lagrangean function and duality P. Daniele. On dynamical equilibrium problems and variational inequalities P. Daniele, A. Maugeri. Nonlinear programming methods for solving optimal control problems C. Durazzi, E. Galligani. Optimal flow pattern in road networks P. Ferrari. On the strong solvability of a unilateral boundary value problem for nonlinear discontinuous operators in the plane S. Giuffre. Most likely traffic equilibrium route flows analysis and computation T. Larsson, et al. Existence of solutions to bilevel variational problems in Banach spaces M.B. Lignola, J. Morgan. On the existence of solutions to vector optimization problems G. Mastroeni, M. Pappalardo. Equilibrium problems and variational inequalities A. Maugeri. Axiomatization for approximate solutions in optimization H. Norde, F. Patrone. Necessary and sufficient conditions of Wardrop type for vectorial traffic equilibria W. Oettli. Approximate solutions and Tikhonov well-posedness for Nash equilibria L.P. Chicco. Equilibrium in time dependent traffic networks with delay F. Raciti. New results on local minima and their applications B. Ricceri. An overview on projection-type methods for convex large-scale quadratic programs V. Ruggiero, L. Zanni.
337 citations
01 Aug 2004
TL;DR: A novel method for controlling physics-based fluid simulations through gradient-based nonlinear optimization and the first method for the full control of free-surface liquids is introduced.
Abstract: We describe a novel method for controlling physics-based fluid simulations through gradient-based nonlinear optimization. Using a technique known as the adjoint method, derivatives can be computed efficiently, even for large 3D simulations with millions of control parameters. In addition, we introduce the first method for the full control of free-surface liquids. We show how to compute adjoint derivatives through each step of the simulation, including the fast marching algorithm, and describe a new set of control parameters specifically designed for liquids.
316 citations
TL;DR: This paper formulates and analyzes a pattern search method for general constrained optimization based on filter methods for step acceptance that preserves the division into SEARCH and local POLL steps, which allows the explicit use of inexpensive surrogates or random search heuristics in the SEARCH step.
Abstract: This paper formulates and analyzes a pattern search method for general constrained optimization based on filter methods for step acceptance. Roughly, a filter method accepts a step that improves either the objective function value or the value of some function that measures the constraint violation. The new algorithm does not compute or approximate any derivatives, penalty constants, or Lagrange multipliers. A key feature of the new algorithm is that it preserves the division into SEARCH and local POLL steps, which allows the explicit use of inexpensive surrogates or random search heuristics in the SEARCH step. It is shown here that the algorithm identifies limit points at which optimality conditions depend on local smoothness of the functions and, to a greater extent, on the choice of a certain set of directions. Stronger optimality conditions are guaranteed for smoother functions and, in the constrained case, for a fortunate choice of the directions on which the algorithm depends. These directional conditions generalize those given previously for linear constraints, but they do not require a feasible starting point. In the absence of general constraints, the proposed algorithm and its convergence analysis generalize previous work on unconstrained, bound constrained, and linearly constrained generalized pattern search. The algorithm is illustrated on some test examples and on an industrial wing planform engineering design application.
TL;DR: A necessary and sufficient condition for quadratic stability of this class of switched systems is derived by using Karush-Kuhn-Tucker condition for nonlinear programming problems.
Abstract: Quadratic stability of a class of switched nonlinear systems is studied in this note. We first transform quadratic stability problem into an equivalent nonlinear programming problem. Then, we derive a necessary and sufficient condition for quadratic stability of this class of switched systems by using Karush-Kuhn-Tucker condition for nonlinear programming problems. The necessary and sufficient condition is given in terms of the strict completeness of a certain set of functions on a subset of the state space, which is much easier to check.
TL;DR: In this paper, an efficient genetic algorithm (GA) is presented to solve the problem of multistage and coordinated transmission expansion planning, which is a mixed integer nonlinear programming problem, difficult for systems of medium and large size and high complexity.
Abstract: In this paper, an efficient genetic algorithm (GA) is presented to solve the problem of multistage and coordinated transmission expansion planning. This is a mixed integer nonlinear programming problem, difficult for systems of medium and large size and high complexity. The GA presented has a set of specialized genetic operators and an efficient form of generation of the initial population that finds high quality suboptimal topologies for large size and high complexity systems. In these systems, multistage and coordinated planning present a lower investment than static planning. Tests results are shown in one medium complexity system and one large size high complexity system.
TL;DR: In this paper, a generalized canonical nonlinear programming circuit (G-NPC) was proposed to solve a general class of nonsmooth non-linear programming problems, where the objective function and constraints are assumed to satisfy only the weak condition of being regular functions.
Abstract: In 1988 Kennedy and Chua introduced the dynamical canonical nonlinear programming circuit (NPC) to solve in real time nonlinear programming problems where the objective function and the constraints are smooth (twice continuously differentiable) functions. In this paper, a generalized circuit is introduced (G-NPC), which is aimed at solving in real time a much wider class of nonsmooth nonlinear programming problems where the objective function and the constraints are assumed to satisfy only the weak condition of being regular functions. G-NPC, which derives from a natural extension of NPC, has a neural-like architecture and also features the presence of constraint neurons modeled by ideal diodes with infinite slope in the conducting region. By using the Clarke's generalized gradient of the involved functions, G-NPC is shown to obey a gradient system of differential inclusions, and its dynamical behavior and optimization capabilities, both for convex and nonconvex problems, are rigorously analyzed in the framework of nonsmooth analysis and the theory of differential inclusions. In the special important case of linear and quadratic programming problems, salient dynamical features of G-NPC, namely the presence of sliding modes , trajectory convergence in finite time, and the ability to compute the exact optimal solution of the problem being modeled, are uncovered and explained in the developed analytical framework.
TL;DR: Experience indicates that sequential quadratic programming (SQP) methods are very well suited for solving MPCCs and at present outperform interior-point solvers both in terms of speed and reliability.
Abstract: We consider solving mathematical programs with complementarity constraints (MPCCs) as nonlinear programs (NLPs) using standard NLP solvers. This approach is appealing because it allows existing off-the-shelf NLP solvers to tackle large instances of MPCCs. Numerical experience on MacMPEC, a large collection of MPCC test problems is presented. Our experience indicates that sequential quadratic programming (SQP) methods are very well suited for solving MPCCs and at present outperform interior-point solvers both in terms of speed and reliability. All NLP solvers also compare very favorably to special MPCC solvers on tests published in the literature.
TL;DR: Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.
Abstract: Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a second-order sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by a constraint involving a positive parameter that can be decreased to zero. In the penalized formulation, the complementarity constraint appears as a penalty term in the objective. The existence and uniqueness of solutions for these formulations are investigated, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.
TL;DR: In this paper, the primal-dual interior-point filter is used to solve the problem of global convergence to first-order critical points in nonlinear nonlinear programs, without the use of merit functions and penalty parameters.
Abstract: In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primal-dual interior-point algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters.The new algorithm decomposes the primal-dual step obtained from the perturbed first-order necessary conditions into a normal and a tangential step, whose sizes are controlled by a trust-region type parameter. Each entry in the filter is a pair of coordinates: one resulting from feasibility and centrality, and associated with the normal step; the other resulting from optimality (complementarity and duality), and related with the tangential step.Global convergence to first-order critical points is proved for the new primal-dual interior-point filter algorithm.
01 Jan 2004
TL;DR: This chapter deals with a new approach which will utilize a log-dynamic penalty function method in the NES algorithm that has been proposed and tested in the previous chapter.
Abstract: Although evolutionary algorithms have proved useful in general function optimization, they appeared particularly apt for addressing nonlinearly constrained optimization problems. Constrained optimization problems present the difficulties with potentially nonconvex or even disjoint feasible regions. Classic linear programming and nonlinear programming methods are often either unsuitable or impractical when applied to these constrained problems [76]. Unfortunately, most of the real-world problems often pose such difficulties. Evolutionary algorithms are global methods, which aim at complex objective functions (e.g., non-differentiable or discontinuous) and they can be constructed to cope effectively with these difficulties. There are, however, no well-established guidelines on how to deal with infeasible solutions. Contemporary evolution strategies usually use “death penalty” heuristic for infeasible solutions. This death penalty offers a few simplifications of the algorithm: for example, there is no need to evaluate infeasible solutions and to compare them with feasible ones. Fortunately, this method may work reasonably well when the feasible search space is convex and it constitutes a reasonable part of the whole search space. Otherwise, such an approach has serious limitations. For example, for many search problems where the initial population consists of infeasible individuals only, it might be essential to improve them [101]. Moreover, quite often the system can reach the optimum solution easier if it is possible to “cross” an infeasible region especially in non-convex feasible search spaces. This chapter deals with a new approach which will utilize a log-dynamic penalty function method in the NES algorithm [61, 62] that has been proposed and tested in the previous chapter.
29 Nov 2004
TL;DR: This work forms a constrained multivariable nonlinear programming problem to determine both the locations of the sensor nodes and data transmission pattern, and shows that the optimal node placement strategies provide significant benefit over a commonly used uniform placement scheme.
Abstract: One of the main design issues for wireless sensor networks is the sensor placement problem. We formulate a constrained multivariable nonlinear programming problem to determine both the locations of the sensor nodes and data transmission pattern. Our two objectives are to maximize the network lifetime and to minimize the application-specific total cost, given a fixed number of sensor nodes in a region with a certain coverage requirement. We first study a linear network, and find optimal placement strategies numerically. Through numerical results, we show that the optimal node placement strategies provide significant benefit over a commonly used uniform placement scheme. Furthermore, we also present a performance bound as a benchmark. Lastly, we extend the results to a more sophisticated planar network, and use numerical results to evaluate the performance of the proposed strategies.
TL;DR: A novel recurrent neural network for solving nonlinear convex programming problems subject to nonlinear inequality constraints is presented and is proved to be stable in the sense of Lyapunov and globally convergent to an exact optimal solution.
Abstract: This paper presents a novel recurrent neural network for solving nonlinear convex programming problems subject to nonlinear inequality constraints. Under the condition that the objective function is convex and all constraint functions are strictly convex or that the objective function is strictly convex and the constraint function is convex, the proposed neural network is proved to be stable in the sense of Lyapunov and globally convergent to an exact optimal solution. Compared with the existing neural networks for solving such nonlinear optimization problems, the proposed neural network has two major advantages. One is that it can solve convex programming problems with general convex inequality constraints. Another is that it does not require a Lipschitz condition on the objective function and constraint function. Simulation results are given to illustrate further the global convergence and performance of the proposed neural network for constrained nonlinear optimization.
TL;DR: In this paper, a mathematical programming model for optimal highway pavement rehabilitation planning is presented, which minimizes the life cycle cost for a finite horizon by solving the problem of multiple rehabilitation activities on multiple facilities.
Abstract: This paper presents a mathematical programming model for optimal highway pavement rehabilitation planning which minimizes the life-cycle cost for a finite horizon. It extends previous researches in this area by solving the problem of multiple rehabilitation activities on multiple facilities, with realistic empirical models of deterioration and rehabilitation effectiveness. The formulation is based on discrete control theory. A nonlinear pavement performance model and integer decision variables are incorporated into a mixed-integer nonlinear programming (MINLP). Two solution approaches, a branch-and-bound algorithm and a greedy heuristic, are proposed for this model. It is shown that the heuristic results provide a good approximation to the exact optima, but with much lower computational costs.
01 Jan 2004
TL;DR: The objective is to present the overall design and describe how to efficiently model a problem in TOMLAB using the standard structures and assign statements.
Abstract: The TOMLAB Optimization Environment is a powerful optimization tool in MATLAB, which incooperates many results from the last 40 years of research in the field. More than 70 different algorithms for linear, discrete, global and nonlinear optimization are implemented in TOMLAB, and a large number of C and Fortran solvers are also fully integrated. The environment is call-compatible with Math-Works’ Optimization Toolbox, and supports problems formulated in AMPL. This chapter discusses the design and contents of TOMLAB, and examplifies its usage on a practical optimization problem. The objective is to present the overall design and describe how to efficiently model a problem in TOMLAB using the standard structures and assign statements. More information about TOMLAB is available at URL: http: //tomlab. BIZ.
TL;DR: In this article, a method of assessing the optimum pumping rates of coastal aquifers based on nonlinear optimization and evolutionary algorithms (EA) was developed to maximize the total pumping rate while protecting the wells from sea water intrusion.
TL;DR: The methods considered cover the following areas: tailoring of nonlinear programming algorithms to the structure of the online optimization, use of the optimal control formulation of the receding horizon problem, constraint and cost approximations based on state space partitioning, and reparameterization of the degrees of freedom in predictions.
TL;DR: This paper addresses two second-best toll pricing problems, one with fixed and the other with elastic travel demands, as mathematical programs with equilibrium constraints, and several equivalent nonlinear programming formulations for the two problems are discussed.
Abstract: This paper addresses two second-best toll pricing problems, one with fixed and the other with elastic travel demands, as mathematical programs with equilibrium constraints. Several equivalent nonlinear programming formulations for the two problems are discussed. One formulation leads to properties that are of interest to transportation economists. Another produces an algorithm that is capable of solving large problems and easy to implement with existing software for linear and nonlinear programming problems. Numerical results using transportation networks from the literature are also presented.
TL;DR: In this paper, a genetic algorithm solution to the hydrothermal coordination problem is presented, where the generation scheduling of the hydro production system is formulated as a mixed-integer, nonlinear optimization problem and solved with an enhanced genetic algorithm featuring a set of problem-specific genetic operators.
Abstract: In this paper, a genetic algorithm solution to the hydrothermal coordination problem is presented. The generation scheduling of the hydro production system is formulated as a mixed-integer, nonlinear optimization problem and solved with an enhanced genetic algorithm featuring a set of problem-specific genetic operators. The thermal subproblem is solved by means of a priority list method, incorporating the majority of thermal unit constraints. The results of the application of the proposed solution approach to the operation scheduling of the Greek Power System, comprising 13 hydroplants and 28 thermal units, demonstrate the effectiveness of the proposed algorithm.
TL;DR: The problem of optimal cooperative three-dimensional conflict resolution involving multiple aircraft is addressed by the rigorous numerical trajectory optimization methods and the ability of the mathematical programming framework to accommodate detailed dynamic models is addressed.
Abstract: Free flight is an emerging paradigm in air traffic management. Conflict detection and resolution is the heart of any free-flight concept. The problem of optimal cooperative three-dimensional conflict resolution involving multiple aircraft is addressed by the rigorous numerical trajectory optimization methods. The conflict problem is posed as an optimal control problem of finding trajectories that minimize a certain objective function while the safe separation between each aircraft pair is maintained. The initial and final positions of the aircraft are known and aircraft models with detailed nonlinear point-mass dynamics are considered. The protection zone around the aircraft is modeled to be cylindrical in shape. A novel formulation of the cylindrical protection zone is proposed by the use of continuous variables. The optimal control problem is converted to a finite dimensional nonlinear program (NLP) by the use of collocation on finite elements. The NLP is solved by the use of an interior point algorithm that incorporates a novel line search method. A reliable initialization strategy that yields a feasible solution on simple models is also proposed and adapted to detailed models. Several resolution scenarios are illustrated. The practical issue of flyability of the generated trajectories is addressed by the ability of our mathematical programming framework to accommodate detailed dynamic models.
TL;DR: This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza and incorporates a trust-region constraint.
Abstract: This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [10]. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the l1 penalty function inside a trust region. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active at the solution of the linear program. The EQP incorporates a trust-region constraint and is solved (inexactly) by means of a projected conjugate gradient method. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEr [1, 15] test set.
TL;DR: Theoretical and numerical properties of the proposed filled function are investigated and a solution algorithm is proposed for identifying a global minimum point for a general class of nonlinear programming problems with a closed bounded domain.
Abstract: A novel filled function is suggested in this paper for identifying a global minimum point for a general class of nonlinear programming problems with a closed bounded domain. Theoretical and numerical properties of the proposed filled function are investigated and a solution algorithm is proposed. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.
18 May 2004
TL;DR: In this paper, the authors investigate the performance of two different position estimation methods based on the estimate of the time-of-arrival (TOA) of the UWB signal at a set of receivers/sensors.
Abstract: The paper reports on the development of a low-cost device for low data rate communications with tracking and positioning capabilities. We investigate the performance of two different position estimation methods based on the estimate of the time-of-arrival (TOA) of the UWB signal at a set of receivers/sensors. The performance evaluation is performed in terms of the root mean-squared (RMS) error of the position coordinates estimation and the failure rate. We first study the direct-calculation method which gives exact solutions of a set of simultaneous equations. We then study one of the classical nonlinear optimization techniques, the Davidon-Fletcher-Powell (DFP) quasi-Newton algorithm. Both the direct-calculation method and the nonlinear optimization algorithms do not require any knowledge of the TOA estimation error variance or distribution. This advantage would be attractive for practical applications.