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29
TrustRegion InteriorPoint SQP Algorithms For A Class Of Nonlinear Programming Problems
 SIAM J. CONTROL OPTIM
, 1997
"... In this paper a family of trustregion interiorpoint SQP algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise e.g. from the discretization of optimal co ..."
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Cited by 35 (8 self)
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In this paper a family of trustregion interiorpoint SQP algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise e.g. from the discretization of optimal control problems. The algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints, but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial differential equations. The algorithms keep strict feasibility with respect to the bound constraints by using an affine scaling method proposed for a different class of problems by Coleman and Li and they exploit trustregion techniques for equalityconstrained optimizatio...
Reconstructing The Unknown Local Volatility Function
 Journal of Computational Finance
, 1998
"... Using market European option prices, a method for computing a smooth local volatility function in a 1factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphas ..."
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Cited by 35 (5 self)
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Using market European option prices, a method for computing a smooth local volatility function in a 1factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphasized that accurately approximating the true local volatility function is crucial in hedging even simple European options, and pricing exotic options. A spline functional approach is used: the local volatility function is represented by a spline whose values at chosen knots are determined by solving a constrained nonlinear optimization problem. The optimization formulation is amenable to various option evaluation methods; a partial differential equation implementation is discussed. Using a synthetic European call option example, we illustrate the capability of the proposed method in reconstructing the unknown local volatility function. Accuracy of pricing and hedging is also illustrated. Moreover, it is demonstrated that, using a different constant implied volatility for an option with different strike/maturity can produce erroneous hedge factors. In addition, real market European call option data on the S&P 500 stock index is used to compute the local volatility function; stability of the approach is demonstrated.
A new active set algorithm for box constrained Optimization
 SIAM Journal on Optimization
, 2006
"... Abstract. An active set algorithm (ASA) for box constrained optimization is developed. The algorithm consists of a nonmonotone gradient projection step, an unconstrained optimization step, and a set of rules for branching between the two steps. Global convergence to a stationary point is established ..."
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Cited by 26 (6 self)
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Abstract. An active set algorithm (ASA) for box constrained optimization is developed. The algorithm consists of a nonmonotone gradient projection step, an unconstrained optimization step, and a set of rules for branching between the two steps. Global convergence to a stationary point is established. For a nondegenerate stationary point, the algorithm eventually reduces to unconstrained optimization without restarts. Similarly, for a degenerate stationary point, where the strong secondorder sufficient optimality condition holds, the algorithm eventually reduces to unconstrained optimization without restarts. A specific implementation of the ASA is given which exploits the recently developed cyclic Barzilai–Borwein (CBB) algorithm for the gradient projection step and the recently developed conjugate gradient algorithm CG DESCENT for unconstrained optimization. Numerical experiments are presented using box constrained problems in the CUTEr and MINPACK2 test problem libraries. Key words. nonmonotone gradient projection, box constrained optimization, active set algorithm,
Hardwareassisted feature analysis and visualization of procedurally encoded multifield volumetric data
 IEEE Computer Graphics & Applications
, 2005
"... Figure 1: Four visualizations of RBF encoded datasets. Left image: Interactively extracted and volume rendered vorticity from the Tornado dataset encoded with 2,100 RBFs. Second: Traces of 110 particles tracked in experimentally obtained Channel dataset encoded with 2,105 RBFs. Third image: Volume r ..."
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Cited by 10 (3 self)
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Figure 1: Four visualizations of RBF encoded datasets. Left image: Interactively extracted and volume rendered vorticity from the Tornado dataset encoded with 2,100 RBFs. Second: Traces of 110 particles tracked in experimentally obtained Channel dataset encoded with 2,105 RBFs. Third image: Volume rendering of water pressure for an injection well. The 156,642 tetrahedra dataset of a simulated blackoil reservoir is encoded using 141 RBFs. Fourth image: Isosurface rendering of vorticity magnitude. Positive helicity has been mapped to red colors and negative helicity to blue colors. Procedural encoding of scattered and unstructured scalar datasets using Radial Basis Functions (RBF) is an active area of research with great potential for compactly representing large datasets. This reduced storage requirement allows the compressed datasets to completely reside in the local memory of the graphics card, thus, enabling accurate and efficient processing and visualization without data transfer problems. We have developed new hierarchical techniques that effectively encode data on arbitrary grids including volumetric scalar, vector,
MODEL REDUCTION FOR LARGESCALE SYSTEMS WITH HIGHDIMENSIONAL PARAMETRIC INPUT SPACE
, 2007
"... Abstract. A modelconstrained adaptive sampling methodology is proposed for reduction of largescale systems with highdimensional parametric input spaces. Our model reduction method uses a reduced basis approach, which requires the computation of highfidelity solutions at a number of sample points ..."
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Cited by 10 (3 self)
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Abstract. A modelconstrained adaptive sampling methodology is proposed for reduction of largescale systems with highdimensional parametric input spaces. Our model reduction method uses a reduced basis approach, which requires the computation of highfidelity solutions at a number of sample points throughout the parametric input space. A key challenge that must be addressed in the optimization, control, and probabilistic settings is the need for the reduced models to capture variation over this parametric input space, which, for many applications, will be of high dimension. We pose the task of determining appropriate sample points as a PDEconstrained optimization problem, which is implemented using an efficient adaptive algorithm that scales well to systems with a large number of parameters. The methodology is demonstrated for examples with parametric input spaces of dimension 11 and 21, which describe thermal analysis and design of a heat conduction fin, and compared with statisticallybased sampling methods. For this example, the modelconstrained adaptive sampling leads to reduced models that, for a given basis size, have error several orders of magnitude smaller than that obtained using the other methods.
Superlinear Convergence of AffineScaling InteriorPoint Newton Methods for InfiniteDimensional Nonlinear Problems with Pointwise Bounds
, 1999
"... We develop and analyze a superlinearly convergent affinescaling interiorpoint Newton method for infinitedimensional problems with pointwise bounds in L p space. The problem formulation is motivated by optimal control problems with L p controls and pointwise control constraints. The finite ..."
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Cited by 9 (6 self)
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We develop and analyze a superlinearly convergent affinescaling interiorpoint Newton method for infinitedimensional problems with pointwise bounds in L p space. The problem formulation is motivated by optimal control problems with L p controls and pointwise control constraints. The finitedimensional convergence theory by Coleman and Li (SIAM J. Optim., 6 (1996), pp. 418445) makes essential use of the equivalence of norms and the exact identifiability of the active constraints close to an optimizer with strict complementarity. Since these features are not available in our infinitedimensional framework, algorithmic changes are necessary to ensure fast local convergence. The main building block is a Newtonlike iteration for an affinescaling formulation of the KKTcondition. We demonstrate in an example that a stepsize rule to obtain an interior iterate may require very small stepsizes even arbitrarily close to a nondegenerate solution. Using a pointwise projection instead ...
On InteriorPoint Newton Algorithms For Discretized Optimal Control Problems With State Constraints
 OPTIM. METHODS SOFTW
, 1998
"... In this paper we consider a class of nonlinear programming problems that arise from the discretization of optimal control problems with bounds on both the state and the control variables. For this class of problems, we analyze constraint qualifications and optimality conditions in detail. We derive ..."
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Cited by 7 (2 self)
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In this paper we consider a class of nonlinear programming problems that arise from the discretization of optimal control problems with bounds on both the state and the control variables. For this class of problems, we analyze constraint qualifications and optimality conditions in detail. We derive an affinescaling and two primaldual interiorpoint Newton algorithms by applying, in an interiorpoint way, Newton's method to equivalent forms of the firstorder optimality conditions. Under appropriate assumptions, the interiorpoint Newton algorithms are shown to be locally welldefined with a qquadratic rate of local convergence. By using the structure of the problem, the linear algebra of these algorithms can be reduced to the null space of the Jacobian of the equality constraints. The similarities between the three algorithms are pointed out, and their corresponding versions for the general nonlinear programming problem are discussed.
Local Convergence of the AffineScaling InteriorPoint Algorithm for Nonlinear Programming
 COMPUT. OPTIM. AND APPL
, 1999
"... This paper addresses the local convergence properties of the anescaling interiorpoint algorithm for nonlinear programming. The analysis of local convergence is developed in terms of parameters that control the interiorpoint scheme and the size of the residual of the linear system that provides the ..."
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Cited by 5 (2 self)
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This paper addresses the local convergence properties of the anescaling interiorpoint algorithm for nonlinear programming. The analysis of local convergence is developed in terms of parameters that control the interiorpoint scheme and the size of the residual of the linear system that provides the step direction. The analysis follows the classical theory for quasiNewton methods and addresses qlinear, qsuperlinear, and qquadratic rates of convergence.
Parametric ReducedOrder Models for Probabilistic Analysis of Unsteady Aerodynamic Applications
"... Methodology is presented to derive reducedorder models for largescale parametric applications in unsteady aerodynamics. The specific case considered in this paper is a computational fluid dynamic (CFD) model with parametric dependence that arises from geometric shape variations. The first key cont ..."
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Cited by 2 (0 self)
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Methodology is presented to derive reducedorder models for largescale parametric applications in unsteady aerodynamics. The specific case considered in this paper is a computational fluid dynamic (CFD) model with parametric dependence that arises from geometric shape variations. The first key contribution of the methodology is the derivation of a linearized model that permits the effects of geometry variations to be represented with an explicit affine function. The second key contribution is an adaptive sampling method that utilizes an optimization formulation to derive a reduced basis that spans the space of geometric input parameters. The method is applied to derive efficient reducedorder models for probabilistic analysis of the effects of blade geometry variation for a twodimensional model problem governed by the Euler equations. Reducedorder models that achieve three orders of magnitude reduction in the number of states are shown to accurately reproduce CFD Monte Carlo simulation results at a fraction of the computational cost. I.
Multiuser Margin Optimization in Digital Subscriber Line (DSL) Channels
"... Abstract—This paper presents efficient multiuser margin optimization algorithms suitable for multicarrier digital subscriber line (DSL) systems using Dynamic Spectrum Management (DSM). The favorable monotonicity and fairness properties of multiuser margin are employed to formulate a boxconstrained ..."
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Cited by 1 (0 self)
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Abstract—This paper presents efficient multiuser margin optimization algorithms suitable for multicarrier digital subscriber line (DSL) systems using Dynamic Spectrum Management (DSM). The favorable monotonicity and fairness properties of multiuser margin are employed to formulate a boxconstrained nonlinear least squares (NLSQ) problem for multiuser margin maximization, which is efficiently solved by using a scaledgradient trustregion approach with Broyden Jacobian update. Based on this NLSQ formulation, a multiuser harmonized margin (MHM) optimization algorithm for resource allocation is developed. A Newton–Raphson method is also developed for fast margin estimation and used within the MHM. The MHM algorithm converges efficiently to a solution for the best common equal margin to all users, while explicitly guaranteeing their target rate requirements. (This is the reason for the term harmonized.) Furthermore, its predominantly distributed structure can be implemented in DSL/DSM scenarios with only Level 1 coordination. Simulation results of various cases verify the convergence to the unique optimal solution within 5–10 iterations. Index Terms—Broyden update, digital subscriber line (DSL), dynamic spectrum management (DSM), iterated waterfilling, loading algorithms, margin adaptive, trust region methods. I.