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63
TrustRegion InteriorPoint Algorithms For Minimization Problems With Simple Bounds
 SIAM J. CONTROL AND OPTIMIZATION
, 1995
"... Two trustregion interiorpoint algorithms for the solution of minimization problems with simple bounds are analyzed and tested. The algorithms scale the local model in a way similar to Coleman and Li [1]. The first algorithm is more usual in that the trust region and the local quadratic model are c ..."
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Cited by 56 (18 self)
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Two trustregion interiorpoint algorithms for the solution of minimization problems with simple bounds are analyzed and tested. The algorithms scale the local model in a way similar to Coleman and Li [1]. The first algorithm is more usual in that the trust region and the local quadratic model are consistently scaled. The second algorithm proposed here uses an unscaled trust region. A global convergence result for these algorithms is given and dogleg and conjugategradient algorithms to compute trial steps are introduced. Some numerical examples that show the advantages of the second algorithm are presented.
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 51 (11 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.
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 49 (7 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.
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 46 (9 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...
A new active set algorithm for box constrained optimizaiton
 SIAM J. Optim
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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 17 (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 ...
Iterative methods for finding a trustregion step
, 2007
"... Abstract. We consider the problem of finding an approximate minimizer of a general quadratic function subject to a twonorm constraint. The SteihaugToint method minimizes the quadratic over a sequence of expanding subspaces until the iterates either converge to an interior point or cross the constr ..."
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Cited by 14 (3 self)
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Abstract. We consider the problem of finding an approximate minimizer of a general quadratic function subject to a twonorm constraint. The SteihaugToint method minimizes the quadratic over a sequence of expanding subspaces until the iterates either converge to an interior point or cross the constraint boundary. The benefit of this approach is that an approximate solution may be obtained with minimal work and storage. However, the method does not allow the accuracy of a constrained solution to be specified. We propose an extension of the SteihaugToint method that allows a solution to be calculated to any prescribed accuracy. If the SteihaugToint point lies on the boundary, the constrained problem is solved on a sequence of evolving lowdimensional subspaces. Each subspace includes an accelerator direction obtained from a regularized Newton method applied to the constrained problem. A crucial property of this direction is that it can be computed by applying the conjugategradient method to a positivedefinite system in both the primal and dual variables of the constrained problem. The method includes a parameter that allows the user to take advantage of the tradeoff between the overall number of function evaluations and matrixvector products associated with the underlying trustregion method. At one extreme, a lowaccuracy solution is obtained that is comparable to the SteihaugToint point. At the other extreme, a highaccuracy solution can be specified that minimizes the overall number of function evaluations at the expense of more matrixvector products. Key words. Largescale unconstrained optimization, trustregion methods, conjugategradient method, Lanczos tridiagonalization process AMS subject classifications. 49J20, 49J15, 49M37, 49D37, 65F05, 65K05, 90C30
Hardwareassisted feature analysis and visualization of procedurally encoded multifield volumetric data
 IEEE Comput. Graph. Appl
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S.: A new minimization protocol for solving nonlinear PoissonBoltzmann mortar finite element equation
 BIT
, 2007
"... The nonlinear Poisson–Boltzmann equation (PBE) is a widelyused implicit solvent model in biomolecular simulations. This paper formulates a new PBE nonlinear algebraic system from a mortar finite element approximation, and proposes a new minimization protocol to solve it efficiently. In particular ..."
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Cited by 11 (1 self)
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The nonlinear Poisson–Boltzmann equation (PBE) is a widelyused implicit solvent model in biomolecular simulations. This paper formulates a new PBE nonlinear algebraic system from a mortar finite element approximation, and proposes a new minimization protocol to solve it efficiently. In particular, the PBE mortar nonlinear algebraic system is proved to have a unique solution, and is equivalent to a unconstrained minimization problem. It is then solved as the unconstrained minimization problem by the subspace trust region Newton method. Numerical results show that the new minimization protocol is more efficient than the traditional merit least squares approach in solving the nonlinear system. At least 80 percent of the total CPU time was saved for a PBE model problem. AMS subject classification (2000): 65N30, 65H10, 65K10, 9208. Key words: Poisson–Boltzmann equation, mortar finite element, nonlinear system,