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Nearinterpolation to arbitrary constraints
 Curve and Surface Fitting: St. Malo 2002
, 2003
"... Abstract. This paper is concerned with the computation of solutions to the problem of best nearinterpolation by parametric curves to constraints of the form f(ti) ∈ Ki for general sets Ki, including derivative constraints when ti has multiplicity greater than one. The minimizers are spline curves ..."
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Abstract. This paper is concerned with the computation of solutions to the problem of best nearinterpolation by parametric curves to constraints of the form f(ti) ∈ Ki for general sets Ki, including derivative constraints when ti has multiplicity greater than one. The minimizers are spline curves that we represent as smoothing splines with weights that are determined from the Lagrange multipliers corresponding to the constraints. To compute approximate solutions, we generalize a particular fixed point iteration used previously in nearinterpolation; to compute “exact ” solutions, we apply a nonsmooth Newton solver. The construction leads to optimality conditions for nearinterpolation with variable data sites for these more general constraints. Standard methods are used to update these data sites. Let H1[a, b]: = L (m) 2 ([a, b]−→IR) denote the Sobolev space of functions f:
Variational Inequalities and Economic Equilibrium
"... Variational inequality representations are set up for a general Walrasian model of consumption and production with trading in a market. The variational inequalities are of functional rather than geometric type and therefore are able to accommodate a wider range of utility functions than has been cov ..."
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Variational inequality representations are set up for a general Walrasian model of consumption and production with trading in a market. The variational inequalities are of functional rather than geometric type and therefore are able to accommodate a wider range of utility functions than has been covered satisfactorily in the past. They incorporate Lagrange multipliers for budget constraints, which are shown to lead to an enhanced equilibrium framework with features of collective optimization. Existence of such an enhanced equilibrium is confirmed through a new result about solutions to nonmonotone variational inequalities over bounded domains. Truncation arguments with specific estimates, based on the data in the economic model, are devised to transform the unbounded variational inequality that naturally comes up into a bounded one having the same solutions. Key words: Walrasian economic equilibrium; functional variational inequalities; equilibrium computations; equilibrium constraints; complementarity problems
Numerical Issues and Influences in the Design of Algebraic Modeling Languages for Optimization
"... The idea of a modeling language is to describe mathematical problems symbolically in a way that is familiar to people but that allows for processing by computer systems. In particular the concept of an algebraic modeling language, based on objective and constraint expressions in terms of decision va ..."
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The idea of a modeling language is to describe mathematical problems symbolically in a way that is familiar to people but that allows for processing by computer systems. In particular the concept of an algebraic modeling language, based on objective and constraint expressions in terms of decision variables, has proved to be valuable for a broad range of optimization and related problems. One modeling language can work with numerous solvers, each of which implements one or more optimization algorithms. The separation of model specification from solver execution is thus a key tenet of modeling language design. Nevertheless, several issues in numerical analysis that are critical to solvers are also important in implementations of modeling languages. Socalled presolve procedures, which tighten bounds with the aim of eliminating some variables and constraints, are numerical algorithms that require carefully chosen tolerances and can benefit from directed roundings. Correctly rounded binarydecimal conversion is valuable in portably conveying problem instances and in debugging. Further rounding options offer tradeoffs between accuracy, convenience, and readability in displaying