22,815 research outputs found
Piecewise-linear pathways to the optimal solution set in linear programming
This paper takes a fresh look at the application of quadratic penalty functions to linear programming. Recently, Madsen et al. (Ref. 1) described a continuation algorithm for linear programming based on smoothing a dual l1-formulation of a linear program with unit bounds. The present paper is prompted by the observation that this is equivalent to applying a quadratic penalty function to the dual of a linear program in standard canonical form, in the sense that both approaches generate continuous, piecewise-linear paths leading to the optimal solution set. These paths lead to new characterizations of optimal solutions in linear programming. An important product of this analysis is a finite penalty algorithm for linear programming closely related to the least-norm algorithm of Mangasarian (Ref. 2) and to the continuation algorithm of Madsen et al. (Ref. 1). The algorithm is implemented, and promising numerical results are given
A method for pricing American options using semi-infinite linear programming
We introduce a new approach for the numerical pricing of American options.
The main idea is to choose a finite number of suitable excessive functions
(randomly) and to find the smallest majorant of the gain function in the span
of these functions. The resulting problem is a linear semi-infinite programming
problem, that can be solved using standard algorithms. This leads to good upper
bounds for the original problem. For our algorithms no discretization of space
and time and no simulation is necessary. Furthermore it is applicable even for
high-dimensional problems. The algorithm provides an approximation of the value
not only for one starting point, but for the complete value function on the
continuation set, so that the optimal exercise region and e.g. the Greeks can
be calculated. We apply the algorithm to (one- and) multidimensional diffusions
and to L\'evy processes, and show it to be fast and accurate
A penalty continuation method for the ℓ∞ solution of overdetermined linear systems
A new algorithm for the ℓ∞ solution of overdetermined linear systems is given. The algorithm is based on the application of quadratic penalty functions to a primal linear programming formulation of the ℓ∞ problem. The minimizers of the quadratic penalty function generate piecewise-linear non-interior paths to the set of ℓ∞ solutions. It is shown that the entire set of ℓ∞ solutions is obtained from the paths for sufficiently small values of a scalar parameter. This leads to a finite penalty/continuation algorithm for ℓ∞ problems. The algorithm is implemented and extensively tested using random and function approximation problems. Comparisons with the Barrodale-Phillips simplex based algorithm and the more recent predictor-corrector primal-dual interior point algorithm are given. The results indicate that the new algorithm shows a promising performance on random (non-function approximation) problems
Online-Computation Approach to Optimal Control of Noise-Affected Nonlinear Systems with Continuous State and Control Spaces
© 2007 EUCA.A novel online-computation approach to optimal control of nonlinear, noise-affected systems with continuous state and control spaces is presented. In the proposed algorithm, system noise is explicitly incorporated into the control decision. This leads to superior results compared to state-of-the-art nonlinear controllers that neglect this influence. The solution of an optimal nonlinear controller for a corresponding deterministic system is employed to find a meaningful state space restriction. This restriction is obtained by means of approximate state prediction using the noisy system equation. Within this constrained state space, an optimal closed-loop solution for a finite decision-making horizon (prediction horizon) is determined within an adaptively restricted optimization space. Interleaving stochastic dynamic programming and value function approximation yields a solution to the considered optimal control problem. The enhanced performance of the proposed discrete-time controller is illustrated by means of a scalar example system. Nonlinear model predictive control is applied to address approximate treatment of infinite-horizon problems by the finite-horizon controller
Automating embedded analysis capabilities and managing software complexity in multiphysics simulation part II: application to partial differential equations
A template-based generic programming approach was presented in a previous
paper that separates the development effort of programming a physical model
from that of computing additional quantities, such as derivatives, needed for
embedded analysis algorithms. In this paper, we describe the implementation
details for using the template-based generic programming approach for
simulation and analysis of partial differential equations (PDEs). We detail
several of the hurdles that we have encountered, and some of the software
infrastructure developed to overcome them. We end with a demonstration where we
present shape optimization and uncertainty quantification results for a 3D PDE
application
Deflation for semismooth equations
Variational inequalities can in general support distinct solutions. In this
paper we study an algorithm for computing distinct solutions of a variational
inequality, without varying the initial guess supplied to the solver. The
central idea is the combination of a semismooth Newton method with a deflation
operator that eliminates known solutions from consideration. Given one root of
a semismooth residual, deflation constructs a new problem for which a
semismooth Newton method will not converge to the known root, even from the
same initial guess. This enables the discovery of other roots. We prove the
effectiveness of the deflation technique under the same assumptions that
guarantee locally superlinear convergence of a semismooth Newton method. We
demonstrate its utility on various finite- and infinite-dimensional examples
drawn from constrained optimization, game theory, economics and solid
mechanics.Comment: 24 pages, 3 figure
Pattern matching in compilers
In this thesis we develop tools for effective and flexible pattern matching.
We introduce a new pattern matching system called amethyst. Amethyst is not
only a generator of parsers of programming languages, but can also serve as an
alternative to tools for matching regular expressions.
Our framework also produces dynamic parsers. Its intended use is in the
context of IDE (accurate syntax highlighting and error detection on the fly).
Amethyst offers pattern matching of general data structures. This makes it a
useful tool for implementing compiler optimizations such as constant folding,
instruction scheduling, and dataflow analysis in general.
The parsers produced are essentially top-down parsers. Linear time complexity
is obtained by introducing the novel notion of structured grammars and
regularized regular expressions. Amethyst uses techniques known from compiler
optimizations to produce effective parsers.Comment: master thesi
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