2,725 research outputs found
Bounding Option Prices Using SDP With Change Of Numeraire
Recently, given the first few moments, tight upper and lower bounds of the no arbitrage prices can be obtained by solving semidefinite programming (SDP) or linear programming (LP) problems. In this paper, we compare SDP and LP formulations of the European-style options pricing problem and prefer SDP formulations due to the simplicity of moments constraints. We propose to employ the technique of change of numeraire when using SDP to bound the European type of options. In fact, this problem can then be cast as a truncated Hausdorff moment problem which has necessary and sufficient moment conditions expressed by positive semidefinite moment and localizing matrices. With four moments information we show stable numerical results for bounding European call options and exchange options. Moreover, A hedging strategy is also identified by the dual formulation.moments of measures, semidefinite programming, linear programming, options pricing, change of numeraire
Nonlinear Integer Programming
Research efforts of the past fifty years have led to a development of linear
integer programming as a mature discipline of mathematical optimization. Such a
level of maturity has not been reached when one considers nonlinear systems
subject to integrality requirements for the variables. This chapter is
dedicated to this topic.
The primary goal is a study of a simple version of general nonlinear integer
problems, where all constraints are still linear. Our focus is on the
computational complexity of the problem, which varies significantly with the
type of nonlinear objective function in combination with the underlying
combinatorial structure. Numerous boundary cases of complexity emerge, which
sometimes surprisingly lead even to polynomial time algorithms.
We also cover recent successful approaches for more general classes of
problems. Though no positive theoretical efficiency results are available, nor
are they likely to ever be available, these seem to be the currently most
successful and interesting approaches for solving practical problems.
It is our belief that the study of algorithms motivated by theoretical
considerations and those motivated by our desire to solve practical instances
should and do inform one another. So it is with this viewpoint that we present
the subject, and it is in this direction that we hope to spark further
research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G.
Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50
Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art
Surveys, Springer-Verlag, 2009, ISBN 354068274
An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging
This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions
Column-Randomized Linear Programs: Performance Guarantees and Applications
We propose a randomized method for solving linear programs with a large
number of columns but a relatively small number of constraints. Since
enumerating all the columns is usually unrealistic, such linear programs are
commonly solved by column generation, which is often still computationally
challenging due to the intractability of the subproblem in many applications.
Instead of iteratively introducing one column at a time as in column
generation, our proposed method involves sampling a collection of columns
according to a user-specified randomization scheme and solving the linear
program consisting of the sampled columns. While similar methods for solving
large-scale linear programs by sampling columns (or, equivalently, sampling
constraints in the dual) have been proposed in the literature, in this paper we
derive an upper bound on the optimality gap that holds with high probability
and converges with rate , where is the number of sampled
columns, to the value of a linear program related to the sampling distribution.
To the best of our knowledge, this is the first paper addressing the
convergence of the optimality gap for sampling columns/constraints in generic
linear programs without additional assumptions on the problem structure and
sampling distribution. We further apply the proposed method to various
applications, such as linear programs with totally unimodular constraints,
Markov decision processes, covering problems and packing problems, and derive
problem-specific performance guarantees. We also generalize the method to the
case that the sampled columns may not be statistically independent. Finally, we
numerically demonstrate the effectiveness of the proposed method in the
cutting-stock problem and in nonparametric choice model estimation
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