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Recent studies of exosolar planet detection methods with a space-based visible light coronagraph have shown the feasibility of this approach. However, the telescope optical precision requirements are extremely demanding-a few Angstroms... more
Recent studies of exosolar planet detection methods with a space-based visible light coronagraph have shown the feasibility of this approach. However, the telescope optical precision requirements are extremely demanding-a few Angstroms residual wavefront error-which is beyond current capabilities for large optical surfaces. Secondly, the coronagraph depends upon use of masks located at either the pupil or a focus to reject the starlight and image the exosolar planet. Effects of diffraction and light scatter place precision ...
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Abstract This paper describes an implementation of the one-phase primal-dual path-following algorithm for solving linear programming problems. The design is intended to be simple, portable and robust. These design goals are achieved... more
Abstract This paper describes an implementation of the one-phase primal-dual path-following algorithm for solving linear programming problems. The design is intended to be simple, portable and robust. These design goals are achieved without sacrificing state-of-the-art performance. We give a brief description of the algorithm and the implementation focusing on the features which most distinguish this implementation from others available.
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So far, we have avoided using matrix notation to present linear programming problems and the simplex method. In this chapter, we shall recast everything into matrix notation. At the same time, we will emphasize the close relations between... more
So far, we have avoided using matrix notation to present linear programming problems and the simplex method. In this chapter, we shall recast everything into matrix notation. At the same time, we will emphasize the close relations between the primal and the dual problems.
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Research Interests: Mathematics and Physics
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Up until now, we have always considered our problems to be given in standard form. However, for real-world problems it is often convenient to formulate problems in the following form: In this chapter, we shall show how to modify the... more
Up until now, we have always considered our problems to be given in standard form. However, for real-world problems it is often convenient to formulate problems in the following form: In this chapter, we shall show how to modify the simplex method to handle problems presented in this form.
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The aim of this chapter is to discuss several applications of metric space ideas to some classical problems of engineering analysis.
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The most time-consuming aspect of each iteration of the path-following method is solving the system of equations that defines the step direction vectors Δx, Δy, Δw, and Δz: $$\displaystyle\begin{array}{rcl} A\Delta x + \Delta w&... more
The most time-consuming aspect of each iteration of the path-following method is solving the system of equations that defines the step direction vectors Δx, Δy, Δw, and Δz: $$\displaystyle\begin{array}{rcl} A\Delta x + \Delta w& =& \rho {}\end{array}$$ (19.1) $$\displaystyle\begin{array}{rcl}{ A}^{T}\Delta y - \Delta z& =& \sigma {}\end{array}$$ (19.2) $$\displaystyle\begin{array}{rcl} Z\Delta x + X\Delta z& =& \mu e - XZe{}\end{array}$$ (19.3) $$\displaystyle\begin{array}{rcl} W\Delta y + Y \Delta w& =& \mu e - Y We.{}\end{array}$$ (19.4)
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ABSTRACT
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For mappings from one metric space to another we employ either notations like T, S, U or notations like f, g, h. Generally, the transformation notation is cleaner: we write Tx for the image of x under T, which becomes f(x) in the standard... more
For mappings from one metric space to another we employ either notations like T, S, U or notations like f, g, h. Generally, the transformation notation is cleaner: we write Tx for the image of x under T, which becomes f(x) in the standard function notation.
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So far, we have avoided using matrix notation to present linear programming problems and the simplex method. In this chapter, we shall recast everything into matrix notation. At the same time, we will emphasize the close relations between... more
So far, we have avoided using matrix notation to present linear programming problems and the simplex method. In this chapter, we shall recast everything into matrix notation. At the same time, we will emphasize the close relations between the primal and the dual problems.
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In this chapter, we consider two related subjects. The first, called sensitivity analysis (or postoptimality analysis) addresses the following question: having found an optimal solution to a given linear programming problem, how much can... more
In this chapter, we consider two related subjects. The first, called sensitivity analysis (or postoptimality analysis) addresses the following question: having found an optimal solution to a given linear programming problem, how much can we change the data and have the current partition into basic and nonbasic variables remain optimal? The second subject addresses situations in which one wishes to solve not just one linear program, but a whole family of problems parametrized by a single real variable.
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In this chapter, we begin our study of an alternative to the simplex method for solving linear programming problems. The algorithm we are going to introduce is called a path-following method. It belongs to a class of methods called... more
In this chapter, we begin our study of an alternative to the simplex method for solving linear programming problems. The algorithm we are going to introduce is called a path-following method. It belongs to a class of methods called interior-point methods. The path-following method seems to be the simplest and most natural of all the methods in this class, so in this book we focus primarily on it. Before we can introduce this method, we must define the path that appears in the name of the method. This path is called the central path and is the subject of this chapter. Before discussing the central path, we must lay some groundwork by analyzing a nonlinear problem, called the barrier problem, associated with the linear programming problem that we wish to solve.