# In the exercises following Chapter 1, we formulated the absolute-value regression problem: In…

In the exercises following Chapter 1, we formulated the absolute-value regression problem:

In this formulation, the yi are measurements of the dependent variable (e.g., income), which is assumed to be explained by independent variables (e.g., level of education, parents’ income, and so forth), which are measured as xi1, xi2, . . . , xin. A linear model assumes that y depends linearly upon the β’s, as:

Given any choice of the parameters β1, β2, . . . , βn, yˆi is an estimate of yi . The above formulation aims to minimize the deviations of the estimates of yˆi from yi as measured by the sum of absolute values of the deviations. The variables in the linear-programming model are the parameters β1, β2, . . . , βn as well as the Pi and Ni . The quantities yi, xi1, xi2, . . . , xin are known data found by measuring several values for the dependent and independent variables for the linear model (29).

In practice, the number of observations m frequently is much larger than the number of parameters n. Show how we can take advantage of this property by formulating the dual to the above linear program in the dual variables u1, u2, . . . , um. How can the special structure of the dual problem be exploited computationally?

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