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Are the data points, (This aces not necessarily guarantee that the polyl hernial will be convex throughout the interval. ) that Gx - w = h where w ~ _O . Introduce slack variables w such 36 Introducing Lagrange multipliers as before, we may write the system as: h i O 0 G -I 0 I A 0 r b GT AT 0 0 x 0 w At the solution, we must have _• ~ o , w~o, T _z_w=0. This implies that when a Lagrange multiplier is non-zero then the corresponding constraint holds with equality. Conversely, corresponding to a non-zero w i the Lagrange multiplier must be zero.
Pereyra, "Solution of Vandermonde Systems of Equations", Publicaion 70-02, Universidad Central de Venezuela, Caracas, Venezuela, 1970.  Cottle, R. W. and @. B. Dantzig, "Complementary Pivot Theory of Mathematical Programming", Mathematics of the Decision Sclences~ Part 1, G. B. Dantzig and A. F. ), American Mathematical Societ 2 (1968), pp. 115-136.  Dantzig, G. , R. P. Harvey, R. D. McKnight, and S. S. ymposium on Sparse Matrices and Their Appllcations, T. J. Watson Research Publications RAI, no.
Therefore, if we know which constraints held with equality at the solution, we could treat the problem as a linear least-squares problem with linear equality constraints. A technique, due to Cottle and Dantzig , exists for solving the problem inthis way. inear Programming, J. ). John Wiley, New York, 1967;  pp. 133-205. , "Iterative Refinement of Linear Least Squares Solutions II", BIT 8 (1968), pp. 8-30.  and G. H. Golub, "Iterative Refinement of Linear Least Squares Solutions by Householder Transformations", BIT 7 (1967), pp.