# Differentiable optimization and equation solving by John L. Nazareth

By John L. Nazareth

An summary of the dramatic reorganization in response to N. Karmakar’s seminal 1984 paper on algorithmic linear programming within the zone of algorithmic differentiable optimization and equation-solving, or, extra easily, algorithmic differentiable programming. aimed toward readers accustomed to complex calculus and numerical research.

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**Example text**

It is often convenient to straddle this “two-lane highway,” so to speak, and to formulate algorithms based on a “middle-of-the-road” approach. We now describe the traditional synthesis based on positive deﬁnite, unconstrained models and a new synthesis, called the NC method, based on M+ k -metric trust regions. 18) + + + + where M+ k is one of the following: I; Dk ; Mk ; L-Mk ; L-RH-Hk ; L-RH+ + Mk ; Hk . 17), and indeed, the foregoing quadratic model can be obtained from a Lagrangian relaxation of the latter expression.

7) i=1 with n = 10 and starting point (0, . . , 0). When the above routine DNEQNJ is run from the starting point, it again terminates with an error message as in the previous example. 084. The quotation below from Dennis and Schnabel [1983, p. 152], which we have transcribed to use our notation, highlights the diﬃculties encountered in these two examples: There is one signiﬁcant case when the global algorithms for nonlinear equations can fail. It is when a local minimizer of F (x) is not a root of h(x).

0). When the above routine DNEQNJ is run from the starting point, it again terminates with an error message as in the previous example. 084. The quotation below from Dennis and Schnabel [1983, p. 152], which we have transcribed to use our notation, highlights the diﬃculties encountered in these two examples: There is one signiﬁcant case when the global algorithms for nonlinear equations can fail. It is when a local minimizer of F (x) is not a root of h(x). A global1 minimization routine, started close to such a point, may converge to it.