Solution of Equations using Optimization

It is sometimes convenient to use optimization to solve equations, or sets of simultaneous equations. This arises from the availability of general-purpose optimization 

The objective of the optimization would be for the equality constraint given by Equation 3.37 to be satisfied as nearly as possible. There are a number of ways in which this objective can be made specific for optimization. Three possibilities are12 

Whereas, slack variable S2 in Equation 3.41 for S2 0 ensures that  

Equations 3.43 and 3.44 are only satisfied simultaneously by Equation 3.37. Thus, x, S and S2 can be varied 

The approach can be extended to solve sets of simultaneous equations. For example, suppose a solution is required for xi and x2 such that  

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