Nonsmooth exact penalty function

Theoretical studies (Nocedal and Wright, 2000 Bazaraa et al, 2006) and many practical applications have demonstrated that the following function allows constrained optimization to be solved using smaller weights than the ones required by a quadratic penalty Junction  

The function (12.12) is called the nonsmooth exact penalty Junction and is often 

This function has the defect of being nonsmooth in the first derivative. 

Note that, in the literature, a single common weight, k, for equality and inequality constraints is usually adopted  

This formulation must be avoided when, as it normally happens, the constraints have different orders of magnitude to other. 

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Despite the exactness feature of Pv no general-purpose, widely available NLP solver is based solely on the Lx exact penalty function Pv This is because Px also has a negative characteristic it is nonsmooth. The term hj(x) has a discontinuous derivative at any point x where hj (x) = 0, that is, at any point satisfying the y th equality constraint in addition, max 0, gj (x) has a discontinuous derivative at any x where gj (x) = 0, that is, whenever the yth inequality constraint is active, as illustrated in Figure 8.6. These discontinuities occur at any feasible or partially feasible point, so none of the efficient unconstrained minimizers for smooth problems considered in Chapter 6 can be applied, because they eventually encounter points where Px is nonsmooth. 

This weighted sum of absolute values in e(x) was also discussed in Section 8.4 as a way of measuring constraint violations in an exact penalty function. We proceed as we did in that section, eliminating the nonsmooth absolute value function by introducing positive and negative deviation variables dpt and dnt and converting this nonsmooth unconstrained problem into an equivalent smooth constrained problem, which is... 

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