On the \(L_ 1\) exact penalty function with locally Lipschitz functions (Q1116898)

We discuss the following inequality constrained optimization problem \[ (P)\quad \min f(x)\quad subject\quad to\quad g(x)\leq 0,\quad g(x)=(g_ 1(x),..,g_ r(x))^ T, \] where f(x), \(g_ j(x)\) \((j=1,...,r)\) are locally Lipschitz functions. The \(L_ 1\) exact penalty function of the problem (P) is \[ (PC)\quad \min f(x)+cp(x)\quad subject\quad to\quad x\in R^ n, \] where \(p(x)=\max \{0,g_ 1(x),...,g_ r(x)\}\), \(c>0\). We will discuss the relationships between (P) and (PC). In particular, we will prove that under some (mild) conditions a local minimum of (PC) is also a local minimum of (P).