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Chapter 1 Lower-Order Penalty Function Algorithm 1.1 Optimal Condition Based on Lower-Order Penalty Function In this section， by assuming a non-Lipschitz penalty function is exact， new conditions for the existence of Lagrange multipliers are established for a continuously differentiable optimization problem with both inequality and equality constrains (Yang et al.， 2007). This is done by virtue of a first-order necessary optimality condition of the penalty problem， which is obtained by estimating Dini upper directional derivatives of the penalty function in terms of Taylor expansions， and a Farkas lemma. Relations among the obtained results and some well-known constraint qualifications are discussed. continuously differentiable functions and Consider the following optimization problem with both inequality and equality constrains and the corresponding lp penalty problem where ρ >0 is the penalty parameter and F(x) the lp penalty function， and when p = 0， we will take the convention 00 = 0. Let be the feasible set. The lp penalty function F(x) is said to be exact if there is a finite value of ρ such that any local minimum of (P) is a local one of (Pρ). The earliest work on the l1 exact penalty function for convex programming was by Zangwill (1967). For the exactness of F(x)， we have Theorem 1.1.1 Let 0 < p 1. Penalty function F(x) is exact if and only if the following generalized calmness-type condition holds (1.1.1) where the perturbation function defined to be the optimal value of the optimization problem When p = 1， Theorem 1.1.1 is established (Burke， 1991; Clarke， 1983)， (1.1.1) is known as the Clarke calmness condition and F(x) the l1 penalty function， which is locally Lipschitz. The l1 penalty function has played a key role in mathematical programming. One important application of the calmness condition is that it， together with a calculus rule of an appropriate subdifferential， is sufficient to the existence of Lagrange multipliers for (P) (Clarke， 1983; Burke， 1991)， and the references therein. When 0 < p < 1， Theorem 1.1.1 is established (Rubinov et al.， 2003) and F(x) is referred to as a non-Lipschitz penalty function since it may not be locally Lipschitz at the point either gi(x) = 0 or hj(x) = 0. Other types of non-Lipschitz penalty functions have been employed in the study of mathematical programs with equilibrium constraints and error bounds (Luo et al.， 1996; Pang， 1997). Recently an augmented Lagrangian function with a level bounded augmenting function is introduced (Huang et al.， 2003) which extends the work of Rockafellar and Wets(1998) with a convex augmenting function and provides a unified scheme for duality and penalization. In particular， this scheme includes penalty functions in Luo et al. (1996)， Pang (1997) and Rubinov and Yang (2003) as spe cases. When p = 0， F(x) is not continuous as it has jumps on the boundary of X0. Condition (1.1.1) is equivalent to that lim inf u→0 β(u) is bounded as from below. Under the condition that f is lower semicontinuous， it is easy to say that F(x) is an exact penalty function， but simple examples can be given to show that condition (1.1.1) may not hold. Thus， when p = 0， the equivalence in Theorem 1.1.1 is in general not true. However， under the assumptions that f and gi(i ∈ I) are lower semicontinuous， hj(j ∈ J) are continuous， the set-valued mapping is upper semicontinuous at 0， and X0 is compact， the perturbation function β is lower semi-continuous at u = 0， see Rubinov et al. (2002). Thus， under these conditions， Theorem 1.1.1 trivially holds. One advantage of the non-Lipschitz penalty function is that it in general requires weaker conditions than those for l1 penalty function for the exact penalty representation (Huang and Yang， 2003; Luo et al.， 1996). For example， consider a simple problem The l1 penalty function for this problem is not exact， but the l0.5 penalty function is exact. One type of non-Lipschitz penalty functions is also shown to be effective， together with some non-gradient based optimization method， in solving a class of concave programming problems (Rubinov and Huang， 2003). On the other hand， under the linear independence constraint qualification， first-order and second-order optimality conditions of (Pρ) and their convergence to that of the original optimization problem (P) were obtained in Rubinov et al. (2003). In this section， under the assumption that F(x) with 0 p 1 is an exact penalty function， we establish the existence of Lagrange multipliers for (P) without employing a subdifferential rule. This is done by virtue of necessary optimality conditions of (Pρ) and a Farkas lemma. In order to derive necessary optimality conditions of problem (Pρ) which meet our need， we estimate Dini lower and upper directional derivatives of the non-Lipschitzian penalty terms in F(x) as per Taylor ex
