ON A REFINEMENT OF THE CONVERGENCE ANALYSIS FOR THE NEW EXACT PENALTY FUNCTION METHOD FOR CONTINUOUS INEQUALITY CONSTRAINED OPTIMIZATION PROBLEM

This note is to provide a refinement of the convergence analysis of the new exact penalty function method proposed recently.

1. Introduction.As in [1], we consider a class of functional inequality constrained optimization problems given below.min f (x) (1a) subject to φ j (x, ω) ≤ 0, ∀ ω ∈ Ω, j = 1, . . ., m, (1b) where the vector R n is the parameter vector to be found, Ω is a compact interval in R, f : R n → R is continuously differentiable in x, and for each j = 1, . . ., m, φ j : R n × R → R is a continuously differentiable function in x and ω.Let this problem be referred to as Problem (P ).Define where R + = {α ∈ R : α ≥ 0}, W j ∈ (0, 1), j = 1, . . ., m, are fixed constants and γ is a positive real number.Clearly, Problem (P ) is equivalent to the following problem, which is denoted as Problem ( P ).
subject to (x, ) ∈ S 0 (3b) where S 0 = S with = 0. We assume that the following conditions are satisfied: • There exists a global minimizer of Problem (P ), implying that f (x) is bounded from below on S 0 .

CHANGJUN YU, KOK LAY TEO, LIANSHENG ZHANG AND YANQIN BAI
• The number of distinct local minimum values of the objective function of Problem (P ) is finite.A new exact penalty function f σ (x, ) defined below is introduced in [1].
where ∆(x, ), which is referred to as the constraint violation, is defined by α and γ are positive real numbers, β > 2, and σ > 0 is a penalty parameter.The surrogate optimization problem, which is referred to as Problem (P σ ), is as follows.
2. Convergence analysis.For every positive integer k, let (x (k), * , (k), * ) be a local minimizer of Problem (P σ k ).For the proof of the convergence results, the definition of constraint qualification given in Definition 2.2 of [1] should be changed to the one given below.
Theorem 2.3 of [1] is modified as follows.
Theorem 2. Suppose that and the constraint qualification is satisfied for the continuous inequality constraints (1b) at x = x * , then * = 0 and x * ∈ S 0 .
For the proof of Theorem 2, it is basically the same as that given for Theorem 2.3 of [1], except Definition 1, rather than Definition 2.2 of [1], is used.
Remark 1.The existence of an accumulating point of the sequence (x (k), * , (k), * ) is assured if the following condition is satisfied Where • denotes the usual Euclidean norm.
The proof of Theorem 3 is similar to Theorem 2.4 of [1], except with the changes listed below.
• Equation (2.16) of [1] should be changed to: Here, J denotes the index set such that for any j ∈ J , max 0, [1] should be changed to: • Equation (2.20) of [1] should be changed to: • Equation (2.21) of [1] should be changed to: x (k), * →x * ∈S 0 Theorems 2.5 and 2.6 of [1] are combined as one theorem given below.
Proof.On the contrary, we assume that the conclusion is false.Then, there exists a subsequence of {(x (k), * , (k), * )}, which is denoted by the original sequence such that for any k 0 > 0, there exists a k > k 0 satisfying (k ), * = 0.By Theorem 2, we have Since (k), * = 0 for all k, it follows from dividing (2.10) in [1] by (13) This is equivalent to Rearranging ( 14) yields From ( 16) and (17), we have Define Note that x (k), * → x * as k → +∞ and that ∂f (x) ∂x and, for each j = 1, . . ., m, φ j and ∂φ j (• , ω) ∂x are continuous in R n for each ω ∈ Ω, where Ω is a compact set.
Then, it can be shown that there exist constants K and K, independent of k, such that, for all k = 1, 2, • • • , ∂f (x (k), * ) ∂x ≤ K (22) For sufficiently large k, every local minimizer (x (k), * , (k), * ) has the form (x * , 0).It is obvious from Theorem 2 that x * is a feasible point of Problem (P ).This indicates that there is a neighborhood of x * , such that for any feasible x of Problem (P ) f (x) = f σ k (x, 0) ≥ f σ k (x * , 0) = f (x * ).Therefore, x * is a local minimizer of Problem (P ).This completes the proof.