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Nonlinear Lagrangian algorithm plays an important role in solving constrained optimization problems. It is known that, under appropriate conditions, the sequence generated by the first-order multiplier iteration converges superlinearly. This paper aims at analyzing the second-order multiplier iteration based on a class of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints. It is suggested that the sequence generated by the second-order multiplier iteration converges superlinearly with order at least two if in addition the Hessians of functions involved in problem are Lipschitz continuous.

Lagrangians play an important role for solving constrained optimization problems. Hestenes [

Based on the above Lagrangians, Bertsekas [

Consider the following inequality constrained optimization problem:

As nonlinear Lagrangians can be used to develop dual algorithms for nonlinear programming, requiring no restrictions on primal feasibility, important contributions on this topic have been done by many authors.

Polyak and Teboulle [

Ren and Zhang [

Given

Solve (approximately)

If

Update Lagrange multiplier

Set

It was shown that, under a set of conditions, dual algorithm based on this class of Lagrange is locally convergent when the penalty parameter is larger than a threshold.

In view of interpretation of the multiplier iteration as the steepest ascent method, it is natural to consider Newton’s method for maximizing the dual functional. Using known results for Newton’s method, we expect that the second-order iteration will yield a vector

We introduce the following notation to end this section:

Consider the inequality constrained optimization problem

For the convenience of description in the sequel, we list the following assumptions, some of which will be used somewhere.

Functions

For convenience of statement, we assume

Let

Strict complementary condition holds; that is,

The set of vectors

For all

Let function

The following proposition concerns properties of

Assume that (a)–(f) and (H1)–(H3) hold. For any

Let

Assume that (a)–(f) and (H1)–(H4) hold. Then there exists

There exists a vector

For

Function

Based on the nonlinear Lagrange function

Assume that conditions (a)–(f) and (H1)–(H4) hold; then for any fixed

Obviously, for

Let

In view of the interpretation of the multiplier iteration as the steepest ascent method, it is natural to consider Newton’s method for maximizing the dual functional

For

Considering the extension of Newton’s method, given

If

For a triple

Define

Let

Furthermore,

By calculating, we have

Substituting (

In view of (

Now, we know that

Assume (a)–(f) hold, and let

In view of Theorem

From the above analysis, we know that the sequence generated by the second-order multiplier iteration converges superlinearly with order at least two if the Hessians of functions involved in problem are Lipschitz continuous.

The author declares that there is no conflict of interests regarding the publication of this paper.

This project is supported by the National Natural Science Foundation of China (Grant no. 11171138).